The total Johnson homomorphism on the homology cylinder and the bracket-quantization HOMFLY-PT skein algebra

Abstract.

A homology cylinder of a surface induces an automorphism of the completed group ring of the fundamental group of the surface. We introduce a new method of computing the automorphism by using the Goldman Lie algebra of the surface or some skein algebra. In particular, we give a refinement of a formula by Kuno and Massuyeau [7].

Acknowledgment

The author is grateful to Kazuo Habiro, Nariya Kawazumi, Yusuke Kuno, Gwénaël Massuyeau, Jun Murakami and Tomotada Ohtsuki for helpful comments and discussions. This work was partially supported by JSPS KAKENHI Grant Number 18J00305, the Research Institute for Mathematical Sciences and an International Joint Usage/Research Center located at Kyoto University.

1. Introduction and the main result

1.1. Back ground

The mapping class group $\mathcal{M}(\Sigma)$ of a compact connected oriented surface $\Sigma$ with boundary is the set of the isotopy classes of diffeomorphism fixing the boundary $\partial\Sigma$ pointwise. The group $\mathcal{M}(\Sigma)$ acts naturally on the fundamental group $\pi_{1}(\Sigma,*)$ of the surface $\Sigma$ with basepoint $*\in\partial\Sigma$ . The action is faithful by a theorem of Dehn and Nielsen, and so provides us with an algebraic method of studying the mapping class group.

We denote by $\Sigma_{g,1}$ a compact connected oriented surface of genus $g$ with one boundary component. In the study of the action of $\mathcal{M}(\Sigma_{g,1})$ on $\pi_{1}(\Sigma_{g,1})$ , Johnson introduced a series of homomorphisms, which we call the Johnson homomorphisms. Morita [10] [11] discovered an explicit relationship between the Johnson homomorphisms and the Casson invariant, which is the finite type invariant of $3$ -manifold of order $1$ . Furthermore, Garoufalidis and Levine [1] made us understand the Johnson homomorphisms as the full tree parts of finite type invariants for “homology cobordisms”.

Kawazumi, Kuno, Massuyeau, and Turaev [5] [6] [9] studied the Goldman Lie algebra using a grading defined by an augmentation ideal. The Torelli group $\mathcal{I}(\Sigma_{g,1})$ is the kernel of the action of $\mathcal{M}(\Sigma_{g,1})$ on $H_{1}(\Sigma_{g,1},\mathbb{Z})$ . Furthermore, Kawazumi and Kuno constructed an injective map from the Torelli group $\mathcal{I}(\Sigma_{g,1})$ to the completed Goldman Lie algebra in terms of the grading, which recovers the Johnson homomorphisms. In other words, we can understand the Johnson homomorphisms as the associated graded quotients.

We denote by $\mathcal{C}(\Sigma)$ the set of diffeomorphism classes of homology cobordisms of the surface, which will be defined later. By “stacking sums”, we may consider $\mathcal{C}(\Sigma)$ as a monoid.

Let $\{(\pi_{1}(\Sigma_{g,1}))_{n}\}_{n\geq 0}$ be the lower central series of $\pi_{1}(\Sigma_{g,1})$ . For any $N\in\mathbb{Z}_{\geq 1}$ , using the Stallings’s theorem [13], Garoufalidis and Levine [1] introduced a monoid homomorphism

\Phi^{\prime}_{N}:\mathcal{C}(\Sigma_{g,1})\to\mathrm{Aut}(\pi_{1}(\Sigma_{g,1})/((\pi_{1}(\Sigma_{g,1}))_{N}).

We can extend it to

\Phi=(\cdot)_{*}:\mathcal{C}(\Sigma_{g,1})\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma_{g,1})),

where $\widehat{\mathbb{Q}\pi_{1}}(\Sigma_{g,1})$ is the completed group ring of $\pi_{1}(\Sigma_{g,1})$ . The monoid homomorphisms provides us with an algebraic method of studying $\mathcal{C}(\Sigma_{g,1})$ in terms of the finite type invariants of 3-manifolds and the Goldman Lie algebra. We remark that we can also define a monoid homomorphism

\Phi=(\cdot)_{*}:\mathcal{C}(\Sigma)\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma)),

for the surface $\Sigma$ .

In the case $\Sigma=\Sigma_{g,1}$ , Garoufalidis and Levine [1] derived the Johnson homomorphisms of $\mathcal{C}(\Sigma_{g,1})$ from the action $\Phi^{\prime}_{N}$ . By their paper, we understand the Johnson homomorphisms as the full tree parts of finite type invariants for homology cobordisms. In terms of the Goldman Lie algebra, Kuno and Massuyeau [7] have recently obtained a formula for the Johnson homomorphisms of $\mathcal{C}(\Sigma_{g,1})$ using “ generalized Dehn twists”.

We can define a submonoid $\mathcal{IC}(\Sigma_{g,1})$ of $\mathcal{C}(\Sigma_{g,1})$ analogous to the Torelli group $\mathcal{I}(\Sigma_{g,1})$ (Definition 1.1) and construct a map from $\mathcal{IC}(\Sigma_{g,1})$ to the completed Goldman Lie algebra. The action $\Phi=(\cdot)_{*}:\mathcal{C}(\Sigma_{g,1})\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma_{g,1}))$ restricted to $\mathcal{IC}(\Sigma_{g,1})$ can be regarded as the exponential of the map to the completed Goldman Lie algebra (Theorem 1.4). Furthermore, we can understand the Johnson homomorphisms as the associated graded quotients of the Goldman Lie algebra.

In §1.2 and §1.3, we introduce the notion of the Goldman Lie algebra and homology cylinders of the surface $\Sigma$ . In §1.4, we present our main result. In §1.5, we explain that we understand the Johnson homomorphisms as the associated graded quotients of the Goldman Lie algebra.

1.2. The Goldman Lie algebra

We set up notation for the Goldman Lie algebra on the surface $\Sigma$ . By using the Goldman Lie algebra, we will describe the action $\Phi$ of a homology cylinder of $\Sigma$ on the completed group ring in §1.4.

Let $\lvert\pi_{1}\rvert(\Sigma)$ denote the set of conjugacy classes in $\pi_{1}(\Sigma)$ . Goldman defined a Lie bracket on the free $\mathbb{Q}$ -vector space $\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)$ over the set $\lvert\pi_{1}\rvert(\Sigma)$ , which we call the Goldman Lie bracket. Kawazumi and Kuno [5] introduced Lie actions $\sigma:\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\times\mathbb{Q}\pi_{1}(\Sigma,*)\to\mathbb{Q}\pi_{1}(\Sigma,*)$ and $\sigma:\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\times\mathbb{Q}\pi_{1}(\Sigma,*_{1},*_{2})\to\mathbb{Q}\pi_{1}(\Sigma,*_{1},*_{2})$ for $*,*_{1},*_{2}\in\partial\Sigma$ . The Lie bracket and the Lie actions satisfy the condition

	$\displaystyle[\lvert(\ker\varepsilon)^{n}\rvert,\lvert(\ker\varepsilon)^{m}\rvert]\subset\lvert(\ker\varepsilon)^{n+m-2}\rvert,$
	$\displaystyle\sigma(\lvert(\ker\varepsilon)^{n}\rvert)((\ker\varepsilon)^{m}\mathbb{Q}\pi_{1}(\Sigma,_{1},_{2}))\subset(\ker\varepsilon)^{n+m-2}\mathbb{Q}\pi_{1}(\Sigma,_{1},_{2}),$

where $\lvert\cdot\rvert:\mathbb{Q}\pi_{1}(\Sigma)\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)$ is the quotient map. Here, for any group $G$ , $\varepsilon:\mathbb{Q}G\to\mathbb{Q}$ is the augmentation map defined by $g\in G\mapsto 1$ .

To define the logarithm and the exponential on the group ring, we define filtrations

	$\displaystyle F^{n}\mathbb{Q}\pi_{1}(\Sigma,*)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}(\ker\varepsilon)^{n},$
	$\displaystyle F^{n}\mathbb{Q}\pi_{1}(\Sigma,_{1},_{2}),\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}(\ker\varepsilon)^{n}\mathbb{Q}\pi_{1}(\Sigma,_{1},_{2}),$
	$\displaystyle F^{n}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\lvert(\ker\varepsilon)^{n}\rvert.$

and completions

	$\displaystyle\widehat{\mathbb{Q}\pi_{1}}(\Sigma,)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\underleftarrow{\lim}_{i\rightarrow\infty}\mathbb{Q}\pi_{1}(\Sigma,)/F^{i}\mathbb{Q}\pi_{1}(\Sigma,*),$
	$\displaystyle\widehat{\mathbb{Q}\pi_{1}}(\Sigma,_{1},_{2})\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\underleftarrow{\lim}_{i\rightarrow\infty}\mathbb{Q}\pi_{1}(\Sigma,_{1},_{2})/F^{i}\mathbb{Q}\pi_{1}(\Sigma,_{1},_{2}),$
	$\displaystyle\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\underleftarrow{\lim}_{i\rightarrow\infty}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)/F^{i}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma).$

The completions also have filtrations such that

F^{n}\widehat{V}=\ker(\widehat{V}\to V/F^{n}V)

where

V=\mathbb{Q}\pi_{1}(\Sigma,*),\mathbb{Q}\pi_{1}(\Sigma,*_{1},*_{2}),\ \mathrm{and}\ \mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)

and $\widehat{V}$ is the completion of $V$ . Using the Baker-Campbell-Hausdorff series, we consider $F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ as a group.

Using the filtrations and the completions, we can discuss a relationship between the Goldman Lie algebra and the mapping class group. Here we denote by $t_{c}$ the Dehn twist along a simple closed curve $c$ . Let $\mathcal{I}^{\prime}(\Sigma)$ be the subgroup of the mapping class group $\mathcal{M}(\Sigma)$ generated by

\displaystyle\{t_{c_{1}}t_{c_{2}}^{-1}|(c_{1},c_{2})\mathrm{\ bounds\ a\ surface}\}\ \mathrm{and}\ \{t_{c_{0}}|c_{0}\mathrm{\ bounds\ a\ surface}\}.

Then there is an embedding $\zeta$ from $\mathcal{I}^{\prime}(\Sigma)$ to $F^{3}\widehat{\mathbb{Q}\pi_{1}}(\Sigma)$ . The map $\zeta$ satisfies

\xi_{*}=\exp(\zeta(\xi))\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\sum_{i=0}^{\infty}\frac{1}{i!}(\sigma(\zeta(\xi)))^{i}:\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2}).

For details, see §2 or [5] [6] [9].

1.3. Homology cylinders

We recall the definition of homology cobordisms. Let $(M,\alpha)$ be a pair of a compact connected oriented $3$ -manifold and a diffeomorphism $\partial(\Sigma\times I)\to\partial M$ , where $I$ is the unit interval $[0,1]$ . If the embedding maps

\alpha_{0}:\Sigma\to M,p\mapsto\alpha(p,0)\ \mathrm{and}\ \alpha_{1}:\Sigma\to M,p\mapsto\alpha(p,1)

induce the isomorphisms

\alpha_{0*}:H_{1}(\Sigma,\mathbb{Z})\to H_{1}(M,\mathbb{Z})\ \mathrm{and}\ \alpha_{1*}:H_{1}(\Sigma,\mathbb{Z})\to H_{1}(M,\mathbb{Z}),

we call it a homology cobordism.

The set of the diffeomorphism classes of homology cobordisms

\mathcal{C}(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\{\mathrm{homology\ cobordisms}\}/\mathrm{diffeomorpic}

is a monoid by “stacking sums”. The action $\mathcal{C}(\Sigma)\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2}))$ and the Johnson homomorphisms introduced by Garoufalidis and Levine are monoid homomorphism.

We define the action $\Phi(\cdot)=(\cdot)_{*}:\mathcal{C}(\Sigma)\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2}))$ as follows. Let $(M,\alpha)$ be a homology cobordism of the surface $\Sigma$ . By Stallings’ theorem, the embedding maps $\alpha_{0},\alpha_{1}$ induce the isomorphisms

	$\displaystyle\alpha_{0}:\widehat{\mathbb{Q}\pi_{1}}(\Sigma,_{1},_{2})\to\widehat{\mathbb{Q}\pi_{1}}(M,\alpha(_{1},0),\alpha(*_{2},0)),$
	$\displaystyle\alpha_{1}:\widehat{\mathbb{Q}\pi_{1}}(\Sigma,_{1},_{2})\to\widehat{\mathbb{Q}\pi_{1}}(M,\alpha(_{1},1),\alpha(*_{2},1)).$

Here, for any $i=0,1$ , we denote by $\widehat{\mathbb{Q}\pi_{1}}(M,\alpha(*_{1},i),\alpha(*_{2},i))$ the completion

\underleftarrow{\lim}_{n\rightarrow\infty}\mathbb{Q}\pi_{1}(M,\alpha(*_{1},i),\alpha(*_{2},i))/(\ker\varepsilon)^{n}\mathbb{Q}\pi_{1}(M,\alpha(*_{1},i),\alpha(*_{2},i)).

Let $\Diamond:\widehat{\mathbb{Q}\pi_{1}}(M,\alpha(*_{1},0),\alpha(*_{2},0))\to\widehat{\mathbb{Q}\pi_{1}}(M,\alpha(*_{1},1),\alpha(*_{2},1))$ be the automorphism defined as $\Diamond(\gamma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}(\gamma_{*01})^{-1}\gamma\gamma_{*01}$ , where the continuous map $I\to M,t\mapsto\alpha(*,t)$ represents the path $\gamma_{*01}$ . The composite $\alpha_{1*}^{-1}\circ\Diamond\circ\alpha_{0*}$ defines a monoid homomorphism

\Phi=(\cdot)_{*}:\mathcal{C}(\Sigma)\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})),(M,\alpha)\mapsto(M,\alpha)_{*}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\alpha_{1*}^{-1}\circ\Diamond\circ\alpha_{0*},

which we call the action $\Phi$ of $\mathcal{C}(\Sigma)$ on $\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})$ .

Definition 1.1.

We denote by $\mathcal{IC}(\Sigma)$ the subset of $\mathcal{C}(\Sigma)$ consisting of all elements $\Xi$ satisfying

(\mathrm{id}-\Xi_{*})(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2}))\subset F^{2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})

for any $*_{1},*_{2}\in\partial\Sigma$ . We call a homology cobordism representing an element of $\mathcal{IC}(\Sigma)$ a homology cylinder.

To state our main result, we introduce a homology cylinder version of a Heegaard splitting of a 3-manifold. We choose disjoint closed disks $D_{1},\cdots,D_{N}$ on $\Sigma$ where $N$ is at least $1$ . Let $\Sigma_{\mathrm{st}}$ be the closure of $\Sigma\backslash(D_{1}\sqcup\cdots\sqcup D_{N})$ . We denote the extended surface $\partial(\Sigma_{\mathrm{st}}\times I)\backslash(\partial\Sigma\times(0,1))=\Sigma_{\mathrm{st}}\times\{0,1\}\cup(\partial(D_{1}\sqcup\cdots\sqcup D_{N})\times I)$ by $\widetilde{\Sigma}_{\mathrm{st}}$ . For an element $\xi\in\mathcal{I}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})$ , we take a representative of $\xi$ and denote it by the same symbol $\xi$ . Here $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I,(p,t)\mapsto(p,\frac{1+t}{3})$ is the standard embedding map. We define a homology cylinder $(\Sigma\times I)(e_{\mathrm{st}},\xi)$ by

((\mathrm{the\ closure\ of\ }(\Sigma\times I\backslash e_{\mathrm{st}}(\Sigma_{\mathrm{st}}\times I)))\cup_{e_{\mathrm{st}}\circ\xi}(\Sigma_{\mathrm{st}}\times I),\mathrm{id}_{\partial(\Sigma\times I)}).

Proposition 1.2 (Proposition 4.9).

For any homology cylinder $(M,\alpha)$ of the surface $\Sigma$ , there exist some disjoint closed disks $D_{1},\cdots,D_{N}$ and an element $\xi\in\mathcal{I}^{\prime}(\widetilde{\Sigma_{\mathrm{st}}})$ such that the diffeomorphism class of $(M,\alpha)$ equals that of $(\Sigma\times I)(e_{\mathrm{st}},\xi)$ .

For details, see §4.

To state our main result, we need to define a map $v:F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma}_{\mathrm{st}})\to F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ as follows. The maps

\widetilde{\Sigma_{\mathrm{st}}}\to\Sigma_{\mathrm{st}}\times I\twoheadrightarrow\Sigma_{\mathrm{st}},\ \Sigma_{\mathrm{st}}\to\widetilde{\Sigma}_{\mathrm{st}},p\mapsto(p,1),\ \mathrm{and}\ \Sigma_{\mathrm{st}}\to\widetilde{\Sigma}_{\mathrm{st}},p\mapsto(p,0)

induce linear maps

	$\displaystyle\kappa_{}:\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma_{\mathrm{st}}})\to\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}}),\ \iota_{0}:\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})\to\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma}_{\mathrm{st}}),$
	$\displaystyle\mathrm{and}\ \iota_{1*}:\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})\to\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma_{\mathrm{st}}}).$

We remark that the second map is a Lie algebra homomorphisms, but the first one and the third one are not.

Definition 1.3.

For $x\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma}_{\mathrm{st}})$ , a sequence $\{v_{n}(x)\}_{n\in\mathbb{Z}_{\geq 1}}\subset F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ is defined by

\displaystyle v_{1}(x)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\kappa_{*}(x),\ v_{n+1}(x)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}v_{n}(x)+\kappa_{*}(\mathrm{bch}(-\iota_{1*}(v_{n}(x)),x)).

Furthermore, $v(x)$ is defined by

v(x)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\lim_{n\to\infty}v_{n}(x).

The element $v(x)$ is a unique one satisfying $\kappa_{*}(\mathrm{bch}(-\iota_{1*}(v(x)),x))=0$ . In other words, $v(x)$ is the unique solution of $\kappa_{*}(\mathrm{bch}(-\iota_{1*}(\cdot),x))=0$ . There is a topological definition of $v$ using some skein algebra in §3. Using this map, we state our main theorem proved in §6.

1.4. Main result

The paper aims to construct a monoid homomorphism

\widetilde{\zeta}:\mathcal{IC}(\Sigma)\to F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)

having the following property

\Phi(\cdot)=\exp(\sigma(\widetilde{\zeta}(\cdot)))\in\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2}))

as follows.

Theorem 1.4.

We have

(\Sigma\times I)(e_{\mathrm{st}},\xi)_{*}=\exp(e^{\prime}_{*}(v(\zeta(\xi)))):\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})

for $*_{1},*_{2}\in\partial\Sigma$ , where $e^{\prime}_{*}:\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})\to\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ is induced by $\Sigma_{\mathrm{st}}\hookrightarrow\Sigma.$ Using the formula, the map

\widetilde{\zeta}:\mathcal{IC}(\Sigma)\to(F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma),\mathrm{bch}),

defined by

(\Sigma\times I)(e_{\mathrm{st}},\xi)\mapsto e^{\prime}_{*}(v(\zeta(\xi))),

is a well-defined monoid homomorphism.

We construct the map $\widetilde{\zeta}$ in this section in an algebraic method. On the other hand, in §3, we reconstruct the map in $\widetilde{\zeta}$ a topological one. In other words, “a surgery formula” of some skein algebra explained later defines $\widetilde{\zeta}$ in a similar way to the WRT invariant. See, for details, Theorem 5.5.

1.5. An application

In this subsection, we assume that $\Sigma=\Sigma_{g,1}$ . By the monoid homomorphism $\widetilde{\zeta}:\mathcal{IC}(\Sigma_{g,1})\to F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{g,1})$ , we recover the Johnson homomorphisms of homology cobordisms defined by Garoufalidis and Levine.

We reformulate the Johnson-like filtration $\{F^{n}\mathcal{C}(\Sigma_{g,1})\}_{n\geq 0}$ of $\mathcal{C}(\Sigma_{g,1})$ introduced by Garoufalidis and Levine. The subset $F^{n}\mathcal{C}(\Sigma_{g,1})$ consists of all elements $\Xi$ satisfying

(\Xi_{*}-\mathrm{id})(\widehat{\mathbb{Q}\pi_{1}}(\Sigma_{g,1}))\subset F^{n+1}\widehat{\mathbb{Q}\pi_{1}}(\Sigma_{g,1}).

Here we set the linear maps

	$\displaystyle\lambda_{n+2}:$	$\displaystyle F^{n+2}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{g,1})/F^{n+3}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{g,1})\to(H_{1}(\Sigma_{g,1},\mathbb{Q}))^{\otimes n+2},$
		$\displaystyle\lvert(\gamma_{1}-1)\cdots(\gamma_{n+2}-1)\rvert\mapsto c_{n+2}([\gamma_{1}]\otimes\cdots\otimes[\gamma_{n+2}]),$
	$\displaystyle c_{n+2}:$	$\displaystyle(H_{1}(\Sigma_{g,1},\mathbb{Q}))^{\otimes n+2}\to(H_{1}(\Sigma_{g,1},\mathbb{Q}))^{\otimes n+2},$
		$\displaystyle a_{1}\otimes\cdots\otimes a_{n+2}\mapsto\sum_{i=1}^{n+2}a_{i}\otimes\cdots\otimes a_{n+2}\otimes a_{1}\otimes\cdots\otimes a_{i-1}.$

For any $n\in\mathbb{Z}_{\geq 1}$ , the composite

F^{n}\mathcal{C}(\Sigma_{g,1})\stackrel{{\scriptstyle\widetilde{\zeta}}}{{\to}}F^{n+2}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{g,1})/F^{n+3}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{g,1})\stackrel{{\scriptstyle\lambda_{n+2}}}{{\to}}c_{n+2}(H_{1}(\Sigma_{g,1},\mathbb{Q})^{\otimes n+2})

is the $n$ -th Johnson homomorphism introduced by Garoufalidis and Levine. For details, see [1]. Since we can recover each $\tau_{n}$ from $\widetilde{\zeta}$ , we call $\widetilde{\zeta}$ the total Johnson homomorphism.

2. Review of the Goldman Lie algebra

We denote by $\Sigma$ a compact connected oriented surface as in §1.2. In this section, we review some facts on the Goldman Lie algebra of $\Sigma$ . In particular, we discuss a relationship between the Goldman Lie algebra and the mapping class group. We use the propositions in this section to prove our main theorem in §5.

Now we recall the Lie bracket and the Lie action introduced by Goldman and Kawazumi-Kuno, respectively. Let $\delta$ be a free loop in $\Sigma$ and $\gamma$ a path from $*_{1}\in\partial\Sigma$ to $*_{2}\in\partial\Sigma$ . We assume $\delta$ and $\gamma$ are in general position. For an intersection $p\in\delta\cap\gamma$ , we denote

•

by $\gamma_{*,p}$ and $\gamma_{p,*}$ a path from $*$ to $p$ and one from $p$ to $*$ along $\gamma$ .
•

by $\delta_{p}$ a based loop along $\delta$ whose basepoint is $p$ , and
•

by $\epsilon(p,\delta,\gamma)$ the local intersection number of $\delta$ and $\gamma$ at $p$ .

We set $\sigma(\delta)(\gamma)$ by

\sigma(\delta)(\gamma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\sum_{p\in\delta\cap\gamma}\epsilon(p,\delta,\gamma)\gamma_{*,p}\delta_{p}\gamma_{p,*}.

For $\delta\in\lvert\pi_{1}\rvert(\Sigma)$ and $\delta^{\prime}\in\pi_{1}(\Sigma,*)$ , Goldman defined the bracket as $[\delta,\lvert\delta^{\prime}\rvert]\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\lvert\sigma(\delta)(\delta^{\prime})\rvert$ . Kawazumi and Kuno proved that the action $\sigma$ makes $\mathbb{Q}\pi_{1}(\Sigma,*_{1},*_{2})$ a Lie module of the Goldman Lie algebra.

To discuss the relationship between the Goldman Lie algebra and the mapping class group, Kawazumi, Kuno, Massuyeau, and Turaev used the filtration and the completion of the Goldman Lie algebra defined as the equations in §1.2. They obtained a formula for the action of the Dehn twist $t_{c}$ along a simple closed curve $c$ in terms of the one of the Goldman Lie algebra.

Theorem 2.1 ([5] [6] [9]).

For a simple closed curve $c$ , we set an element

L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\lvert\frac{1}{2}(\log\gamma)^{2}\rvert\in\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)

of the completed Goldman Lie algebra where $\gamma\in\pi_{1}(\Sigma)$ satisfies $\lvert\gamma\rvert=c$ . We have

t_{c*}=\exp(\sigma(L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c)))\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\sum_{i=0}^{\infty}\frac{1}{i!}(\sigma(L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c)))^{i}:\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})

for any $*_{1},*_{2}\in\partial\Sigma$ .

We need the following two propositions Proposition 2.2 and Proposition 2.3 to state our main theorem Theorem 1.4. We use the first one to prove that the homomorphism $\widetilde{\zeta}$ is well-defined. We describe the automorphism induced by a homology cylinder using the second one.

We can prove the following proposition by “the symplectic expansion”. For details, see [5] [6] [9]. In this paper, we use it without proof.

Proposition 2.2.

(1)

For $x\in F^{1}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ , if $\sigma(x)(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2}))=\{0\}$ for any $*_{1},*_{2}\in\partial\Sigma$ , we have $x=0$ .

(2)

We fix a subset $B\subset\partial\Sigma$ such that the map $B\to\pi_{0}(\partial\Sigma)$ is bijective. Let $\xi$ be a bijection $\coprod_{\bullet,*\in B}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*)\to\coprod_{\bullet,*\in B}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*)$ satisfying the following

•

For any $\bullet,*\in B$ , $\xi(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*))=\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*)$ .
•

For any $\star_{1},\star_{2},\star_{3}\in B$ and $x\in\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})$ , $x^{\prime}\in\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{2},\star_{3})$ , $\xi(xx^{\prime})=\xi(x)\xi(x^{\prime})$ .

•

For any $\bullet,*\in B$ ,

	$\displaystyle\xi(I_{\mathbb{Q}\pi_{1}(\Sigma,\bullet)}^{n}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,))=I_{\mathbb{Q}\pi_{1}(\Sigma,\bullet)}^{n}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,),$
	$\displaystyle(\xi-\mathrm{id})(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*))\subset I_{\mathbb{Q}\pi_{1}(\Sigma,\bullet)}^{2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,\star).$

•

For any $\gamma\in\pi_{1}(\Sigma,\bullet,*)$ represented by a continuous map $I\to\partial\Sigma$ ,

$\xi(\gamma)=\gamma$

Then there exists a unique element $\zeta_{\mathrm{Aut}}(\xi)\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ such that

\xi=\exp(\zeta_{\mathrm{Aut}}(\xi)):\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star,\star^{\prime})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star,\star^{\prime})

for any $\star,\star^{\prime}\in\partial\Sigma$ .

We use Theorem 2.1 and Proposition 2.2 to prove the following.

Proposition 2.3.

The homomorphism $\zeta:\mathcal{I}^{\prime}(\Sigma)\to(F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma),\mathrm{bch})$ is defined by

\displaystyle\zeta(t_{c_{0}})=L(c_{0})\ \mathrm{and}\ \zeta(t_{c_{1}}t_{c_{2}}^{-1})=L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{1})-L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{2})

where a simple closed curve $c_{0}$ and the pair $(c_{1},c_{2})$ of simple closed curves bound surfaces. Then the homomorphism is well-defined. Furthermore, $\zeta$ satisfies the property

\xi_{*}=\exp(\sigma(\zeta(\xi))):\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})

for any $\xi\in\mathcal{I}^{\prime}(\Sigma)$ and $*_{1},*_{2}\in\partial\Sigma$ .

Proof.

Let

\zeta^{\prime}:\{t_{c_{0}}|c_{0}\mathrm{\ bounds\ a\ surface.}\}\cup\{t_{c_{1}}t_{c_{2}}^{-1}|(c_{1},c_{2})\mathrm{\ bpounds\ s\ surface.}\}\to F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)

be a map defined by

t_{c_{0}}\mapsto L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{0})\mathrm{\ and\ }t_{c_{1}}t_{c_{2}}^{-1}\mapsto L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{1})-L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{2}).

We prove the proposition by two steps. We use Theorem 2.1 and Proposition 2.2 in the first and second steps, respectively.

In the first step, we prove that, for any

\xi_{1},\cdots,\xi_{j}\in\{t_{c_{0}}|c_{0}\mathrm{\ bounds\ a\ surface.}\}\cup\{t_{c_{1}}t_{c_{2}}^{-1}|(c_{1},c_{2})\mathrm{\ bpounds\ s\ surface.}\},

(\xi_{1}\cdots\xi_{j})_{*}=\exp(\sigma(\mathrm{bch}(\zeta^{\prime}(\xi_{1}),\cdots,\zeta^{\prime}(\xi_{j})))):\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,*_{1},*_{2})

holds. Here we denote $\mathrm{bch}(x_{1},\mathrm{bch}(x_{2},\cdots,\mathrm{bch}(x_{n-1},x_{n})))$ by $\mathrm{bch}(x_{1},\cdots,x_{n})$ for any $x_{1},\cdots,x_{n}\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ . By Theorem 2.1, we have

	$\displaystyle\exp(\sigma(\mathrm{bch}(\zeta^{\prime}(\xi_{1}),\cdots,\zeta^{\prime}(\xi_{j}))))$
	$\displaystyle=\exp(\sigma(\zeta^{\prime}(\xi_{1})))\circ\cdots\circ\exp(\sigma(\zeta^{\prime}(\xi_{j})))$
	$\displaystyle=\xi_{1}\circ\cdots\circ\xi_{j}=(\xi_{1}\cdots\xi_{j})_{*}.$

In the second step, we prove that, for any

\xi_{1},\cdots,\xi_{j},\xi^{\prime}_{1},\cdots,\xi^{\prime}_{j^{\prime}}\in\{t_{c_{0}}|c_{0}\mathrm{\ bounds\ a\ surface.}\}\cup\{t_{c_{1}}t_{c_{2}}^{-1}|(c_{1},c_{2})\mathrm{\ bpounds\ s\ surface.}\},

\xi=\xi_{1}\cdots\xi_{j}=\xi^{\prime}_{1}\cdots\xi^{\prime}_{j^{\prime}},

\mathrm{bch}(\zeta^{\prime}(\xi_{1}),\cdots,\zeta^{\prime}(\xi_{j}))=\mathrm{bch}(\zeta^{\prime}(\xi^{\prime}_{1}),\cdots,\zeta^{\prime}(\xi^{\prime}_{j^{\prime}}))

holds. By the first statement, we have

	$\displaystyle\sigma(\mathrm{bch}(\zeta^{\prime}(\xi_{1}),\cdots,\zeta^{\prime}(\xi_{j}))=\sigma(\mathrm{bch}(\zeta^{\prime}(\xi^{\prime}_{1}),\cdots,\zeta^{\prime}(\xi^{\prime}_{j^{\prime}})))=\sum_{i=1}^{\infty}\frac{(-1)^{i}}{i}(\xi_{*}-\mathrm{id})^{i}$
	$\displaystyle:\widehat{\mathbb{Q}\pi_{1}}(\Sigma,_{1},_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,_{1},_{2})$

for any $*_{1},*_{2}\in\partial\Sigma$ . Using it, by Proposition 2.2, we obtain

\mathrm{bch}(\zeta^{\prime}(\xi_{1}),\cdots,\zeta^{\prime}(\xi_{j}))=\mathrm{bch}(\zeta^{\prime}(\xi^{\prime}_{1}),\cdots,\zeta^{\prime}(\xi^{\prime}_{j^{\prime}})).

The above two statements prove the proposition.

∎

3. The bracket-quantization HOMFLY-PT skein algebra

In this section, we set a skein module $\mathcal{A}^{\prime}(M)$ of a oriented 3-manifold $M$ . We call $\mathcal{A}^{\prime}(M)$ the bracket-quantization HOMFLY-PT skein module in this paper. If the 3-manifold is a cylinder $S\times I$ of a compact oriented surface $S$ , we consider an algebraic structure of the module and call $\mathcal{A}^{\prime}(S)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\mathcal{A}^{\prime}(S\times I)$ the quantum bracket-quantization HOMFLY-PT skein module.

For a compact oriented 3-manifold $M$ , let $\mathcal{E}^{\prime}(M)$ be a set consisting of all embeddings $E:(\coprod S^{1})\to M$ satisfying the condition.

•

$E(\coprod(S^{1}))\subset M\backslash\partial M.$

For two elements $E_{1}$ and $E_{2}$ of $\mathcal{E}^{\prime}(M)$ , $E_{1}$ and $E_{2}$ are isotopic, if and only if there exists an isotopy $Y:(\coprod S^{1})\times I\to M$ having the properties.

•

For any $t\in I$ , $Y(\cdot,t)\in\mathcal{E}^{\prime}(M).$
•

$E_{0}=Y(\cdot,0)$ , $E_{1}=Y(\cdot,1)$ .

We denote by $\mathcal{T}^{\prime}(M)$ the set of all tangles, which are isotopy classes of elements of $\mathcal{E}^{\prime}(M)$ .

In this paper, we also use tangles. For two points $\star_{0},\star_{1}\in\partial\Sigma$ , let $\mathcal{E}^{\prime}(M,\star_{1},\star_{2})$ be a set consisting of all embeddings $E:I\sqcup(\coprod S^{1})\to M$ satisfying the condition.

(1)

$E((0,1)\sqcup\coprod(S^{1}))\subset M\backslash\partial M.$
(2)

$E(\{0\in I\})=\{\star_{0}\}$ .
(3)

$E(\{1\in I\})=\{\star_{1}\}$ .

For two elements $E_{1}$ and $E_{2}$ of $\mathcal{E}^{\prime}(M,\star_{1},\star_{2})$ , $E_{1}$ and $E_{2}$ are isotopic, if and only if there exists an isotopy $Y:(I\sqcup\coprod(S^{1}))\times I$ having the properties.

•

For any $t\in I$ , $Y(\cdot,t)\in\mathcal{E}^{\prime}(M,\star_{0},\star_{1}).$
•

$E_{0}=Y(\cdot,0)$ , $E_{1}=Y(\cdot,1)$ .

We denote by $\mathcal{T}^{\prime}(M,\star_{0},\star_{1})$ the set of all tangles in $M$ with basepoint set $\star_{0},\star_{1}$ , which are isotopy classes of $\mathcal{E}^{\prime}(M,\star_{0},\star_{1})$ .

To define the quantum-bracket skein module, we will consider a Conway triple who are elements of $\mathcal{T}^{\prime}(M)$ or $\mathcal{T}^{\prime}(M,\star_{0},\star_{1})$ and maps

	$\displaystyle\lvert\cdot\rvert:\mathcal{T}^{\prime}(M,\star_{0},\star_{1})\to\mathbb{Z}_{\geq 0},T\mapsto(\mathrm{the\ number\ of\ components\ of\ }T),$
	$\displaystyle\lvert\cdot\rvert:\mathcal{T}^{\prime}(M)\to\mathbb{Z}_{\geq 0},L\mapsto(\mathrm{the\ number\ of\ components\ of\ }L).$

Conway triples need to define skein relations. There exist two types of the skein relations of the quantum-bracket skein module in this paper.

Definition 3.1.

Let $(E_{(C+)},E_{(C-)},E_{(C0)})\in\mathcal{E}^{\prime}(M,\star_{0},\star_{1})^{\times 3}\mathrm{\ or\ }\mathcal{E}^{\prime}(M)^{\times 3}$ be embeddings having the two properties.

•

The image of $E_{(C+)},E_{(C-)},E_{(C0)}$ are equals except for a closed ball as oriented submanifolds.
•

In the ball, the images of them are two lines presented by Figure3, Figure3, and Figure3.

Then the three elements $(T_{(C+)},T_{(C-)},T_{(C0)})\in\mathcal{T}^{\prime}(M,\star_{0},\star_{1})^{\times 3}\mathrm{\ or\ }\mathcal{T}^{\prime}(M)^{\times 3}$ represented by $(E_{(C+)},E_{(C-)},E_{(C0)})$ is a Conway triple.

Figure 1.

T_{(C+)}

Figure 2.

T_{(C-)}

Figure 3.

T_{(C0)}

Definition 3.2.

For a compact oriented 3-manifold $M$ and $\star_{0},\star_{1}\in\partial M$ , we set $\mathcal{A}^{\prime}(M)$ and $\mathcal{A}^{\prime}(M,\star_{0},\star_{1})$ as the quotients of the free $\mathbb{Q}[h]$ -modules $\mathbb{Q}[h]\mathcal{T}^{\prime}(M)$ and $\mathbb{Q}[h]\mathcal{T}^{\prime}(M,\star_{0},\star_{1})$ by the relations

•

For a Conway triple $(T_{(C+)},T_{(C-)},T_{(C0)})\in\mathcal{T}^{\prime}(M)^{\times 3}\mathrm{\ or\ }\mathcal{T}^{\prime}(M,\star_{0},\star_{1})^{\times 3}$ such that

$\lvert T_{(C+)}\rvert=\lvert T_{(C-)}\rvert=\lvert T_{C0}\rvert-1,$

we consider the relation

$T_{(C+)}-T_{(C-)}=hT_{(C0)}.$
•

For a Conway triple $(T_{(C+)},T_{(C-)},T_{(C0)})\in\mathcal{T}^{\prime}(M)^{\times 3}\mathrm{\ or\ }\mathcal{T}^{\prime}(M,\star_{0},\star_{1})^{\times 3}$ such that

$\lvert T_{(C+)}\rvert=\lvert T_{(C-)}\rvert=\lvert T_{C0}\rvert+1,$

we consider the relation

$T_{(C+)}-T_{(C-)}=0.$

In other words, the figure visualizes relations defining the skein modules $\mathcal{A}^{\prime}(M)$ and $\mathcal{A}^{\prime}(M,\star_{0},\star_{1})$ .

Let $S$ be a compact oriented surface. We denote by $\mathcal{A}^{\prime}(S)$ and $\mathcal{A}^{\prime}(S,\star_{0},\star_{1})$ the skein modules $\mathcal{A}((\Sigma\times I)$ and $\mathcal{A}^{\prime}(\Sigma\times I,(\star_{0},1),(\star_{1},0))$ for $\star_{0},\star_{1}\in\partial\Sigma$ . For a tangle $T\in\mathcal{T}^{\prime}(S\times I,(\star_{0},1),(\star_{1},0))$ represented by $E_{\gamma}\sqcup E:I\sqcup\coprod(S^{1})\to S\times I$ and $t_{0},t_{1}\in I$ , we set $E_{t_{0}\cdot\gamma\cdot t_{1}}$ as

E_{t_{0}\cdot\gamma\cdot t_{1}}(t)\begin{cases}(\star_{0},t_{0}(1-3t)+3t)\mathrm{\ if\ }t\in[0,\frac{1}{3}]\\ E_{\gamma}(3t-1)\mathrm{\ if\ }t\in[\frac{1}{3},\frac{2}{3}]\\ (\star_{1},t_{1}(3t-2)\mathrm{\ if\ }t\in[\frac{2}{3},1]\end{cases}

and $\diamondsuit_{(1,0)}^{(t_{0},t_{1})}(T)$ as an element of $\mathcal{T}^{\prime}(S\times I,(\star_{0},t_{0}),(\star_{1},t_{1}))$ represented by one of $\mathcal{E}^{\prime}(S\times I,(\star_{0},t_{0}),(\star_{1},t_{1}))$ isotopic to the embedding $E_{t_{0},\gamma,t_{1}}\sqcup E$ . The map induces a bijection $\diamondsuit_{(1,0)}^{(t_{0},t_{1})}:\mathcal{A}(S,\star_{0},\star_{1})\to\mathcal{A}(S\times I,(\star_{0},t_{0}),(\star_{1},t_{1}))$ , whose reverse map $\diamondsuit_{(t_{0},t_{1})}^{(1,0)}$ denotes. We will define a multiple of $\mathcal{A}^{\prime}(S)$ and right and left actions of $\mathcal{A}^{\prime}(S)$ on $\mathcal{A}^{\prime}(S,\star_{0},\star_{1})$ . We set two embeddings $e_{\mathrm{over}},e_{\mathrm{under}}:S\times I\to S\times I$ as

e_{\mathrm{over}}(p,t)=(p,\frac{1+t}{2}),\ e_{\mathrm{under}}(p,t)=(p,\frac{t}{2}).

Let $L_{1}$ and $L_{2}$ be two links in $S\times I$ and $T$ be a tangle represented by $E_{1}$ , $E_{2}$ , and $E$ . We set the multiple $L_{1}L_{2}$ of $L_{1}$ and $L_{2}$ , the right and left actions $L_{1}T,TL_{1}$ of $L_{1}$ on $T$ as the following. The multiple $L_{1}L_{2}$ is a link represented by the embedding $e_{\mathrm{over}}\circ E_{1}\sqcup e_{\mathrm{under}}\circ E_{2}$ . The right and left actions $L_{1}T,TL_{1}$ are the tangles as

L_{1}T\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\diamondsuit_{(\frac{1}{2},0)}^{(1,0)}((L_{1}T)^{\prime}),\ TL_{1}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\diamondsuit_{(1,\frac{1}{2})}^{(1,0)}((TL_{1})^{\prime})

where the embeddings $e_{\mathrm{over}}\circ E_{1}\sqcup e_{\mathrm{under}}\circ E$ and $e_{\mathrm{over}}\circ E\sqcup e_{\mathrm{under}}\circ E_{1}$ represent $(L_{1}T)^{\prime}$ and $(TL_{1})^{\prime}$ .

Furthermore, we consider a Lie bracket $[\cdot,\cdot]:\mathcal{A}^{\prime}(\Sigma)\times\mathcal{A}^{\prime}(\Sigma)\to\mathcal{A}^{\prime}(\Sigma)$ and a Lie action $\sigma(\cdot)(\cdot):\mathcal{A}^{\prime}(\Sigma)\times\mathcal{A}^{\prime}(\Sigma,\star_{0},\star_{1})$ as the following. By Theorem 3.3, the maps

	$\displaystyle h(\cdot):\mathcal{A}^{\prime}(\Sigma)\to\mathcal{A}^{\prime}(\Sigma),x\mapsto hx$
	$\displaystyle h(\cdot):\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2})\to\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2}),y\mapsto hy$

are injective, and by the skein relation, we have

	$\displaystyle x_{1}x_{2}-x_{2}x_{1}\in h\mathcal{A}^{\prime}(\Sigma),$
	$\displaystyle x_{1}y-yx_{1}\in h\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2}).$

for $x_{1},x_{2}\in\mathcal{A}^{\prime}(\Sigma)$ and $y\in\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2}).$ We can define

	$\displaystyle[x_{1},x_{2}]\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\frac{1}{h}(x_{1}x_{2}-x_{2}x_{1}),$
	$\displaystyle\sigma(x_{1})(y)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\frac{1}{h}(x_{1}y-yx_{1}).$

Let $\Sigma$ be a compact connected oriented surface. We set $\mathcal{U}_{h}(\Sigma)$ as the quotient of the tensor algebra $\oplus_{i=0}^{\infty}(\mathbb{Q}[h]\lvert\pi_{1}\rvert(\Sigma))^{\otimes_{\mathbb{Q}[h]}i}$ by the relation

•

For any $x,y\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)$ ,

$x\otimes y-y\otimes x=h[x,y].$

We define the Lie bracket $[\cdot,\cdot]:\mathcal{U}_{h}(\Sigma)\times\mathcal{U}_{h}(\Sigma)\to\mathcal{U}_{h}(\Sigma)$ by the Leibniz rule, which means that we have

	$\displaystyle[x_{1}\otimes\cdots\otimes x_{i},y_{1}\otimes\cdots\otimes y_{j}]$	$\displaystyle=$
	$\displaystyle\sum_{i^{\prime}\in\{1,\cdots,i\},j^{\prime}\in\{1,\cdots,j\}}$	$\displaystyle x_{1}\otimes\cdots\otimes x_{i^{\prime}-1}\otimes y_{1}\otimes\cdots\otimes y_{j^{\prime}-1}$
		$\displaystyle\otimes[x_{i^{\prime}},y_{j^{\prime}}]\otimes y_{j^{\prime}+1}\otimes\cdots\otimes y_{j}\otimes x_{i^{\prime}+1}\otimes\cdots\otimes x_{i}$

for $x_{1}\otimes\cdots\otimes x_{i}\in(\mathbb{Q}[h]\lvert\pi_{1}\rvert(\Sigma))^{\otimes i}$ and $y_{1}\otimes\cdots\otimes y_{j}\in(\mathbb{Q}[h]\lvert\pi_{1}\rvert(\Sigma))^{\otimes j}$ . It satisfies the formula

[x,y]=\frac{1}{h}(x\otimes y-y\otimes x)

for $x$ and $y\in\mathcal{U}_{h}(\Sigma)$ . We denote by $\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}$ the natural injection $\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\to\mathcal{U}_{h}(\Sigma)$ and by $\Psi_{\mathcal{U}_{h}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}$ the natural surjection

\mathcal{U}_{h}(\Sigma)\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma),x\mapsto\begin{cases}x\mathrm{\ if\ }x\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\\ 0\mathrm{\ if\ }x\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)^{\otimes k},k\neq 1.\end{cases}

We also set $\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})$ as the quotient of the tensor module $\mathcal{U}_{h}(\Sigma)\otimes_{\mathbb{Q}[h]}\mathbb{Q}[h]\pi_{1}(\Sigma,\star_{1},\star_{2})\otimes_{\mathbb{Q}[h]}\mathcal{U}_{h}(\Sigma)$ by the relation

•

For $x_{1},\cdots,x_{i}\in\mathbb{Q}[h]\lvert\pi_{1}\rvert(\Sigma)$ and $y\in\mathbb{Q}[h]\pi_{1}(\Sigma,\star_{1},\star_{2})$ ,

	$\displaystyle x_{1}\otimes\cdots\otimes x_{i^{\prime}}\otimes y\otimes x_{i^{\prime}+1}\otimes\cdots\otimes x_{i}$
	$\displaystyle-x_{1}\otimes\cdots\otimes x_{i^{\prime}-1}\otimes y\otimes x_{i^{\prime}}\otimes\cdots\otimes x_{i}$
	$\displaystyle=x_{1}\otimes\cdots\otimes x_{i^{\prime}-1}\otimes\sigma(x_{i^{\prime}})(y)\otimes x_{i^{\prime}+1}\otimes\cdots\otimes x_{i}$

We define the Lie action $\sigma(\cdot)(\cdot):\mathcal{U}_{h}(\Sigma)\to\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})$ by the Leibniz rule, which means that we have

	$\displaystyle\sigma(x_{1}\otimes\cdots\otimes x_{i})(x^{\prime}\otimes y\otimes x^{\prime\prime})=[x_{1}\otimes\cdots\otimes x_{i},x^{\prime}]\otimes y\otimes x^{\prime\prime}$
	$\displaystyle+\sum_{i^{\prime}\in\{1,\cdots,i\}}x^{\prime}\otimes x_{1}\otimes\cdots\otimes x_{i-1}\otimes\sigma(x_{i})(y)\otimes x_{i^{\prime}+1}\otimes\cdots\otimes x_{i}\otimes x$
	$\displaystyle+x^{\prime}\otimes y\otimes[x_{1}\otimes\cdots\otimes x_{i},x^{\prime\prime}]$

for $x_{1},\cdots,x_{i}\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)$ , $x,x^{\prime}\in\mathcal{U}_{h}(\Sigma)$ , and $y\in\mathbb{Q}\pi_{1}(\Sigma)$ . It also satisfies

\sigma(x)(y)=\frac{1}{h}(x\otimes y-y\otimes x).

We denote by $\Psi_{\mathbb{Q}\pi_{1}}^{\mathcal{U}_{h}}$ the natural injection $\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})$ and by $\Psi_{\mathcal{U}_{h}}^{\mathbb{Q}\pi_{1}}$ the natural surjection

	$\displaystyle\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle x\otimes y\otimes x^{\prime}\mapsto\begin{cases}(x)(x^{\prime})y\mathrm{\ if\ }x,x^{\prime}\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)^{\otimes 0}=\mathbb{Q}\\ 0\mathrm{\ if\ }y\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)^{\otimes k},y^{\prime}\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)^{\otimes k^{\prime}},k+k^{\prime}\geq 1\end{cases}$

for $y\in\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})$ .

We set the $\mathbb{Q}[h]$ -algebra homomorphism $\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}$ and the $\mathbb{Q}[h]$ -module one $\Psi_{\mathcal{U}_{h}}^{\mathcal{A}}$ by

	$\displaystyle\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}:\mathcal{U}_{h}(\Sigma)\to\mathcal{A}^{\prime}(\Sigma),\lvert\gamma\rvert\in\lvert\pi_{1}\rvert(\Sigma)\mapsto K_{\lvert\gamma\rvert},$
	$\displaystyle\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}:\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})\to\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2}),x\otimes\gamma\otimes x^{\prime}\mapsto\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(x)T_{\gamma}\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(x)$

for $\gamma\in\pi_{1}(\Sigma)$ and $x,x^{\prime}\in\mathcal{U}_{h}(\Sigma)$ , where $K_{\lvert\gamma\rvert}$ and $T_{\gamma}$ are a knot and a a tangle whose homotopy class is $\lvert\gamma\rvert$ and $\gamma$ , respectively. We can prove the following theorem in the same way as the proof of Turaev [15, Theorem 3.3].

Theorem 3.3.

For a compact connected oriented surface $\Sigma$ and $\star_{1},\star_{2}\in\partial\Sigma$ , the algebra homomorphism $\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}:\mathcal{U}_{h}(\Sigma)\to\mathcal{A}^{\prime}(\Sigma)$ and module one $\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}:\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})\to\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2})$ are bijective. Furthermore, we have

	$\displaystyle[\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(x),\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(x^{\prime})]=\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}([x,x^{\prime}]),$
	$\displaystyle\sigma(\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(x))(\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(y))=\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(\sigma(x)(y))$

for $x,x^{\prime}\in\mathcal{U}_{h}(\Sigma)$ and $y\in\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})$ .

We denote by $\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}$ the reverse map of $\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}$ , and consider the composites

	$\displaystyle\Psi_{\mathbb{Q}\pi_{1}}^{\mathcal{A}^{\prime}}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathbb{Q}\pi_{1}}^{\mathcal{U}_{h}}\circ\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}},\ \Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}\circ\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}},$
	$\displaystyle\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\pi_{1}}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}\circ\Psi_{\mathcal{U}_{h}}^{\mathbb{Q}\pi_{1}},\ \Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}\circ\Psi_{\mathcal{U}_{h}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}.$

Let $\{F^{n}\mathcal{U}_{h}(\Sigma)\}_{n\geq 0}$ and $\{F^{n}\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})\}_{n\geq 0}$ be the filtrations of $\mathcal{U}_{h}(\Sigma)$ and $\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})$ such as

	$\displaystyle F^{n}\mathcal{U}_{h}(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\sum_{2i_{0}+i_{1}+\cdots+i_{j}\geq n}h^{i_{0}}(F^{i_{1}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma))\otimes\cdots\otimes(F^{i_{j}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)),$
	$\displaystyle F^{n}\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\sum_{i_{1}+i_{2}+i_{3}\geq n}F^{i_{1}}\mathcal{U}_{h}(\Sigma)\otimes(F^{i_{2}}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}))\otimes F^{i_{3}}\mathcal{U}_{h}(\Sigma).$

We consider the completed algebra and module

	$\displaystyle\widehat{\mathcal{U}_{h}}(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\underleftarrow{\lim}_{i\rightarrow\infty}\mathcal{U}_{h}(\Sigma)/F^{i}\mathcal{U}_{h}(\Sigma),$
	$\displaystyle\widehat{\mathcal{U}_{h}}(\Sigma,\star_{1},\star_{2})\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\underleftarrow{\lim}_{i\rightarrow\infty}\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})/F^{i}\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})$

of $\mathcal{U}_{h}(\Sigma)$ and $\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})$ in terms of the filtrations. Furthermore, they also define $\{F^{n}\mathcal{A}^{\prime}(\Sigma)\}_{n\geq 0}$ and $\{F^{n}\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2})\}_{n\geq 0}$ of $\mathcal{A}^{\prime}(\Sigma)$ and $\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2})$ such as

	$\displaystyle F^{n}\mathcal{A}^{\prime}(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(F^{n}\mathcal{U}_{h}(\Sigma)),$
	$\displaystyle F^{n}\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(F^{n}(\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2})).$

We also consider the completed algebra and module

	$\displaystyle\widehat{\mathcal{A}}^{\prime}(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\underleftarrow{\lim}_{i\rightarrow\infty}\mathcal{A}^{\prime}(\Sigma)/F^{i}\mathcal{A}^{\prime}(\Sigma),$
	$\displaystyle\widehat{\mathcal{A}}^{\prime}(\Sigma,\star_{1},\star_{2})\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\underleftarrow{\lim}_{i\rightarrow\infty}\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2})/F^{i}\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2}),$

of $\mathcal{A}^{\prime}(\Sigma)$ and $\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2})$ . For a compact oriented surface $S$ having components $\Sigma^{(1)},\cdots,\Sigma^{(k)}$ , using the isomorphism

e_{\coprod_{i}\Sigma^{(i)},S*}:\mathcal{A}^{\prime}(\Sigma^{(1)})\otimes\cdots\otimes\mathcal{A}^{\prime}(\Sigma^{(k)})\to\mathcal{A}^{\prime}(S),

we define the filtration $\{F^{n}\mathcal{A}^{\prime}(S)\}_{n\geq 0}$ as

F^{n}\mathcal{A}^{\prime}(S)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\sum_{i_{1}+\cdots+i_{k}\geq n}e_{\coprod_{i}\Sigma^{(i)},S*}(F^{i_{1}}\mathcal{A}^{\prime}(\Sigma^{(1)})\otimes\cdots\otimes F^{i_{k}}(\Sigma^{(i_{k})})).

We also set the filtrations $\{F^{n}\widehat{V}\}_{n\geq 0}$ of completions by

F^{n}\widehat{V}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\ker(\widehat{V}\to V/F^{n}V)

for $V=\mathcal{U}_{h}(\Sigma),\mathcal{U}_{h}(\Sigma,\star_{1},\star_{2}),\mathcal{A}^{\prime}(\Sigma),\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2})$ and $\mathcal{A}^{\prime}(S)$ . The filtration has the following property.

Proposition 3.4.

For compact connected surfaces $S$ and $S^{\prime}$ , the induced map $e_{*}:\mathcal{A}^{\prime}(S^{\prime})\to\mathcal{A}^{\prime}(S)$ of an embedding $e:S^{\prime}\times I\to S\times I$ satisfies

e_{*}(F^{N}\mathcal{A}^{\prime}(S^{\prime}))\subset F^{N}\mathcal{A}^{\prime}(S)

for any $N\in\mathbb{Z}_{\geq 0}.$

Proof.

We can prove the statement in the same way as our paper [14, §4]. Furthermore, it is easy to prove it by Lemma 7.6 in this paper in case the sequence is trivial, which means $\mathbf{b}=\{0,0,0,\cdots\}$ .

∎

4. Homology cylinders

Let $\Sigma$ be a compact connected oriented surface with boundary. We define a homology cobordism or cylinder not only in the case that $\Sigma$ has one boundary component but in the general case.

Definition 4.1.

Let $M$ be a compact connected oriented 3-manifold and $\alpha:\partial(\Sigma\times I)\to\partial M$ a diffeomorphism. We call the pair $(M,\alpha)$ a homology cylinder, if and only if the embeddings

\displaystyle\alpha_{0}:\Sigma\to M,p\mapsto\alpha(p,0),\ \alpha_{1}:\Sigma\to M,p\mapsto\alpha(p,1)

induce isomorphisms $\alpha_{0*}:H_{1}(\Sigma,\mathbb{Z})\to H_{1}(M,\mathbb{Z})$ and $\alpha_{1*}:H_{1}(\Sigma,\mathbb{Z})\to H_{1}(M,\mathbb{Z})$ in homology groups over the integers.

Let $(M^{1},\alpha^{1})$ and $(M^{2},\alpha^{2})$ be two homology cobordisms. If there exists a diffeomorphism $\chi:M^{1}\to M^{2}$ satisfying $\chi\circ\alpha^{1}=\alpha^{2}$ , such pairs are called diffeomorphic. We denote by $\mathcal{C}(\Sigma)$ the set of diffeomorphic classes of homology cobordisms of the surface $\Sigma$ .

We consider the stacking sum of homology cobordisms defined as follows, which makes $\mathcal{C}(\Sigma)$ a monoid. For two homology cobordisms $(M^{1},\alpha^{1})$ and $(M^{2},\alpha^{2})$ , we set a $3$ -manifold $M^{1}\circ_{\alpha^{1},\alpha^{2}}M^{2}$ and a diffeomorphism $\alpha^{1}\sqcup\alpha^{2}:\partial(\Sigma\times I)\to\partial(M^{1}\circ_{\alpha^{1},\alpha^{2}}M^{2})$ as the following. The $3$ -manifold $M^{1}\circ_{\alpha^{1},\alpha^{2}}M^{2}$ is the quotient of $M^{1}\sqcup M^{2}$ by the relation

\alpha^{1}_{0}(p)\sim\alpha^{2}_{0}(p)\mathrm{\ for\ }p\in\Sigma.

We define the diffeomorphism $\alpha^{1}\sqcup\alpha^{2}$ as

(\alpha^{1}\sqcup\alpha^{2})(p,t)=\begin{cases}\alpha^{1}(p,1)\ &\\ \alpha^{1}(p,2t-1)\ &\mathrm{if}\ t\in[\frac{1}{2},1]\\ \alpha^{2}(p,2t)\ &\mathrm{if}\ t\in[0,\frac{1}{2}]\\ \alpha^{2}(p,0).\ &\end{cases}

Then the pair $(M^{1}\circ_{\alpha^{1},\alpha^{2}}M^{2},\alpha^{1}\sqcup\alpha^{2})$ is also a homology cobordism. The operator

(\cdot)\circ(\cdot):\mathcal{C}(\Sigma)\times\mathcal{C}(\Sigma)\to\mathcal{C}(\Sigma),((M^{1},\alpha^{1}),(M^{2},\alpha^{2}))\mapsto(M^{1}\circ_{\alpha^{1},\alpha^{2}}M^{2},\alpha^{1}\sqcup\alpha^{2})

makes $\mathcal{C}(\Sigma)$ a monoid, and we call it the stacking sum.

We will consider the action

\Phi:\mathcal{C}(\Sigma)\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,\star))

of $\mathcal{C}(\Sigma)$ on $\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})$ as follows. We set isomorphisms

	$\displaystyle\diamondsuit_{0}^{1}:\pi_{1}(M,\alpha(\bullet,0),\alpha(,0))\to\pi_{1}(M,\alpha(\bullet,1),\alpha(,1)),\gamma\mapsto\gamma_{10\bullet}\gamma\gamma_{01*},$
	$\displaystyle\diamondsuit_{1}^{0}:\pi_{1}(M,\alpha(\bullet,1),\alpha(,1))\to\pi_{1}(M,\alpha(\bullet,0),\alpha(,0)),\gamma\mapsto\gamma_{01\bullet}\gamma\gamma_{10*},$

where the continuous maps

\displaystyle[0,1]\to M,t\mapsto\alpha(\star,t),\ \ [0,1]\to M,t\mapsto\alpha(\star,1-t),

represent the paths $\gamma_{01\star}\in\pi_{1}(M,\alpha(\star,0),\alpha(\star,1)),$ $\gamma_{10\star}\in\pi_{1}(M,\alpha(\star,1),\alpha(\star,0))$ , respectively, for any $\star\in\partial\Sigma$ . By Stallings’s theorem [13], the embeddings $\alpha_{0}$ and $\alpha_{1}$ induced isomorphisms

	$\displaystyle\alpha_{0}:\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,)\to\widehat{\mathbb{Q}\pi_{1}}(M,\pi_{1}(M,\alpha(\bullet,0),\alpha(*,0)),$
	$\displaystyle\alpha_{1}:\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,)\to\widehat{\mathbb{Q}\pi_{1}}(M,\alpha(\bullet,1),\alpha(*,1)).$

Then the action $\Phi((M,\alpha))=(M,\alpha)_{*}\in\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*))$ is defined as the composite map

\Phi((M,\alpha))=(M,\alpha)_{*}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\alpha_{1*}^{-1}\circ\diamondsuit_{0}^{1}\circ\alpha_{0*}.

The map

\Phi=(\cdot)_{*}:\mathcal{C}(\Sigma)\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*))

is a monoid homomorphism.

To make the definition of homology cylinders, we will check that the two conditions in the following proposition are equivalent.

Proposition 4.2.

For a homology cobordism $(M,\alpha)$ , the following two conditions are equivalent.

(1)

We have

\ker(\iota_{*}:H_{1}(\partial(\Sigma\times I),\mathbb{Z})\to H_{1}(\Sigma\times I,\mathbb{Z}))=\ker(\alpha_{*}:H_{1}(\partial(\Sigma\times I),\mathbb{Z})\to H_{1}(M,\mathbb{Z})),

where the natural embedding $\iota:\partial(\Sigma\times I)\to\Sigma\times I$ and $\alpha:\partial(\Sigma\times I)\to M$ induce the group homomorphisms

\displaystyle\iota_{*}:H_{1}(\partial(\Sigma\times I),\mathbb{Z})\to H_{1}(\Sigma\times I,\mathbb{Z}),\ \alpha_{*}:H_{1}(\partial(\Sigma\times I),\mathbb{Z})\to H_{1}(M,\mathbb{Z}).

(2)

For any $\star_{1},\star_{2}\in\partial\Sigma$ , the action $(M,\alpha)_{*}$ satisfies

((M,\alpha)_{*}-\mathrm{id}_{\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*)})(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2}))\subset F^{2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2}).

Proof.

To prove it, we need some notation. We choose some base points $\star,\star_{1},\star_{2}\in\partial\Sigma$ . Let

\displaystyle\gamma^{\partial(\Sigma\times I)}_{01\star}\in\pi_{1}(\partial(\Sigma\times I),(\star,0),(\star,1)),

\displaystyle\gamma^{\partial(\Sigma\times I)}_{10\star}\in\pi_{1}(\partial(\Sigma\times I),(\star,1),(\star,0)),

be the paths represented by the continuous maps

\displaystyle I\to\partial(\Sigma\times I),t\mapsto(\star,t),\

\displaystyle I\to\partial(\Sigma\times I),t\mapsto(\star,1-t).

We set the embeddings $\iota^{\prime}_{0},\iota^{\prime}_{1}:\Sigma\to\partial(\Sigma\times I)$ as

\displaystyle\iota^{\prime}_{0}:\Sigma\to\partial(\Sigma\times I),p\mapsto(p,0),\ \iota^{\prime}_{1}:\Sigma\to\partial(\Sigma\times I),p\mapsto(p,1).

They induce

	$\displaystyle\iota^{\prime}_{0*}:\pi_{1}(\Sigma,\star_{1},\star_{2})\to\pi_{1}(\partial(\Sigma\times I),(\star_{1},0),(\star_{2},0)),$
	$\displaystyle\iota^{\prime}_{1*}:\pi_{1}(\Sigma,\star_{1},\star_{2})\to\pi_{1}(\partial(\Sigma\times I),(\star_{1},1),(\star_{2},1)).$

First, we prove $(1)\Rightarrow(2)$ . Let $\gamma$ be an element of $\pi_{1}(\Sigma,\star_{1},\star_{2})$ , and $\bar{\gamma}$ the reverse of $\gamma$ . Then, the homology class represented by the path

\iota^{\prime}_{0*}(\gamma)\gamma^{\partial(\Sigma\times I)}_{01\star_{2}}\iota^{\prime}_{1,*}(\bar{\gamma})\gamma^{\partial(\Sigma\times I)}_{10\star_{1}}

satisfies

\iota_{*}([\iota^{\prime}_{0*}(\gamma)\gamma^{\partial(\Sigma\times I)}_{01\star_{2}}\iota^{\prime}_{1,*}(\bar{\gamma})\gamma^{\partial(\Sigma\times I)}_{10\star_{1}}])=0\in H_{1}(\partial(\Sigma\times I),\mathbb{Z}).

By the first statement, we have

	$\displaystyle\alpha_{}([\iota^{\prime}_{0}(\gamma)\gamma^{\partial(\Sigma\times I)}_{01\star_{2}}\iota^{\prime}_{1,*}(\bar{\gamma})\gamma^{\partial(\Sigma\times I)}_{10\star_{1}}])=0\in H_{1}(M,\mathbb{Z}),$
	$\displaystyle\alpha_{0}(\gamma)\gamma_{01\star_{2}}\alpha_{1}(\bar{\gamma})\gamma_{10\star_{1}}\in[\pi_{1}(M,\alpha(\star_{1},0)),\pi_{1}(M,\alpha(\star_{1},0))]\subset 1+F^{2}\mathbb{Q}\pi_{1}(M,\alpha(\star_{1},0)).$

So we have

	$\displaystyle\alpha_{0}(\gamma)-\gamma_{01\star_{1}}\alpha_{1}({\gamma})\gamma_{10\star_{2}}$
	$\displaystyle=(\alpha_{0}(\gamma)\gamma_{01\star_{2}}\alpha_{1}(\bar{\gamma})\gamma_{10\star_{1}}-1)\gamma_{01\star_{1}}\alpha_{1*}({\gamma})\gamma_{10\star_{2}}$
	$\displaystyle\in F^{2}\mathbb{Q}\pi_{1}(M,\alpha(\star_{1},0),\alpha(\star_{2},0))$

Hence we obtain $(1)\Rightarrow(2)$ .

Next, we prove $(2)\Rightarrow(1)$ . We will prove it in five steps.

(Step 1)

The homology group $H_{1}(\partial(\Sigma\times I),\mathbb{Z})$ is the direct sum $V\oplus\iota^{\prime}_{1*}(H_{1}(\Sigma,\mathbb{Z}))$ , where, for a path $\gamma\in\pi_{1}(\Sigma,\star_{1},\star_{2})$ and the reverse path $\bar{\gamma}$ , the homology classes of

\iota^{\prime}_{0}(\gamma)\gamma_{01\star_{2}}^{\partial(\Sigma\times I)}\iota^{\prime}_{1}(\bar{\gamma})\gamma_{10\star_{1}}^{\partial(\Sigma\times I)}

generate $V$ .

(Step 2)

We have $\ker\iota_{*}=V.$
(Step 3)

The composite $\alpha_{*}\circ\iota^{\prime}_{1*}:H_{1}(\Sigma,\mathbb{Z})\to H_{1}(M,\mathbb{Z})$ is an isomorphism.
(Step 4)

We have $\alpha_{*}(V)=\{0\}.$
(Step 5)

We have $\ker\alpha_{*}=V$ .

(Step 1) The basis in the figure generates the homology group $H_{1}(\partial(\Sigma\times I),\mathbb{Z})$ as a free $\mathbb{Z}$ -module. Some set of it generates $\iota^{\prime}_{1*}(H_{1}(\Sigma,\mathbb{Z}))$ , and one of the others $V$ . This proves (Step 1).

(Step 2) Using the above basis, we have $\iota_{*}(V)=\{0\}$ and $\iota_{*}\circ\iota^{\prime}_{1*}=\mathrm{id}_{H_{1}(\Sigma,\mathbb{Z})}:H_{1}(\Sigma,\mathbb{Z})\to H_{1}(\Sigma,\mathbb{Z})$ . So we obtain (Step 2).

(Step 3) By definition of homology cobordisms, the homomorphism $\alpha_{1*}:H_{1}(\Sigma,\mathbb{Z})\to H_{1}(M,\mathbb{Z})$ is an isomorphism. Since $\alpha_{1}=\alpha\circ\iota^{\prime}_{1}$ , we have (Step 3).

(Step 4) Here we use (2). By it, for a path $\gamma\in\pi_{1}(\Sigma,\star_{1},\star_{2})$ and the reverse $\bar{\gamma}$ , we have

\alpha_{0*}(\gamma)-\gamma_{01\star_{1}}\alpha_{1*}(\gamma)\gamma_{10\star_{2}}\in F^{2}\mathbb{Q}\pi_{1}(M,\star_{1M,0},\star_{2M,0}).

So we obtain

\alpha_{0*}(\gamma)\gamma_{10\star_{2}}\alpha_{1*}(\bar{\gamma})\gamma_{10\star_{1}}-1\in F^{2}\mathbb{Q}\pi_{1}(M,\star_{1M0}).

Since

\displaystyle I_{\mathbb{Q}\pi_{1}(M,\star_{1M0})}/(I_{\mathbb{Q}\pi_{1}(M,\star_{1M0})})^{2}

\displaystyle\simeq H_{1}(M,\mathbb{Q}),x-1\mapsto[x],

where $I_{\mathbb{Q}\pi_{1}(M,\star_{1M0})}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\{\sum_{g\in G}q_{g}g|\sum_{g\in G}q_{g}=0\}$ , the homology class

[\alpha_{0*}(\gamma)\gamma_{10\star_{2}}\alpha_{1*}(\bar{\gamma})\gamma_{10\star_{1}}]

equals $0$ as desired.

(Step 5) The statement (Step 4) gives $\alpha_{*}(v_{1}+\iota^{\prime}_{1*}(v_{2}))=\alpha_{*}(\iota^{\prime}_{1*}(v_{2}))$ for two elements $v_{1}\in V$ and $v_{2}\in H_{1}(\Sigma,\mathbb{Z})$ . By (Step 3), if $\alpha_{*}(v_{1}+\iota^{\prime}_{1*}(v_{2}))=0$ , $v_{2}$ equals $0$ . Using (Step 1), we obtain $\ker\alpha_{*}=V$ as desired.

By the second and fifth steps, we obtain $(2)\Rightarrow(1)$ .

∎

Remark 4.3.

For a group $G$ , let $I_{\mathbb{Q}G}$ be the augmentation ideal $\{\sum_{g\in G}q_{g}g|\sum_{g\in G}q_{g}=0\}$ . We can prove $[G,G]\subset 1+I_{\mathbb{Q}G}^{2}$ by

\prod_{i=1}^{k}[\gamma_{i1},\gamma_{i2}]=\prod_{i=1}^{k}(((\gamma_{i1}-1)(\gamma_{i2}-1)-(\gamma_{i2}-1)(\gamma_{i1}-1))\gamma_{i1}^{-1}\gamma_{i2}^{-1}+1)

for elements $\gamma_{11},\gamma_{21},\cdots,\gamma_{k1},$ $\gamma_{12},\gamma_{22},\cdots,\gamma_{k2}\in G$ . So, the linear map

\displaystyle\mathbb{Q}\otimes G/[G,G]\to I_{\mathbb{Q}G}/(I_{\mathbb{Q}G})^{2},[x]\mapsto x-1

is well-defined. It is bijection because

(x-1)(y-1)=(xy-1)-(x-1)-(y-1).

Hence we have the isomorphism

\displaystyle\mathbb{Q}\otimes G/[G,G]\simeq I_{\mathbb{Q}G}/(I_{\mathbb{Q}G})^{2}.

For a homology cobordism $(M,\alpha)$ , if it has one of the properties in Proposition 4.2, we call it a homology cylinder. Then, it has both of them.

We introduce two ways to construct homology cylinders. The first way is to use boundary links. The second way is to use standard embeddings, which is an analogy of Heegaard splittings. Both are equivalent to each other, and we can obtain any diffeomorphic class of a homology cylinder in one of both.

“A construction using a boundary link”

First, we prepare notations about boundary links.

Definition 4.4.

For a framed unoriented link $L$ in $\Sigma\times I$ , we call it a boundary link if and only if there exists an embedded surface $S$ satisfying the two conditions.

•

Any component of $S$ has one or two boundary components.
•

The tubular neighborhood of the boundary is isotopic to $L$ .

Here we call $S$ a Seifert surface. A label of $S$ is a map $\pi_{0}(\partial S)\to\{-1,+1\}$ satisfying

\partial_{1},\partial_{2}\subset(\mathrm{a\ component\ of\ }S),[\partial_{1}]\neq[\partial_{2}]\Rightarrow\lambda([\partial_{1}])\lambda([\partial_{2}])=-1.

Let a labeled boundary link $L(\lambda)$ be a framed unoriented link having the two properties.

•

The unframed link of $L$ equals that of $L(\lambda)$ .
•

If $l$ and $l^{\prime}$ are knot components of $L$ and $L(\lambda)$ corresponding to $[\partial]\in\pi_{0}(\partial S)$ , respectively, we have

$w(l^{\prime})-w(l)=\lambda([\partial]).$

Here $w(\cdot)$ is the framing number, which is the difference in the number of positive crossings and that of negative ones.

Let $L$ be a boundary link in $\Sigma\times I$ and $\lambda$ a label of a Seifert surface of $L$ . We choose a 3-manifold representing the diffeomorphic class obtained by the Dehn surgery of $L(\lambda)$ and denote it by $(\Sigma\times I)(L(\lambda))$ . Then the pair $((\Sigma\times I)(L(\lambda)),\mathrm{id}_{\partial(\Sigma\times I)})$ is a homology cylinder. Here we write the same symbol $(\Sigma\times I)(L(\lambda))$ for the diffeomorphic class of $((\Sigma\times I)(L(\lambda)),\mathrm{id}_{\partial(\Sigma\times I)})$ . Let $\mathcal{IC}_{1}(\Sigma)$ be the subset of $\mathcal{IC}(\Sigma)$ consisting of all elements obtained in this way.

Proposition 4.5 ([3] p.4 Theorem 2.5, p.17 Lemma 6.1).

We have $\mathcal{IC}_{1}(\Sigma)=\mathcal{IC}(\Sigma)$ . In other words, for any homology cylinder $(M,\alpha)$ , there exists a boundary link $L\subset\Sigma\times I$ and exists a label $\lambda$ of a Seifert surface of $L$ such that the diffeomorphic class $(\Sigma\times I)(L(\lambda))$ is that of $(M,\alpha)$ .

“A construction using a standard embedding”

We introduce the notion of standard embeddings.

Definition 4.6.

We call $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\hookrightarrow\Sigma\times I$ a standard embedding if and only if there exists an embedding $e^{\prime}_{\mathrm{st}}:\Sigma_{\mathrm{st}}\hookrightarrow\Sigma$ satisfying the two properties.

•

The image $e^{\prime}_{\mathrm{st}}(\Sigma)$ is the closure of $\Sigma$ except for closed disks and includes $\partial\Sigma$ .
•

We have $e_{\mathrm{st}}(p,t)=(e^{\prime}_{\mathrm{st}}(p),\frac{1+t}{3})$ for $p\in\Sigma_{\mathrm{st}}$ and $t\in I$ .

For a diffeomorphism $\chi^{\prime}\in\mathrm{Diff}^{+}(\widetilde{\Sigma}_{\mathrm{st}},\partial\widetilde{\Sigma}_{\mathrm{st}})$ , the quotient space of the set

(\mathrm{the\ closure\ of\ }(\Sigma\times I)\backslash e({\Sigma}_{\mathrm{st}}\times I))\sqcup{\Sigma}_{\mathrm{st}}\times I

by the relation

e_{\mathrm{st}}\circ\chi^{\prime}(p)\sim p\mathrm{\ for\ }p\in\widetilde{\Sigma}_{\mathrm{st}}

is a 3-manifold. We denote by $(\Sigma\times I)(e_{\mathrm{st}},\chi^{\prime})$ the 3-manifold. We recall that $\mathcal{I}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})\subset\mathcal{M}(\widetilde{\Sigma}_{\mathrm{st}})$ is a subgroup generated

\displaystyle\{t_{c_{1}}\circ t_{c_{2}}^{-1}|(c_{1},c_{2})\mathrm{\ bounds\ a\ surface.}\}\cup\{t_{c}|c\mathrm{\ bounds\ a\ surface.}\}.

If $\chi$ represents an element $\xi$ of $\mathcal{I}^{\prime}(\Sigma)$ , the pair $((\Sigma\times I)(e_{\mathrm{st}},\chi^{\prime}),\alpha^{e_{\mathrm{st}}})$ is a homology cylinder, where we set the diffeomorphism $\alpha^{e_{\mathrm{st}}}:\partial(\Sigma\times I)\to\partial((\Sigma\times I)(e_{\mathrm{st}},\chi^{\prime}))$ as

\alpha^{e_{\mathrm{st}}}(p,t)=\begin{cases}(p,t)\mathrm{\ if\ }p\in\Sigma,t\in\{0,1\}\\ (p,t)\mathrm{\ if\ }p\in\partial\Sigma,t\in[0,\frac{1}{3}]\cup[\frac{2}{3},1]\\ e_{\mathrm{st}}^{-1}(p,3t-1)\mathrm{\ if\ }p\in\partial\Sigma,t\in[\frac{1}{3},\frac{2}{3}]\end{cases}

We write $(\Sigma\times I)(e_{\mathrm{st}},\xi)$ for the diffeomorphic class of $((\Sigma\times I)(e_{\mathrm{st}},\chi^{\prime}),\alpha^{e_{\mathrm{st}}})$ . Let $\mathcal{IC}_{2}(\Sigma)$ be the subset of $\mathcal{IC}(\Sigma)$ consisting of all elements obtained in this way. To prove $\mathcal{IC}_{2}(\Sigma)=\mathcal{IC}_{1}(\Sigma)$ , we need the following lemma.

Lemma 4.7.

Let $S$ be a compact oriented surface and $e$ an embedding from $S$ to $\Sigma\times I$ . Then, there exists a standard embedding $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ and exists an embedding $\widetilde{e}:S\to\widetilde{\Sigma}_{\mathrm{st}}$ satisfying that the composite $e_{\mathrm{\circ}}\widetilde{e}$ is isotopic to $e$ .

Proof.

First, we construct a standard embedding $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ by the three steps.

•

We split $S$ into $0$ -handles and 1-handles. For example, see Figure 4.

Figure 4. A $0$ -handle and $1$ -handles
•

We consider $1$ -handles as tangles and illustrate a diagram presenting the isotopy type of the embedding $e$ similar to a tangle diagram. For example, see Figure 5.

Figure 5. A diagram
•

We set $\Sigma_{\mathrm{st}}$ as the closure of $\Sigma$ except for closed disks corresponding to angles for any crossing of this diagram. For example, see Figure 6.

Figure 6. $\Sigma_{\mathrm{st}}$

Furthermore, we set $\widetilde{e}:S\to\widetilde{\Sigma}_{\mathrm{st}}$ as this diagram except for the neighborhood of the crossings, where we set $\widetilde{e}$ as Figure 7. Then the composite $e_{\mathrm{st}}\circ\widetilde{e}$ is isotopic to $e$ . This proves the lemma.

Figure 7.

\widetilde{e}:S\to\partial(\Sigma_{\mathrm{st}}\times I)\backslash e_{\mathrm{st}}^{-1}(\partial\Sigma\times I)

∎

To prove $\mathcal{IC}(\Sigma)=\mathcal{IC}_{2}(\Sigma)$ , we need the following lemma, which we call Lickorish’s trick [8].

Lemma 4.8.

Let $S$ be a compact oriented surface such that each component of $S$ has one or two boundary components, $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ a standard embedding, $\widetilde{e}:S\to\widetilde{\Sigma}_{\mathrm{st}}$ an embedding, and $L_{e_{\mathrm{st}}\circ\widetilde{e}(S)}$ a boundary link whose Seifert surface is $e\circ\widetilde{e}(S)$ . We write $c_{\partial}$ for a closed curve in $S$ parallel to a boundary $\partial$ . For a label $\lambda:\pi_{0}(\partial S)=\pi_{0}(\partial e_{\mathrm{st}}\circ\widetilde{e}(S))\to\{\pm 1\}$ of $e\circ\widetilde{e}(S)$ , we have

(\Sigma\times I)(e_{\mathrm{st}},\prod_{[\partial]\in\pi_{0}(\partial S)}t_{\widetilde{e}(c_{\partial})}^{-\lambda([\partial])})=(\Sigma\times I)(L_{e_{\mathrm{st}}\circ\widetilde{e}(S)}(\lambda)).

Proposition 4.9.

We have $\mathcal{IC}(\Sigma)=\mathcal{IC}_{2}(\Sigma)$ . In other words, for any homology cylinder $(M,\alpha)$ , there exists a standard embedding $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ and exists an element $\xi$ of $\mathcal{I}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})$ such that the diffeomorphic class $(\Sigma\times I)(e_{\mathrm{st}},\xi)$ is that of the pair $(M,\alpha)$ .

Proof.

By Proposition 4.5, it is enough to show that $\mathcal{IC}_{1}(\Sigma)\subset\mathcal{IC}_{2}(\Sigma)$ . Let $L$ be a boundary link in $\Sigma\times I$ and $e:S\to\Sigma\times I$ a Seifert surface of $L$ . By Lemma 4.7, there exists an embedding $e_{S}:S\to\widetilde{\Sigma}_{\mathrm{st}}=\overline{\partial(\Sigma_{\mathrm{st}}\times I)\backslash e^{-1}(\partial\Sigma\times I)}$ such that the composite $e_{\mathrm{st}}\circ e_{S}$ is isotopic to this embedding $e$ . By Lemma 4.8, for a label $\lambda:\pi_{0}(\partial S)\to\{\pm 1\}$ of $S$ , we have

(\Sigma\times I)(e_{\mathrm{st}},\prod_{[\partial]\in\pi_{0}(\partial S)}t_{e_{S}(c_{\partial})}^{-\lambda([\partial])})=(\Sigma\times I)(L(\lambda)),

where we denote by $c_{\partial}$ the simple closed curve parallel to the boundary $\partial$ . Hence we obtain $(\Sigma\times I)(L(\lambda))\in\mathcal{IC}_{2}(\Sigma)$ as desired.

∎

5. The main result

For $\bullet,*\in\partial\Sigma$ , we consider the action $\Phi:\mathcal{IC}(\Sigma)\to\mathrm{Aut}(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*))$ . We obtain

	$\displaystyle\Xi_{}(I_{\mathbb{Q}\pi_{1}(\Sigma,\bullet)}^{n}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,))=I_{\mathbb{Q}\pi_{1}(\Sigma,\bullet)}^{n}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*),$
	$\displaystyle(\Xi_{}-\mathrm{id})(\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,))\subset I_{\mathbb{Q}\pi_{1}(\Sigma,\bullet)}^{2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,\star)$

for any $\Xi$ . By Prop 2.2, there exists a unique element $\widetilde{\zeta}(\Xi)$ of $F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ satisfying

\exp(\sigma(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\Xi)))=\Xi_{*}:\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*)\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\bullet,*).

In this section, we introduce two ways to compute $\widetilde{\zeta}:\mathcal{IC}(\Sigma)\to F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ . The first way is to use a sequence of the Goldman Lie algebra. The second one is to use the skein algebra $\mathcal{A}^{\prime}$ .

To state the main theorems, we introduce an extension of $\mathcal{A}^{\prime}(\Sigma)$ and set the filtration of it as

	$\displaystyle(\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3})\mathcal{A}^{\prime}(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3}\mathcal{A}^{\prime}(\Sigma)\subset\mathbb{Q}[h,\frac{1}{h}]\otimes_{\mathbb{Q}}\mathcal{A}^{\prime}(\Sigma),$
	$\displaystyle F^{n}(\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3})\mathcal{A}^{\prime}(\Sigma)=\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{2+\max(n,*)}\mathcal{A}^{\prime}(\Sigma).$

We consider the completion

\widehat{(\sum_{*\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{*}}F^{3*})}\mathcal{A}^{\prime}(\Sigma)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\underleftarrow{\lim}_{n\rightarrow\infty}(\sum_{*\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{*}}F^{3*})\mathcal{A}^{\prime}(\Sigma)/F^{n}(\sum_{*\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{*}}F^{3*})\mathcal{A}^{\prime}(\Sigma)

in terms of the filtration. Using this extension, we can obtain a formula about the Baker-Campbel-Hausdorff series as follows.

Remark 5.1.

Let $(R,\{F^{n}R\}_{n\geq 0})$ be a filtered completed ring satisfying $F^{j}RF^{k}R\subset F^{j+k}R$ . In this paper, we set the bracket $\{x,y\}$ by $xy-yx$ . Then, for $x_{0},x_{1}\in F^{1}R$ , $\mathrm{bch}^{\prime}(x_{0},x_{1})\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\log(\exp x_{0},\exp x_{1})$ equals

\sum_{j}\sum_{\epsilon_{1},\cdots,\epsilon_{j}\in\{0,1\}}b_{\epsilon_{1},\cdots,\epsilon_{j}}\{x_{\epsilon_{1}},\{x_{\epsilon_{2}},\cdots\{x_{\epsilon_{j-1}},x_{\epsilon_{j}}\}\}\},

where $b_{\epsilon_{1},\cdots,\epsilon_{j}}$ is a rational number. We return to the skein algebra $\mathcal{A}^{\prime}(\Sigma)$ . We set $[x,y]$ as $\frac{1}{h}(xy-yx)$ . Hence we obtain

	$\displaystyle\mathrm{bch}(x_{0},x_{1})$
	$\displaystyle=\sum_{j}\sum_{\epsilon_{1},\cdots,\epsilon_{j}\in\{0,1\}}b_{\epsilon_{1},\cdots,\epsilon_{j}}[x_{\epsilon_{1}},[x_{\epsilon_{2}},\cdots[x_{\epsilon_{j-1}},x_{\epsilon_{j}}]]]$
	$\displaystyle=h\sum_{j}\sum_{\epsilon_{1},\cdots,\epsilon_{j}\in\{0,1\}}b_{\epsilon_{1},\cdots,\epsilon_{j}}\{\frac{1}{h}x_{\epsilon_{1}},\{\frac{1}{h}x_{\epsilon_{2}},\cdots\{\frac{1}{h}x_{\epsilon_{j-1}},\frac{1}{h}x_{\epsilon_{j}}\}\}\}$
	$\displaystyle=h\mathrm{bch}^{\prime}(\frac{1}{h}x_{0},\frac{1}{h}x_{1})=h\log(\exp(\frac{1}{h}x_{0}\frac{1}{h}x_{1}))$

for any two elements $x_{0},x_{1}\in F^{3}\widehat{\mathcal{A}}^{\prime}(\Sigma)$ . In other words, we have

\displaystyle\exp(\frac{1}{h}\mathrm{bch}(x_{0},x_{1}))=\exp(\frac{1}{h}x_{0})\exp(\frac{1}{h}x_{1}).

“A construction using a standard embedding”

We will introduce how to compute $\widetilde{\zeta}:\mathcal{IC}(\Sigma)\to F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ using a sequence of $\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ . We fix a standard embedding $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ and denote by $\widetilde{\Sigma}_{\mathrm{st}}$ the closure of $\partial(\Sigma_{\mathrm{st}}\times I)\backslash e^{-1}(\partial\Sigma\times I)$ .

The embeddings

\iota_{1}:\Sigma_{\mathrm{st}}\to\widetilde{\Sigma}_{\mathrm{st}},p\mapsto(p,0),\ \iota_{0}:\Sigma_{\mathrm{st}}\to\widetilde{\Sigma}_{\mathrm{st}},p\mapsto(p,1),

induce linear maps

\iota_{1*}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}})\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}_{\mathrm{st}}),\ \iota_{0*}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}})\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}_{\mathrm{st}}).\

The first embedding $\iota_{1}$ is an orientation preserving map, but the second one $\iota_{0}$ an orientation reversing one. So the first linear map $\iota_{1*}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}})\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}_{\mathrm{st}})$ is a Lie algebra homomorphism, but the second one $\iota_{0*}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}})\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}_{\mathrm{st}})$ is not. The embeddings

	$\displaystyle\iota_{1}:\Sigma_{\mathrm{st}}\times I\to\widetilde{\Sigma}_{\mathrm{st}}\times I,(p,t)\mapsto((p,0),t),$
	$\displaystyle\iota_{0}:\Sigma_{\mathrm{st}}\times I\to\widetilde{\Sigma}_{\mathrm{st}}\times I,(p,t)\mapsto((p,1),1-t)$

also induce linear maps

\iota_{1*}:\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}})\to\mathcal{A}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}}),\ \iota_{0*}:\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}})\to\mathcal{A}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}}).\

Then we have

	$\displaystyle\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}\circ\iota_{1}=\iota_{1}\circ\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}},$
	$\displaystyle\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}\circ\iota_{0}=\iota_{0}\circ\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}.$

We can extend the linear maps to the completions.

We set the linear map $\kappa_{*}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}_{\mathrm{st}})\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}})$ as the composite of $\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}_{\mathrm{st}})\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}}\times I)$ induced by the embedding $\widetilde{\Sigma}_{\mathrm{st}}\to\Sigma_{\mathrm{st}}\times I$ and the natural isomorphism $\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}}\times I)\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}}).$ It is not a Lie algebra homomorphism. Let $\kappa:\widetilde{\Sigma}_{\mathrm{st}}\times I\to\Sigma_{\mathrm{st}}\times I$ be the tubular neighborhood satisfying the two conditions.

•

For any $p\in\widetilde{\Sigma}_{\mathrm{st}}$ , $e_{\partial}(p,1)=p.$

•

For any $p\in\widetilde{\Sigma}_{\mathrm{st}}\cap e_{\mathrm{st}}^{-1}(\partial\Sigma\times I)$ ,

	$\displaystyle e_{\mathrm{st}}\circ e_{\partial}((p,1),t)=(e^{\prime}_{\mathrm{st}}(p),\frac{2-\epsilon+\epsilon t}{3})$
	$\displaystyle e_{\mathrm{st}}\circ e_{\partial}((p,0),t)=(e^{\prime}_{\mathrm{st}}(p),\frac{1+\epsilon-\epsilon t}{3}).$

The embedding $\kappa$ induces the linear map $\kappa_{*}:\mathcal{A}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})\to\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}}).$ Then we have

\kappa_{*}\circ\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}=\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}\circ\kappa_{*}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}_{\mathrm{st}})\to\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}}).

We can extend the linear maps to the completion.

Furthermore, since the embeddings $\mathrm{id}_{\Sigma\times I}$ , $\kappa\circ\iota_{0}$ , and $\kappa\circ\iota_{1}$ are isotopic, we have $\mathrm{id}_{\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{\mathrm{st}})}=\kappa_{*}\circ\iota_{0*}=\kappa\circ\iota_{1*}$ and $\mathrm{id}_{\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}})}=\kappa_{*}\circ\iota_{0*}=\kappa\circ\iota_{1*}$ .

Definition 5.2.

For an element $x\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma}_{\mathrm{st}})$ , we define a sequence $\{v_{n}(x)\}_{n\in\mathbb{Z}_{\geq 1}}\subset F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ by

\displaystyle v_{1}(x)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\kappa(x),\ v_{n+1}(x)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}v_{n}(x)+\kappa_{*}(\mathrm{bch}(-\iota_{1*}(v_{n}(x)),x))

and an element $v(x)\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ as

v(x)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\lim_{n\to\infty}v_{n}(x).

By the following proposition, $v(x)$ is well-defined as an element of the completed Goldman Lie algebra. We can define $v(x)$ using a computation of the skein algebra $\mathcal{A}^{\prime}$ . Furthermore, $v(x)$ is a unique solution to $\kappa_{*}(\mathrm{bch}(-\iota_{1*}(\cdot),x))=0$ .

Proposition 5.3.

Let $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ be a standard embedding and $\widetilde{\Sigma}_{\mathrm{st}}$ be the closure of $\partial(\Sigma_{\mathrm{st}}\times I)\backslash e^{-1}(\partial\Sigma\times I).$ Then we have the following.

(1)

$v_{n+1}(x)-v_{n}(x)=\kappa_{*}(\mathrm{bch}(-\iota_{1*}(v_{n}(x)),x))\in F^{2+n}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ .

(2)

For $x\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma}_{\mathrm{st}})$ and $y\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ satisfying $\kappa_{*}(\mathrm{bch}(-\iota_{1*}(y),x))=0$ , we have

\exp(\frac{\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y)}{h})=\kappa_{*}(\exp(\frac{\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x)}{h}))\in\widehat{(\sum_{*\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{*}}F^{3*})}\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}}).

Using them, we obtain two statements.

(a)

$v(x)$ is well-defined.
(b)

$v(x)$ is a unique solution to $\kappa_{*}(\mathrm{bch}(-\iota_{1*}(\cdot),x))=0$ .

Proof.

First, we prove the statement $(1)$ by induction. If $n=1$ ,

	$\displaystyle v_{2}(x)-v_{1}(x)=\kappa_{}(\mathrm{bch}(-\iota_{1}(v_{1}(x)),x))$
	$\displaystyle=\kappa_{}(-\iota_{1}(v_{1}(x))+x)F^{4}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$
	$\displaystyle=-\kappa_{}\circ\iota_{1}\kappa_{}(x)+\kappa_{}(x)=0$

holds. We assume

v_{k+1}(x)-v_{k}(x)=\kappa_{*}(\mathrm{bch}(-\iota_{1*}(v_{k}(x)),x))\in F^{2+k}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}}).

For $z,w,w^{\prime}\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ satisfying $w-w^{\prime}\in F^{m}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ , we have

-z-w+\mathrm{bch}(z,w)=-z-w^{\prime}+\mathrm{bch}(z,w^{\prime})\mod F^{m+1}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})

by $[F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}}),F^{m}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})]\subset F^{m+1}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ . Using it, we have

	$\displaystyle v_{k+2}(x)-v_{k+1}(x)=\kappa_{}(\mathrm{bch}(-\iota_{1}(v_{k+1}(x)),x))$
	$\displaystyle=-\kappa_{}\circ\iota_{1}(v_{k+1}(x))+\kappa_{}(x)+(\kappa_{}\circ\iota_{1}(v_{k+1}(x))-\kappa_{}(x)+\kappa_{}(\mathrm{bch}(-\iota_{1}(v_{k+1}(x)),x)))$
	$\displaystyle=-\kappa_{}\circ\iota_{1}(v_{k+1}(x))+\kappa_{*}(x)$
	$\displaystyle+(\kappa_{}\circ\iota_{1}(v_{k}(x))-\kappa_{}(x)+\kappa_{}(\mathrm{bch}(-\iota_{1*}(v_{k}(x)),x)))\mod F^{3+k}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$
	$\displaystyle=-v_{k+1}(x)+\kappa_{}(x)+(v_{k}(x)-\kappa_{}(x)+\kappa(\mathrm{bch}(-\iota_{1*}(v_{k}(x)),x)))$
	$\displaystyle=-v_{k+1}(x)+v_{k}(x)+\kappa(\mathrm{bch}(-\iota_{1*}(v_{k}(x)),x))=0$

and prove (1).

By (1), we have

K\geq k\Rightarrow v_{K}(x)-v_{k}(x)\in F^{k+2}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma),

which means that $\lim_{n\to\infty}v_{n}(x)$ is well-defined. In other words, the statement (a) holds.

We will prove the statement (2). By definition of $\mathrm{bch}$ , we have

	$\displaystyle\kappa_{*}(\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x)))$
	$\displaystyle=\kappa_{}(\exp(\frac{1}{h}\mathrm{bch}(\iota_{1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y)),\mathrm{bch}(-\iota_{1*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y)),\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x)))))$
	$\displaystyle=\kappa_{}(\exp(\frac{1}{h}\iota_{1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y)))\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\mathrm{bch}(-\iota_{1*}(y),x)))).$

Since

\kappa_{*}(x^{\prime})=0\Rightarrow\kappa_{*}(x^{\prime\prime}x^{\prime})=0

for $x^{\prime},x^{\prime\prime}\in\mathcal{A}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})$ , we have

\displaystyle\kappa_{*}(\exp(\frac{1}{h}\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(x)))=\kappa_{*}(\exp(\frac{1}{h}\iota_{1*}(\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(y)))).

Since $\iota_{1*}$ is an algebra homomorphism, we have

\displaystyle\kappa_{*}(\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x)))=\kappa_{*}\circ\iota_{1*}(\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y)))=\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y))

as desired.

Finally, we prove the statement (b). By (1), we have $\kappa_{*}(\mathrm{bch}(-\iota_{1*}(v(x)),x))\in\cap_{i=4}^{\infty}F^{i}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)=\{0\}$ , which means that $v(x)$ is a solution to $\kappa_{*}(\mathrm{bch}(-\iota_{1*}(\cdot),x))=0$ . By (2), we have

\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}(v(x))=\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(h\log(\kappa_{*}(\exp(\frac{\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x)}{h}))))\in\widehat{(\sum_{*\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{*}}F^{3*})}\mathcal{U}_{h}(\Sigma).

Since $\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}$ is an injection, $v(x)$ is a unique solution.

The above statements prove the proposition.

∎

We state Theorem 5.4 using the definition.

Theorem 5.4.

Let $e^{\prime}_{\mathrm{st}};\Sigma_{\mathrm{st}}\hookrightarrow\Sigma$ be an embedding such that the map $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\hookrightarrow\Sigma\times I,(p,t)\mapsto(e^{\prime}(p),\frac{1+t}{3})$ is a standard embedding and $\widetilde{\Sigma}_{\mathrm{st}}$ the closure of $\partial(\Sigma_{\mathrm{st}}\times I)\backslash e^{-1}(\partial\Sigma\times I).$ For an element $\xi\in\mathcal{I}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})$ , we have

\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e_{\mathrm{st}},\xi))=e^{\prime}_{\mathrm{st}*}(v(\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert(\xi)))\in F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma).

In other words, for $\star_{1},\star_{2}\in\partial\Sigma$ , we have

(\Sigma\times I)(e_{\mathrm{st}},\xi)_{*}=\exp(\sigma(e^{\prime}_{\mathrm{st}*}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\xi)))):\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2}).

We will prove the theorem in §6.

“A construction using a boundary link”

For a boundary link $L$ in $\Sigma\times I$ and a label $\lambda$ of a Seifert surface of $L$ , we set an element $L_{\mathcal{A}^{\prime}}(S,\lambda)\in\widehat{\mathcal{A}}^{\prime}(S)$ as

L_{\mathcal{A}^{\prime}}(S,\lambda)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}-\sum_{[\partial]\in\pi_{0}(\partial S)}\lambda([\partial])L_{\mathcal{A}^{\prime}}(c_{\partial}),

where

L_{\mathcal{A}^{\prime}}(c)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c))=\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}\log(\gamma_{c})^{2}\rvert)

for any simple closed curve $c$ . Here we use two notations as follows.

•

For a simple close curve $c$ , $\gamma_{c}$ is an element, where the conjugacy class $\lvert\gamma_{c}\rvert$ equals $c$ .
•

The simple closed curve $c_{\partial}$ is parallel to the boundary component $\partial$ .

The Seifert surface $S$ defines the embedding $e_{S}:S\times I\to\Sigma\times I$ inducing the module homomorphism $e_{S*}:\widehat{\mathcal{A}}^{\prime}(S)\to\widehat{\mathcal{A}}^{\prime}(\Sigma)$ .

Theorem 5.5.

For a boundary link $L$ in $\Sigma\times I$ and a label $\lambda$ of a Seifert surface $S$ of $L$ , we have

	$\displaystyle\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(L(\lambda)))))=e_{S*}(\exp(\frac{1}{h}L_{\mathcal{A}^{\prime}}(S,\lambda)))$
	$\displaystyle=\sum_{i=0}^{\infty}\frac{1}{i!h^{i}}e_{S}(L_{\mathcal{A}^{\prime}}(S,\lambda)^{i})\in\widehat{(\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3})}\mathcal{A}^{\prime}(\Sigma).$

In other words, we have

\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(h\log(e_{S*}(\exp(\frac{1}{h}L_{\mathcal{A}^{\prime}}(S,\lambda)))))=\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(L(\lambda)))\in\widehat{\mathcal{U}_{h}}(\Sigma).

We will prove the theorem in §6.

6. Proof of the main result

In this section, we will prove the theorems in §5. The third statement in Proposition 6.1 plays an important role.

Proposition 6.1.

Let $\Sigma$ be a compact connected oriented surface and $M^{(1)}$ a submanifold of $\partial\Sigma$ . We choose base points $\star_{1},\star_{2}\in M^{(1)}$ and denote by $\widetilde{\Sigma}$ the closure of $\partial(\Sigma\times I)\backslash(M^{(1)}\times I)$ . We consider linear maps

	$\displaystyle\kappa^{\prime}_{}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma})\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma),\ \kappa^{\prime}_{}:\mathbb{Q}\pi_{1}(\widetilde{\Sigma},(\star_{1},i),(\star_{2},i))\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\mathrm{\ for\ }i=0,1,$
	$\displaystyle\iota^{\prime}_{0}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}),\ \iota^{\prime}_{0}:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\widetilde{\Sigma},(\star_{1},0),(\star_{2},0)),$
	$\displaystyle\iota^{\prime}_{1}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma}),\ \iota^{\prime}_{1}:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\widetilde{\Sigma},(\star_{1},1),(\star_{2},1)).$

The homomorphisms $\kappa^{\prime}_{*}:\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma})\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)$ and $\kappa^{\prime}_{*}:\mathbb{Q}\pi_{1}(\widetilde{\Sigma},(\star_{1},i),(\star_{2},i))\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})$ are composites of the ones induced by the embedding $\widetilde{\Sigma}\hookrightarrow\Sigma\times I$ and the natural maps

	$\displaystyle\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma\times I)\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma),$
	$\displaystyle\mathbb{Q}\pi_{1}(\Sigma\times I,(\star_{1},i),(\star_{2},i))\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}).$

The embeddings

\displaystyle\iota_{1}:\Sigma\to\widetilde{\Sigma},p\mapsto(p,1)\iota_{0}:\Sigma\to\widetilde{\Sigma},p\mapsto(p,0)

induce $\iota_{1*},\iota{0*}$ . We can extend them to completions. Then we have the three statements.

(1)

For $i=0,1$ , $x\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma})$ , and $y\in\mathbb{Q}\pi_{1}(\widetilde{\Sigma},(\star_{1},i),(\star_{2},i))$ , we have

\kappa^{\prime}_{*}(\sigma(x)(y))=\kappa^{\prime}_{*}(\sigma(x)(\iota^{\prime}_{1*}\circ\kappa^{\prime}_{*}(y)))+\kappa^{\prime}(\sigma(\iota^{\prime}_{0*}\circ\kappa^{\prime}_{*}(x))(y)).

(2)

For $x\in\mathbb{Q}\lvert{\pi}_{1}\rvert(\widetilde{\Sigma})$ and $y\in\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})$ satisfying $\kappa^{\prime}_{*}(x)=0$ , we have

\kappa^{\prime}_{*}((\sigma(x))^{n}(\iota^{\prime}_{0*}(y)))=\kappa^{\prime}_{*}((\sigma(x))^{n}(\iota^{\prime}_{1*}(y)))

for any $n\in\mathbb{Z}_{\geq 0}$ .

(3)

We choose a diffeomorphism $\chi\in\mathrm{Diff}(\widetilde{\Sigma},\partial\widetilde{\Sigma})$ representing an element of $\mathcal{I}^{\prime}(\widetilde{\Sigma})$ . Let $\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)$ and $v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))$ be elements of $F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma})$ and $F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ , respectively, satisfying

	$\displaystyle\chi_{*}=\exp(\sigma(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))):\widehat{\mathbb{Q}\pi_{1}}(\widetilde{\Sigma},\star^{\prime}_{1},\star^{\prime}_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\widetilde{\Sigma},\star^{\prime}_{1},\star^{\prime}_{2}),$
	$\displaystyle\kappa^{\prime}_{}(\mathrm{bch}(-\iota^{\prime}_{1}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))),\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)))=0.$

Then we have

\kappa^{\prime}_{*}\circ\chi_{*}^{-1}\circ\iota^{\prime}_{0*}(x)=\kappa^{\prime}_{*}\circ\chi_{*}^{-1}\circ\iota^{\prime}_{1*}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x))

for any $x\in\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})$ .

Proof.

(1)We can prove the statement directly. In this paper, we use the skein algebra $\mathcal{A}^{\prime}$ to prove it. We consider three embeddings $\kappa^{\prime}:\widetilde{\Sigma}\times I\to\Sigma\times I$ , $\iota^{\prime}_{1}:\Sigma\times I\to\widetilde{\Sigma}\times I$ , and $\iota^{\prime}_{0}:\Sigma\times I\to\widetilde{\Sigma}\times I$ .

•
The first one satisfies
- –
  
  For $p\in\widetilde{\Sigma}$ , $\kappa^{\prime}(p,1)=p$ .
- –
  
  For $p\in M^{(1)}$ , $\kappa^{\prime}((p,1),t)=(p,1-\epsilon+\epsilon t),\ \kappa^{\prime}((p,0),t)=(p,\epsilon-\epsilon t).$
for enough small $\epsilon>0$ .
•

We define the second one and the third one as

$\displaystyle\iota^{\prime}_{1}(p,t)=((p,1),t),\ \iota^{\prime}_{0}(p,t)=((p,0),1-t).$

There exist isomorphisms $\diamondsuit_{(t_{0},t_{1})}^{(1,0)}:\mathcal{A}^{\prime}(\Sigma^{\prime}\times I,(\star,t_{0}),(\star^{\prime},t_{1}))\to\mathcal{A}^{\prime}(\Sigma^{\prime},\star,\star^{\prime})$ for any $t_{0},t_{1}\in I$ . These embeddings and these isomorphisms induce the linear maps

	$\displaystyle\kappa^{\prime}_{*}:\mathcal{A}^{\prime}(\widetilde{\Sigma})\to\mathcal{A}^{\prime}(\Sigma)$
	$\displaystyle\kappa^{\prime}_{*}:\mathcal{A}^{\prime}(\widetilde{\Sigma},(\star_{1},i),(\star_{2},i))\to\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle\iota^{\prime}_{i*}:\mathcal{A}^{\prime}(\Sigma)\to\mathcal{A}^{\prime}(\widetilde{\Sigma}),$
	$\displaystyle\iota^{\prime}_{i*}:\mathcal{A}^{\prime}(\Sigma,\star_{1},\star_{2})\to\mathcal{A}^{\prime}(\Sigma,(\star_{1},i),(\star_{2},i)).$

We prove the statement using

	$\displaystyle\kappa^{\prime}_{}(xy)=\kappa^{\prime}_{}(x\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y))=\kappa^{\prime}_{}(x\iota^{\prime}_{0}\circ\kappa^{\prime}_{*}(y)),$
	$\displaystyle\kappa^{\prime}_{}(yx)=\kappa^{\prime}_{}(y\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(x))=\kappa^{\prime}_{}(y\iota^{\prime}_{0}\circ\kappa^{\prime}_{*}(x)),$
	$\displaystyle\kappa^{\prime}_{}(x)\kappa^{\prime}_{}(y)=\kappa^{\prime}_{}(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(x)y)=\kappa^{\prime}_{}(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(y)x),$
	$\displaystyle\kappa^{\prime}_{}(y)\kappa^{\prime}_{}(x)=\kappa^{\prime}_{}(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)x)=\kappa^{\prime}_{}(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x)y)$

for any $x\in\mathcal{A}^{\prime}(\widetilde{\Sigma})$ and $y\in\mathcal{A}^{\prime}(\widetilde{\Sigma},(\star_{1},i),(\star_{2},i))$ . We set

	$\displaystyle e_{\mathrm{over}}:\Sigma^{\prime}\times I\to\Sigma^{\prime}\times I,(p,t)\mapsto(p,\frac{2+t}{4}),$
	$\displaystyle e_{\mathrm{under}}:\Sigma^{\prime}\times I\to\Sigma^{\prime}\times I,(p,t)\mapsto(p,\frac{t}{4})$

for any surface $\Sigma^{\prime}$ . Since the embeddings $\kappa^{\prime}\circ(e_{\mathrm{over}}\sqcup e_{\mathrm{under}})$ , $\kappa^{\prime}\circ(e_{\mathrm{over}}\sqcup(e_{\mathrm{under}}\circ\iota^{\prime}_{1}\circ\kappa^{\prime}))$ , and $\kappa^{\prime}\circ(e_{\mathrm{over}}\sqcup(e_{\mathrm{under}}\circ\iota^{\prime}_{0}\circ\kappa^{\prime}))$ are isotopic preserving $\{\star_{1},\star_{2}\}\times I$ , we have the first one and the second one. Furthermore, since the embeddings $e_{\mathrm{over}}\circ\kappa^{\prime}\sqcup e_{\mathrm{under}}\circ\kappa^{\prime}$ , $\kappa^{\prime}\circ((e_{\mathrm{over}}\circ\iota^{\prime}_{1}\circ\kappa^{\prime})\sqcup e_{\mathrm{under}})$ , and $\kappa^{\prime}\circ(e_{\mathrm{under}}\sqcup(e_{\mathrm{over}}\circ\iota^{\prime}_{0}\circ\kappa^{\prime}))$ are isotopic preserving $\{\star_{1},\star_{2}\}\times I$ , we also have the third one and the fourth one.

Using the above formulas, we obtain

	$\displaystyle h\kappa^{\prime}_{*}(\sigma(x)(y))$
	$\displaystyle=\kappa^{\prime}_{*}(xy-yx)$
	$\displaystyle=\kappa^{\prime}_{}(x\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)-y\iota^{\prime}_{0}\circ\kappa^{\prime}_{*}(x))$
	$\displaystyle=(\kappa^{\prime}_{}(x\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y))-\kappa^{\prime}_{}(y)\kappa^{\prime}_{}(x))+(\kappa^{\prime}_{}(y)\kappa^{\prime}_{}(x)-\kappa^{\prime}_{}(y\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x)))$
	$\displaystyle=\kappa^{\prime}_{}(x\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)-\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)x)+\kappa^{\prime}_{}(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x)y-y\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))$
	$\displaystyle=h(\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)))+\kappa^{\prime}_{}(\sigma(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))(y)))$

for any $x\in\mathcal{A}^{\prime}(\widetilde{\Sigma})$ and $y\in\mathcal{A}^{\prime}(\widetilde{\Sigma},(\star_{1},i),(\star_{2},i))$ . By Theorem 3.3, $\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}},\star_{1},\star_{2})$ is a free $\mathbb{Q}[h]$ -module. So we have

\kappa^{\prime}_{*}(\sigma(x)(y))=\kappa^{\prime}_{*}(\sigma(x)(\iota^{\prime}_{1*}\circ\kappa^{\prime}_{*}(y)))+\kappa(\sigma(\iota^{\prime}_{0*}\circ\kappa^{\prime}_{*}(x))(y)).

For $x^{\prime}\in\mathbb{Q}\pi_{1}(\widetilde{\Sigma}_{\mathrm{st}})$ and $y^{\prime}\in\mathbb{Q}\pi_{1}(\widetilde{\Sigma}_{\mathrm{st}},(\star_{1},i),(\star_{2},i))$ , we set $x$ and $y$ as

\displaystyle x\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x^{\prime}),\ y\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathbb{Q}\pi_{1}}^{\mathcal{A}^{\prime}}(y^{\prime}).

We have

	$\displaystyle\kappa^{\prime}_{*}(\sigma(x^{\prime})(y^{\prime}))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\pi_{1}(i,i)}(\kappa^{\prime}_{*}(\sigma(x)(y)))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\pi_{1}(i,i)}(\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)))+\kappa^{\prime}_{}(\sigma(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x^{\prime})(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y^{\prime})))+\kappa^{\prime}_{}(\sigma(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x^{\prime}))(y^{\prime}))$

as desired.

(2)We will prove the statement by induction on $n$ . If $n=0$ , $\kappa^{\prime}_{*}(\iota^{\prime}_{0*}(y))=\kappa^{\prime}_{*}(\iota^{\prime}_{1*}(y))$ holds. We assume

\kappa^{\prime}_{*}((\sigma(x))^{n-1}(\iota^{\prime}_{0*}(y)))=\kappa^{\prime}_{*}((\sigma(x))^{n-1}(\iota^{\prime}_{1*}(y))).

We have

	$\displaystyle\kappa^{\prime}_{}((\sigma(x))^{n}(\iota^{\prime}_{0}(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}((\sigma(x))^{n-1}(\iota^{\prime}_{0}(y)))))+\kappa^{\prime}_{}(\sigma(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))(\sigma(x))^{n-1}(\iota^{\prime}_{0}(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}((\sigma(x))^{n-1}(\iota^{\prime}_{0}(y))))),$
	$\displaystyle\kappa^{\prime}_{}((\sigma(x))^{n}(\iota^{\prime}_{1}(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}((\sigma(x))^{n-1}(\iota^{\prime}_{1}(y)))))+\kappa^{\prime}_{}(\sigma(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))(\sigma(x))^{n-1}(\iota^{\prime}_{1}(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}((\sigma(x))^{n-1}(\iota^{\prime}_{1}(y))))).$

By this inductive assumption, we obtain

\kappa^{\prime}_{*}((\sigma(x))^{n}(\iota^{\prime}_{0*}(y)))=\kappa^{\prime}_{*}((\sigma(x))^{n}(\iota^{\prime}_{1*}(y)))

as desired.

(3)We write $\zeta$ , $v$ , and $v^{\prime}$ for $\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi),$ $v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))$ , and $\mathrm{bch}(-\iota^{\prime}_{1*}(v),\zeta)$ respectively. Using

\chi_{*}=\exp(\sigma(\zeta)):\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma},\star^{\prime}_{1},\star^{\prime}_{2}),

we have

	$\displaystyle\kappa^{\prime}_{}\circ\chi_{}^{-1}\circ\iota^{\prime}_{0*}(x)$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-\zeta))(\iota^{\prime}_{0}(x)))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-\mathrm{bch}(\iota^{\prime}_{1}(v),v^{\prime}))(\iota^{\prime}_{0*}(x)))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-v^{\prime}))\circ\exp(\sigma(-\iota^{\prime}_{1}(v)))(\iota^{\prime}_{0*}(x)))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-v^{\prime}))(\iota^{\prime}_{0}(x))),$

	$\displaystyle\kappa^{\prime}_{}\circ\chi_{}^{-1}\circ\iota^{\prime}_{1*}(\exp(\sigma(v)(x))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-\zeta))(\iota^{\prime}_{1}(\exp(\sigma(v)(x))))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-\mathrm{bch}(\iota^{\prime}_{1}(v),v^{\prime}))(\iota^{\prime}_{1*}(\exp(\sigma(v)(x))))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-v^{\prime}))\circ\exp(\sigma(-\iota^{\prime}_{1}(v)))(\iota^{\prime}_{1*}(\exp(\sigma(v)(x))))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-v^{\prime}))\circ\exp(\sigma(-\iota^{\prime}_{1}(v)))\circ\exp(\sigma(\iota^{\prime}_{1}(v)))(\iota^{\prime}_{1}(x))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-v^{\prime}))(\iota^{\prime}_{1}(x)).$

By (2), we obtain

\kappa^{\prime}_{*}\circ\chi_{*}^{-1}\circ\iota^{\prime}_{0*}(x)=\kappa^{\prime}_{*}\circ\chi_{*}^{-1}\circ\iota^{\prime}_{1*}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x)),

as desired.

The above statements prove the proposition.

∎

Let $e^{\prime}_{\mathrm{st}};\Sigma_{\mathrm{st}}\hookrightarrow\Sigma$ be an embedding such that $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\hookrightarrow\Sigma\times I,(p,t)\mapsto(e^{\prime}_{\mathrm{st}},\frac{1+t}{3})$ is a standard embedding and $\widetilde{\Sigma}_{\mathrm{st}}$ be the closure of $\partial(\Sigma_{\mathrm{st}}\times I)\backslash(e^{\prime-1}(\partial\Sigma)\times I)$ . We choose a diffeomorphism $\chi$ representing an element of $\mathcal{I}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})$ . We set $(\Sigma\times I)(e_{\mathrm{st}},\chi)$ as the 3-manifold that is the quotient of the set $\overline{(\Sigma\times I)\backslash e({\Sigma}_{\mathrm{st}}\times I)}\sqcup{\Sigma}_{\mathrm{st}}\times I$ by the relation $e_{\mathrm{st}}\circ\chi(p)\sim p$ for any $p\in\widetilde{\Sigma}_{\mathrm{st}}$ . We consider the embedding maps.

•

We denote by $\kappa$ the natural embedding $\widetilde{\Sigma}_{\mathrm{st}}\to\Sigma_{\mathrm{st}}\times I$ .
•

The identity map $\Sigma_{\mathrm{st}}\times I\to\Sigma_{\mathrm{st}}\times I$ induces an embedding map

$e_{\mathrm{st}}^{\flat}:\Sigma_{\mathrm{st}}\times I\to(\Sigma\times I)(e_{\mathrm{st}},\chi).$

•

We recall that we set the embeddings $\iota_{1},\iota_{0}:\Sigma_{\mathrm{st}}\to\widetilde{\Sigma}_{\mathrm{st}}$ and $\alpha_{1},\alpha_{0}:\Sigma\to(\Sigma\times I)(e_{\mathrm{st}},\chi)$ as

	$\displaystyle\iota_{1}:\Sigma_{\mathrm{st}}\to\widetilde{\Sigma}_{\mathrm{st}},p\mapsto(p,1)$
	$\displaystyle\iota_{0}:\Sigma_{\mathrm{st}}\to\widetilde{\Sigma}_{\mathrm{st}},p\mapsto(p,0),$
	$\displaystyle\alpha_{1}:\Sigma\to(\Sigma\times I)(e_{\mathrm{st}},\chi),p\mapsto(p,1),$
	$\displaystyle\alpha_{0}:\Sigma\to(\Sigma\times I)(e_{\mathrm{st}},\chi),p\mapsto(p,0).$

We recall that we set the diffeomorphism $\alpha^{e_{\mathrm{st}}}:\partial(\Sigma\times I)\to\partial((\Sigma\times I)(e_{\mathrm{st}},\chi))$ as

\alpha^{e_{\mathrm{st}}}(p,t)=\begin{cases}(p,t)\mathrm{\ if\ }p\in\Sigma,t\in\{0,1\}\\ (p,t)\mathrm{\ if\ }p\in\partial\Sigma,t\in[0,\frac{1}{3}]\cup[\frac{2}{3},1]\\ e_{\mathrm{st}}^{-1}(p,3t-1)\mathrm{\ if\ }p\in\partial\Sigma,t\in[\frac{1}{3},\frac{2}{3}].\end{cases}

For any $\star_{1},\star_{2}\in\partial\Sigma$ and $t_{1},t_{2}\in I$ , we define

	$\displaystyle\diamondsuit_{t_{1}}^{t_{2}}:\pi_{1}((\Sigma\times I)(e_{\mathrm{st}},\chi),\alpha^{e_{\mathrm{st}}-1}(\star_{1},t_{1}),\alpha^{e_{\mathrm{st}}-1}(\star_{2},t_{1}))$
	$\displaystyle\to\pi_{1}((\Sigma\times I)(e_{\mathrm{st}},\chi),\alpha^{e_{\mathrm{st}}-1}(\star_{1},t_{2}),\alpha^{e_{\mathrm{st}}-1}(\star_{2},t_{2}))$

by $(\cdot)\mapsto\gamma_{\star_{1},t_{2},t_{1}}(\cdot)\gamma_{\star_{2},t_{1},t_{2}}$ . Here, for any $\star\in\partial\Sigma$ and $t,t^{\prime}\in I$ , the continuous map $I\to(\Sigma\times I)(e,\chi),s\mapsto\alpha^{e_{\mathrm{st}}-1}(\star,ts+t^{\prime}(1-s))$ represents the path $\gamma_{\star,t,t^{\prime}}\in\pi_{1}((\Sigma\times I)(e,\chi),\alpha^{e_{\mathrm{st}}-1}(\star,t),\alpha^{e_{\mathrm{st}}-1}(\star,t^{\prime}))$ . To prove Theorem 5.4, we need the following lemma.

Lemma 6.2.

(1)

For any $\star_{1},\star_{2}\in\partial\Sigma$ , the homomorphism

e^{\prime}_{\mathrm{st}*}:\pi_{1}(\Sigma_{\mathrm{st}},{e^{\prime}_{\mathrm{st}}}^{-1}(\star_{1}),{e^{\prime}_{\mathrm{st}}}^{-1}(\star_{2}))\to\pi_{1}(\Sigma,\star_{1},\star_{2})

induced by the embedding $e^{\prime}_{\mathrm{st}}:\Sigma_{\mathrm{st}}\to\Sigma$ is surjective.

(2)

We fix $\star_{1},\star_{2}\in\partial\Sigma_{\mathrm{st}}$ . The homomorphisms

	$\displaystyle(\alpha_{1}\circ e_{\mathrm{st}})_{*}:\pi_{1}(\Sigma_{\mathrm{st}},\star_{1},\star_{2})\to\pi_{1}((\Sigma\times I)(e_{\mathrm{st}},\chi),(e^{\prime}_{\mathrm{st}}(\star_{1}),1),(e^{\prime}_{\mathrm{st}}(\star_{2}),1)),$
	$\displaystyle(\alpha_{0}\circ e_{\mathrm{st}})_{*}:\pi_{1}(\Sigma_{\mathrm{st}},\star_{1},\star_{2})\to\pi_{1}((\Sigma\times I)(e_{\mathrm{st}},\chi),(e^{\prime}_{\mathrm{st}}(\star_{1}),0),(e^{\prime}_{\mathrm{st}}(\star_{2}),0)),$
	$\displaystyle(e_{\mathrm{st}}^{\flat}\circ\widetilde{\kappa}\circ\chi^{-1}\circ\iota_{1})_{*}:\pi_{1}(\Sigma_{\mathrm{st}},\star_{1},\star_{2})\to\pi_{1}((\Sigma\times I)(e_{\mathrm{st}},\chi),(e^{\prime}_{\mathrm{st}}(\star_{1}),\frac{2}{3}),(e^{\prime}_{\mathrm{st}}(\star_{2}),\frac{2}{3})),$
	$\displaystyle(e_{\mathrm{st}}^{\flat}\circ\widetilde{\kappa}\circ\chi^{-1}\circ\iota_{0})_{*}:\pi_{1}(\Sigma_{\mathrm{st}},\star_{1},\star_{2})\to\pi_{1}((\Sigma\times I)(e_{\mathrm{st}},\chi),(e^{\prime}_{\mathrm{st}}(\star_{1}),\frac{1}{3}),(e^{\prime}_{\mathrm{st}}(\star_{2}),\frac{1}{3})),$

induced by the embeddings $\alpha_{1}\circ e_{\mathrm{st}}$ , $\alpha_{0}\circ e_{\mathrm{st}}$ , $e_{\mathrm{st}}^{\flat}\circ\widetilde{\kappa}\circ\chi^{-1}\circ\iota_{1}$ , and $e_{\mathrm{st}}^{\flat}\circ\widetilde{\kappa}\circ\chi^{-1}\circ\iota_{0}$ satisfy

	$\displaystyle(\alpha_{1}\circ e_{\mathrm{st}})_{}=\diamondsuit_{\frac{2}{3}}^{1}\circ(e_{\mathrm{st}}^{\flat}\circ\widetilde{\kappa}\circ\chi^{-1}\circ\iota_{1})_{},$
	$\displaystyle(\alpha_{0}\circ e_{\mathrm{st}})_{}=\diamondsuit_{\frac{1}{3}}^{0}\circ(e_{\mathrm{st}}^{\flat}\circ\widetilde{\kappa}\circ\chi^{-1}\circ\iota_{0})_{}.$

(3)

Let $\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)$ and $v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))$ be elements in $F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\widetilde{\Sigma}_{\mathrm{st}})$ and $F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$ satisfying the two properties, respectively.

•

For any $\star^{\prime}_{1},\star^{\prime}_{2}\in\partial\widetilde{\Sigma}_{\mathrm{st}}$ , we have

\chi_{*}=\exp(\sigma(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))):\widehat{\mathbb{Q}\pi_{1}}(\widetilde{\Sigma}_{\mathrm{st}},\star^{\prime}_{1},\star^{\prime}_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\widetilde{\Sigma}_{\mathrm{st}},\star^{\prime}_{1},\star^{\prime}_{2}).

•

We have $\widetilde{\kappa}_{*}(\mathrm{bch}(-\iota_{1*}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))),\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)))=0$ .

Then we have

\widetilde{\kappa}_{*}\circ\chi_{*}^{-1}\circ\iota_{0*}(x)=\diamondsuit_{1}^{0}\circ\widetilde{\kappa}_{*}\circ\chi_{*}^{-1}\circ\iota_{1*}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x))

for any $x\in\widehat{\mathbb{Q}\pi_{1}}(\Sigma_{\mathrm{st}},\star_{1},\star_{2})$ .

(4)

Under the assumption in (3), we have

\alpha_{0*}(x)=\diamondsuit_{1}^{0}\circ\alpha_{1*}(\exp(\sigma(e^{\prime}_{\mathrm{st}*}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)))))(x))

for any $x\in\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})$ .

Proof.

(1)Since $e_{\mathrm{st}}$ is a standard embedding, the closure of $\Sigma\backslash e^{\prime}_{\mathrm{st}}(\Sigma_{\mathrm{st}})$ is a connected sum of closed disks. So there exists an embedding $e^{\prime\prime}:\Sigma\to\Sigma_{\mathrm{st}}$ satisfying the two conditions.

•

We have $e^{\prime}_{\mathrm{st}}\circ e^{\prime\prime}(\star_{1})=\star_{1},e^{\prime}_{\mathrm{st}}\circ e^{\prime\prime}(\star_{2})=\star_{2}$ .
•

The composite $e^{\prime}_{\mathrm{st}}\circ e^{\prime\prime}$ is isotopic to the identity map preserving $\{\star_{1},\star_{2}\}$ .

Then we have $e^{\prime}_{\mathrm{st}*}\circ e^{\prime\prime}_{*}=\mathrm{id}_{\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})}$ . So the homomorphism

e^{\prime}_{\mathrm{st}*}:\pi_{1}(\Sigma_{\mathrm{st}},{e^{\prime}_{\mathrm{st}}}^{-1}(\star_{1}),{e^{\prime}_{\mathrm{st}}}^{-1}(\star_{2}))\to\pi_{1}(\Sigma,\star_{1},\star_{2})

is surjective as desired.

(2)The embeddings $\alpha_{1}\circ e_{\mathrm{st}}$ and $\alpha_{0}\circ e_{\mathrm{st}}$ are isotopic to $e_{\mathrm{st}}^{\flat}\circ\widetilde{\kappa}\circ\chi^{-1}\circ\iota_{1}$ and $e_{\mathrm{st}}^{\flat}\circ\widetilde{\kappa}\circ\chi^{-1}\circ\iota_{0}$ preserving $\star_{1}\times I\cup\star_{2}\times I$ , respectively. It proves (2).

(3)For $i=0,1$ , we denote by $\mathrm{Proj}_{i}$ the natural map $\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma\times I,(\star_{1},i),(\star_{2},i))\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma,\star_{1},\star_{2})$ and have $\mathrm{Proj}_{0}\circ\diamondsuit_{1}^{0}=\mathrm{Proj}_{1}$ . By Proposition 6.1, we obtain

\mathrm{Proj}_{0}\circ\widetilde{\kappa}_{*}\circ\chi_{*}^{-1}\circ\iota_{0*}(x)=\mathrm{Proj}_{1}\circ\widetilde{\kappa}_{*}\circ\chi_{*}^{-1}\circ\iota_{1*}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x)).

So we have

\mathrm{Proj}_{0}\circ\widetilde{\kappa}_{*}\circ\chi_{*}^{-1}\circ\iota_{0*}(x)=\mathrm{Proj}_{0}\circ\diamondsuit_{1}^{0}\circ\widetilde{\kappa}_{*}\circ\chi_{*}^{-1}\circ\iota_{1*}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x)).

Since $\mathrm{Proj}_{0}$ is an isomorphism, we get

\widetilde{\kappa}_{*}\circ\chi_{*}^{-1}\circ\iota_{0*}(x)=\diamondsuit_{1}^{0}\circ\widetilde{\kappa}_{*}\circ\chi_{*}^{-1}\circ\iota_{1*}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x)),

as desired.

(4)By (1), there exists an element $x^{\prime}\in\mathbb{Q}\pi_{1}(\Sigma,{e^{\prime}_{\mathrm{st}}}^{-1}(\star_{1}),{e^{\prime}_{\mathrm{st}}}^{-1}(\star_{2}))$ satisfying $e^{\prime}_{\mathrm{st}*}(x^{\prime})=x$ . Using (2), we obtain

	$\displaystyle\diamondsuit_{1}^{0}\circ\alpha_{1}(\exp(\sigma(e^{\prime}_{\mathrm{st}}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)))))(x))$
	$\displaystyle=\diamondsuit_{1}^{0}\circ\alpha_{1}(\exp(\sigma(e^{\prime}_{\mathrm{st}}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)))))(e^{\prime}_{\mathrm{st}*}(x^{\prime})))$
	$\displaystyle=\diamondsuit_{1}^{0}\circ\alpha_{1}\circ e^{\prime}_{\mathrm{st}}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x^{\prime}))$
	$\displaystyle=\diamondsuit_{1}^{0}\circ\diamondsuit_{\frac{2}{3}}^{1}\circ e^{\flat}_{\mathrm{st}}\circ\widetilde{\kappa}_{}\circ\chi^{-1}_{}\circ\iota_{1}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x^{\prime}))$
	$\displaystyle=\diamondsuit_{\frac{2}{3}}^{0}\circ e^{\flat}_{\mathrm{st}}\circ\widetilde{\kappa}_{}\circ\chi^{-1}_{}\circ\iota_{1}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x^{\prime}))$
	$\displaystyle=\diamondsuit_{\frac{1}{3}}^{0}\circ e^{\flat}_{\mathrm{st}}(\diamondsuit_{1}^{0}\circ\widetilde{\kappa}_{}\circ\chi^{-1}_{}\circ\iota_{1}(\exp(\sigma(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))))(x^{\prime}))),$

	$\displaystyle\alpha_{0*}(x)$
	$\displaystyle=\alpha_{0*}(e^{\prime}_{\mathrm{st}}(x^{\prime}))$
	$\displaystyle=\diamondsuit_{\frac{1}{3}}^{0}\circ e^{\flat}_{\mathrm{st}}\circ\widetilde{\kappa}_{}\circ\chi^{-1}_{}\circ\iota_{0}(x^{\prime})$
	$\displaystyle=\diamondsuit_{\frac{1}{3}}^{0}\circ e^{\flat}_{\mathrm{st}}(\widetilde{\kappa}_{}\circ\chi^{-1}_{}\circ\iota_{0}(x^{\prime})).$

By (3), we have

\alpha_{0*}(x)=\diamondsuit_{1}^{0}\circ\alpha_{1*}(\exp(\sigma(e^{\prime}_{\mathrm{st}*}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)))))(x)),

as desired.

The above statements prove the lemma.

∎

proof of Theorem 5.4.

We use the notations in Lemma 6.2(3). We assume the diffeomorphism $\chi$ represents an element $\xi$ of $\mathcal{I}^{\prime}(\Sigma)$ . By Proposition 2.2, $\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi)$ equals $\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\xi)$ . Furthermore, by Proposition 5.3, $v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\chi))$ equals $v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\xi))$ . Lemma 6.2(4) says

\alpha_{1*}^{-1}\circ\diamondsuit_{0}^{1}\circ\alpha_{0*}(x)=\exp(\sigma(e^{\prime}_{\mathrm{st}*}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\xi)))))(x),

which means

(\Sigma\times I)(e_{\mathrm{st}},\xi)_{*}(x)=\exp(\sigma(e^{\prime}_{\mathrm{st}*}(v(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\xi)))))(x).

The equation proves the theorem.

∎

Next, we will prove Theorem 5.5. To do it, we need the following lemma. We recall that $\kappa:\widetilde{\Sigma}_{\mathrm{st}}\times I\to\Sigma_{\mathrm{st}}\times I$ is the tubular neighborhood satisfying the two conditions for enough small $\epsilon>0$ .

•

For any $p\in\widetilde{\Sigma}_{\mathrm{st}}$ , $e_{\partial}(p,1)=p.$

•

For any $p\in\widetilde{\Sigma}_{\mathrm{st}}\cap e_{\mathrm{st}}^{-1}(\partial\Sigma\times I)$ ,

	$\displaystyle e_{\mathrm{st}}\circ e_{\partial}((p,1),t)=(e^{\prime}_{\mathrm{st}}(p),\frac{2-\epsilon+\epsilon t}{3}),$
	$\displaystyle e_{\mathrm{st}}\circ e_{\partial}((p,0),t)=(e^{\prime}_{\mathrm{st}}(p),\frac{1+\epsilon-\epsilon t}{3}).$

Lemma 6.3.

For a standard embedding $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ and an element $\xi\in\mathcal{I}(\widetilde{\Sigma}_{\mathrm{st}})$ , we have

\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e_{\mathrm{st}},\xi))))=e_{\mathrm{st}*}\circ\kappa_{*}(\exp(\frac{1}{h}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\xi))))).

Proof.

Let $e^{\prime}_{\mathrm{st}}:\Sigma_{\mathrm{st}}\to\Sigma$ be the embedding satisfying $e^{\prime}\times\mathrm{id}_{\{\frac{1}{2}\}}=e_{\Sigma\times\{\frac{1}{2}\}}$ . By Theorem 5.4, we have

	$\displaystyle\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e_{\mathrm{st}},\xi))))=\exp(\frac{1}{h}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(e^{\prime}_{\mathrm{st}*}(v(\zeta(\xi)))))$
	$\displaystyle=\exp(\frac{1}{h}(e_{\mathrm{st}}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(v(\zeta(\xi))))))=e_{\mathrm{st}}(\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(v(\zeta(\xi))))).$

Using Proposition 5.3, we have

\displaystyle e_{\mathrm{st}*}(\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(v(\zeta(\xi)))))=e_{\mathrm{st}*}\circ\kappa_{*}(\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\zeta(\xi)))).

The equation proves the lemma.

∎

proof of Theorem 5.5.

Let $L$ be a boundary link in $\Sigma\times I$ and $e_{S}:S\to\Sigma\times I$ a Seifert surface of $L$ . By Lemma 4.7, there exists a standard embedding $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ and an embedding $e^{\prime}:S\to\widetilde{\Sigma}_{\mathrm{st}}$ such that the composite $e_{\mathrm{st}}\circ e^{\prime}$ is isotopic to $e_{S}$ . Using Lemma 4.8, for any label $\lambda:\pi_{0}(\partial S)\to\{\pm{1}\}$ , we have

(\Sigma\times I)(L(\lambda))=(\Sigma\times I)(e_{\mathrm{st}},\prod_{[\partial]\in\pi_{0}(\partial S)}t_{e^{\prime}(c_{\partial})}^{-\lambda([\partial])}).

By Lemma 6.3, we have

	$\displaystyle\exp(\frac{1}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e_{\mathrm{st}},\xi))))$
	$\displaystyle=e_{\mathrm{st}}\circ\kappa_{}(\exp(\frac{1}{h}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\xi)))))$
	$\displaystyle=e_{\mathrm{st}}\circ\kappa_{}(\exp(\frac{1}{h}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\zeta_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\prod_{[\partial]\in\pi_{0}(\partial S)}t_{e^{\prime}(c_{\partial})}^{-\lambda([\partial])})))))$
	$\displaystyle=e_{\mathrm{st}}\circ\kappa_{}(\exp(\frac{1}{h}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\sum_{[\partial]\in\pi_{0}(\partial S)}-\lambda([\partial])e^{\prime}_{*}(L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{\partial}))))))$
	$\displaystyle=e_{\mathrm{st}}\circ\kappa_{}\circ e^{\prime}_{*}(\exp(\frac{1}{h}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\sum_{[\partial]\in\pi_{0}(\partial S)}-\lambda([\partial])L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{\partial})))))$
	$\displaystyle=e_{S*}(\exp(\frac{1}{h}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\sum_{[\partial]\in\pi_{0}(\partial S)}-\lambda([\partial])L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{\partial}))))).$

The computation proves the theorem.

∎

7. A lemma of $\mathcal{A}^{\prime}$

To give another proof of the formula [7] of Kuno and Massuyeau and to clarify it, we need a lemma of the filtration of the skein algebra $\mathcal{A}^{\prime}$ . In this section, we introduce the lemma and prove it.

Let $\Sigma$ be a compact connected oriented surface and $\star$ a point in $\partial\Sigma$ . We fix an embedding $e:\Sigma\hookrightarrow\widetilde{\Sigma}\times I$ and a non-increasing sequence $\mathbf{b}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\{b_{n}\}_{n\geq 0}$ of non-negative integers. For any $m\in\mathbb{Z}_{\geq 0}$ , we set a $F^{(e,\mathbf{b})m}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\subset\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)$ generated by

\lvert\eta_{1}\eta_{2}\cdots\eta_{k}\rvert

satisfying the properties.

•

$\eta_{1},\eta_{2},\cdots,\eta_{k}\in\mathbb{Q}\pi_{1}(\Sigma,\star)$ .
•
There exist non-negative integers $m_{1},\cdots,m_{k},n_{1},\cdots,c_{k}$ satisfying the conditions.
- –
  
  For any $i$ , $\eta_{i}\in F^{m_{i}}\mathbb{Q}\pi_{1}(\Sigma,\star)$ .
- –
  
  For any $i$ , $e_{*}(\eta_{i})\in F^{m_{i}+n_{i}}\mathbb{Q}\pi_{1}(\Sigma,\star)$ .
- –
  
  We have $\sum_{i=1}^{k}m_{i}\geq m.$
- –
  
  For any subset $\varsigma\subset\{1,\cdots,k\}$ , $\sum_{i\in\varsigma}n_{i}\geq b_{k-\sharp\varsigma}$ .

In this paper, we denote by $\mathbf{b}(N)$ the $N$ -th element of $\mathbf{b}(N)$ for any sequence $\mathbf{b}$ . Let $\mathbf{b}_{1},\mathbf{b}_{2},\cdots$ be some sequences. We set a new one $\Lambda(\mathbf{b}_{1},\mathbf{b}_{2})$ by

\Lambda(\mathbf{b}_{1},\mathbf{b}_{2})(n)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\min\{\mathbf{b}_{1}(i)+\mathbf{b}_{2}(j)|i+j+2=n\}.

Then the operator has the associativity, which means

\Lambda(\mathbf{b}_{1},\Lambda(\mathbf{b}_{2},\mathbf{b}_{3}))(n)=\Lambda(\Lambda(\mathbf{b}_{1},\mathbf{b}_{2}),\mathbf{b}_{3})(n)=\min\{b_{i}+b^{\prime}_{j}+b^{\prime\prime}_{k}|i+j+k=n-4\}.

We denote

\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{j})=\Delta(\mathbf{b}_{1},(\Delta(\mathbf{b}_{2},\cdots,\Delta(\mathbf{b}_{j-1},\mathbf{b}_{j})\cdots))).

In this section, we will prove the following statement Statement $(N)$ for any $N\in\mathbb{Z}_{\geq 1}$ .

Let $\Sigma_{1},\cdots,\Sigma_{N},\Sigma$ be compact connected oriented surfaces and $e=e_{1}\sqcup\cdots\sqcup e_{N}:\Sigma_{1}\times I\sqcup\cdots\sqcup\Sigma_{N}\times I\hookrightarrow\Sigma\times I$ be an embedding that induces homomorphism $e_{*}:\mathcal{A}^{\prime}(\Sigma_{1})\otimes\cdots\otimes\mathcal{A}^{\prime}(\Sigma_{N})\to\mathcal{A}^{\prime}(\Sigma)$ . For any $m_{1},\cdots,m_{N}\in\mathbb{Z}_{\geq 0}$ and non-increasing sequences $\mathbf{b}_{1},\cdots,\mathbf{b}_{N}$ of non-negative integers, we have

e_{*}(\otimes_{i=1}^{j}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(F^{(e_{i},\mathbf{b}_{i})m_{i}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{i})))\subset F^{(\sum_{i=1}^{N}m_{i})+\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N})(0)}\mathcal{A}^{\prime}(\Sigma).

Lemma 7.1.

If $N=1$ , Statement $(1)$ holds.

Proof.

Let $\eta_{1},\cdots,\eta_{k}$ be elements of $\mathbb{Q}\pi_{1}(\Sigma_{1})$ satisfying $e_{*}(\eta_{j})\in I_{\mathbb{Q}\pi_{1}(\Sigma)}^{n_{j}+m_{j}}$ , where

\sum_{j=1,\cdots,k}m_{j}\geq\mathbf{b}_{1}(0).

We remark that we can ignore the self-crossing in the skein algebra. We obtain

e_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\eta_{1}\cdots\eta_{k}\rvert))\in F^{\sum_{j=1,\cdots,k}(m_{j}+n_{j})}\mathcal{A}^{\prime}(\Sigma)\subset F^{\sum_{j=1,\cdots,k}m_{j}+\mathbf{b}_{1}(0)}\mathcal{A}^{\prime}(\Sigma)

as desired.

∎

Lemma 7.2.

Let $\Sigma$ and $\widetilde{\Sigma}$ be compact connected oriented surfaces and $e:\Sigma\hookrightarrow\widetilde{\Sigma}\times I$ an embedding. For non-increasing sequences $\mathbf{b},\mathbf{b}^{\prime}$ of non-negative integers and integers $m,m^{\prime}\in\mathbb{Z}_{\geq 0}$ , we have

[F^{(e,\mathbf{b})m}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma),F^{(e,\mathbf{b}^{\prime})m^{\prime}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)]\subset F^{(e,\Lambda(\mathbf{b},\mathbf{b^{\prime}}))m+m^{\prime}-2}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma).

Proof.

Let $\eta_{1},\cdots,\eta_{k},\eta^{\prime}_{1},\cdots,\eta^{\prime}_{k^{\prime}}$ be elements of $\mathbb{Q}\pi_{1}(\Sigma,\star)$ satisfying the conditions.

•

For any $j\in\{1,\cdots,k\}$ , there exists a set $S_{j}$ , exists $m_{j}\in\mathbb{Z}_{\geq 0}$ , sxists $q_{\alpha_{j}}\in\mathbb{Q}$ for any $\alpha_{j}\in S_{j}$ , and exists $\gamma^{(\alpha_{j})}_{i}\in\pi_{1}(\Sigma,\star)$ for any $\alpha_{j}\in S_{j},i\in\{1,\cdots,m_{j}\}$ satisfying

\eta_{j}=\sum_{\alpha_{j}\in S_{j}}q_{\alpha_{j}}(\gamma^{(\alpha_{j})}_{1}-1)(\gamma^{(\alpha_{j})}_{2}-1)\cdots(\gamma^{(\alpha_{j})}_{m_{j}}-1)\in I_{\mathbb{Q}\pi_{1}(\Sigma,\star)}^{m_{j}}.

•

For any $j^{\prime}\in\{1,\cdots,k^{\prime}\}$ , there exists a set $S^{\prime}_{j^{\prime}}$ , exists $m^{\prime}_{j^{\prime}}\in\mathbb{Z}_{\geq 0}$ , sxists $q^{\prime}_{\alpha^{\prime}_{j^{\prime}}}\in\mathbb{Q}$ for any $\alpha^{\prime}_{j^{\prime}}\in S^{\prime}_{j^{\prime}}$ , and exists $\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i}\in\pi_{1}(\Sigma,\star)$ for any $\alpha^{\prime}_{j^{\prime}}\in S^{\prime}_{j^{\prime}},i\in\{1,\cdots,m^{\prime}_{j^{\prime}}\}$ satisfying

\eta^{\prime}_{j^{\prime}}=\sum_{\alpha^{\prime}_{j^{\prime}}\in S^{\prime}_{j^{\prime}}}q^{\prime}_{\alpha^{\prime}_{j^{\prime}}}(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{1}-1)(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{2}-1)\cdots(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{m^{\prime}_{j^{\prime}}}-1)\in I_{\mathbb{Q}\pi_{1}(\Sigma,\star)}^{m^{\prime}_{j^{\prime}}}.

•

There exist integers $n_{1},\cdots,n_{k},n^{\prime}_{1},\cdots,n^{\prime}_{k^{\prime}}\in\mathbb{Z}_{\geq 0}$ such that

e_{*}(\eta_{k})\in I_{\mathbb{Q}\pi_{1}(\widetilde{\Sigma}\times I)}^{m_{k}+n_{k}}\mathrm{\ and\ }e_{*}(\eta^{\prime}_{k^{\prime}})\in I_{\mathbb{Q}\pi_{1}(\widetilde{\Sigma}\times I)}^{m^{\prime}_{k^{\prime}}+n^{\prime}_{k^{\prime}}}.

•

$\sum_{i\in\{1,\cdots,k\}}m_{i}\geq m$ .
•

$\sum_{i\in\{1,\cdots,k^{\prime}\}}m^{\prime}_{i}\geq m^{\prime}$ .
•

For any subset $\varsigma\subset\{1,\cdots,k\}$ , $\sum_{i\in\varsigma}n_{i}\geq b_{k-\sharp\varsigma}$ .
•

For any subset $\varsigma^{\prime}\subset\{1,\cdots,k^{\prime}\}$ , $\sum_{i\in\varsigma^{\prime}}n^{\prime}_{i}\geq b^{\prime}_{k-\sharp\varsigma^{\prime}}$ .

Here we recall that $I_{\mathbb{Q}G}\subset\mathbb{Q}G$ is the augmentation ideal $\{\sum_{g\in G}q_{g}g|\sum_{g\in G}q_{g}=0\}$ . for any group $G$ . Then we have

	$\displaystyle[\lvert\eta_{1}\eta_{2}\cdots\eta_{k}\rvert,\lvert\eta^{\prime}_{1},\eta^{\prime}_{2},\cdots\eta^{\prime}_{k^{\prime}}\rvert]$
	$\displaystyle=\sum_{j\in\{1,\cdots,k\}}\sum_{j^{\prime}\in\{1,\cdots,k^{\prime}\}}\sum_{\alpha_{j}\in S_{j}}\sum_{\alpha^{\prime}_{j^{\prime}}\in S^{\prime}_{j^{\prime}}}\sum_{i\in\{1,\cdots,m_{k}\}}\sum_{i^{\prime}\in\{1,\cdots,m^{\prime}_{k^{\prime}}\}}\sum_{p\in\gamma^{(\alpha_{j})}_{i}\cap\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i^{\prime}}}$
	$\displaystyle q_{\alpha_{j}}q^{\prime}_{\alpha^{\prime}_{j^{\prime}}}\epsilon(p,\gamma^{(\alpha_{j})}_{i},\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i^{\prime}})\lvert(\gamma^{(\alpha_{j})}_{1}-1)\cdots(\gamma^{(\alpha_{j})}_{i-1}-1)(\gamma^{(\alpha_{j})}_{i})_{\star,p}(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i^{\prime}})_{p,\star}$
	$\displaystyle(\gamma^{(\alpha^{\prime}_{j^{\prime}})}_{i^{\prime}+1}-1)\cdots(\gamma^{(\alpha^{\prime}_{j^{\prime}})}_{m^{\prime}_{j^{\prime}}}-1)\eta^{\prime}_{j^{\prime}+1}\cdots\eta^{\prime}_{k^{\prime}}\eta^{\prime}_{1}\cdots\eta^{\prime}_{j^{\prime}-1}(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{1}-1)\cdots(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i-1}-1)$
	$\displaystyle(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i^{\prime}})_{\star,p}(\gamma^{(\alpha_{j})}_{i})_{p,\star}(\gamma^{(\alpha_{j})}_{i+1}-1)\cdots(\gamma^{(\alpha_{j})}_{m_{j}}-1)\eta_{j+1}\cdots\eta_{k}\eta_{1}\cdots\eta_{j-1}\rvert.$

Now, for any subsets $\varsigma\subset\{1,\cdots,j-1,j+1,\cdots,k\}$ and $\varsigma^{\prime}\subset\{1,\cdots,j^{\prime}-1,j^{\prime}+1,\cdots,k^{\prime}\}$ ,

\sum_{i\in\varsigma}n_{i}+\sum_{i^{\prime}\in\varsigma^{\prime}}n^{\prime}_{i^{\prime}}\geq b_{k-\sharp\varsigma}+b^{\prime}_{k^{\prime}-\sharp\varsigma^{\prime}}\geq\Delta(\mathbf{b},\mathbf{b^{\prime}})(k+k^{\prime}-2-\sharp\varsigma-\sharp^{\prime}\varsigma^{\prime})

holds. By definition, we obtain

	$\displaystyle\lvert(\gamma^{(\alpha_{j})}_{1}-1)\cdots(\gamma^{(\alpha_{j})}_{i-1}-1)(\gamma^{(\alpha_{j})}_{i})_{\star,p}(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i^{\prime}})_{p,\star}$
	$\displaystyle(\gamma^{(\alpha^{\prime}_{j^{\prime}})}_{i^{\prime}+1}-1)\cdots(\gamma^{(\alpha^{\prime}_{j^{\prime}})}_{m^{\prime}_{j^{\prime}}}-1)\eta^{\prime}_{j^{\prime}+1}\cdots\eta^{\prime}_{k^{\prime}}\eta^{\prime}_{1}\cdots\eta^{\prime}_{j^{\prime}-1}(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{1}-1)\cdots(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i-1}-1)$
	$\displaystyle(\gamma^{\prime(\alpha^{\prime}_{j^{\prime}})}_{i^{\prime}})_{\star,p}(\gamma^{(\alpha_{j})}_{i})_{p,\star}(\gamma^{(\alpha_{j})}_{i+1}-1)\cdots(\gamma^{(\alpha_{j})}_{m_{j}}-1)\eta_{j+1}\cdots\eta_{k}\eta_{1}\cdots\eta_{j-1}\rvert$
	$\displaystyle\in F^{(e,\Delta(\mathbf{b},\mathbf{b}^{\prime}))m+m^{\prime}-2}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma).$

It proves the lemma.

∎

Lemma 7.3.

Let $\Sigma,\Sigma_{1},\Sigma_{2},$ and $\widetilde{\Sigma}$ be compact connected oriented surfaces and $e:\Sigma\times I\to\widetilde{\Sigma}\times I$ , $e_{1}:\Sigma_{1}\to\Sigma\times I$ , and $e_{2}:\Sigma_{2}\to\Sigma\times I$ embeddings. We also denote by $e$ the restriction $\Sigma\to\widetilde{\Sigma}\times I,p\mapsto e(p,0)$ . For any elements $x\in\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(F^{(e\circ e_{1},\mathbf{b}_{1})m_{1}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{1}))$ and $y\in\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(F^{(e\circ e_{2},\mathbf{b}_{2})m_{2}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{2}))$ , we obtain

e_{1*}(x)e_{2*}(y)-e_{2*}(y)e_{1*}(x)\in h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(F^{(e,\Delta(\mathbf{b}_{1},\mathbf{b}_{2}))m_{1}+m_{2}-2}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)).

Proof.

We remark that, by definition, we have

\displaystyle e_{i*}(F^{(e\circ e_{i},\mathbf{b}_{i})m_{i}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{i}))\subset F^{(e,\mathbf{b}_{i})m_{i}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)

for $i=1,2$ . By the relation of $\mathcal{U}_{h}(\Sigma)$ , we have

\displaystyle\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(e_{1*}(x)e_{2*}(y)-e_{2*}(y)e_{1*}(x))=h[\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(e_{1*}(x)),\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(e_{2*}(y))].

Using Lemma 7.2, we obtain

\displaystyle h[\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(e_{1*}(x)),\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(e_{2*}(y))]\in hF^{(e,\Delta(\mathbf{b}_{1},\mathbf{b}_{2}))m_{1}+m_{2}-2}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma).

It proves the lemma.

∎

Lemma 7.4.

Let $\Sigma_{0},\cdots,\Sigma_{N+1},\Sigma$ be compact connected oriented surfaces and

	$\displaystyle e^{+}=e^{+}_{1}\sqcup\cdots\sqcup e^{+}_{N+1}$	$\displaystyle:\Sigma_{1}\times I\sqcup\cdots\sqcup\Sigma_{N-1}\times I\sqcup\Sigma_{N}\times I\sqcup\Sigma_{N+1}\times I\hookrightarrow\Sigma\times I$
	$\displaystyle e^{-}=e^{-}_{1}\sqcup\cdots\sqcup e^{-}_{N+1}$	$\displaystyle:\Sigma_{1}\times I\sqcup\cdots\sqcup\Sigma_{N-1}\times I\sqcup\Sigma_{N}\times I\sqcup\Sigma_{N+1}\times I\hookrightarrow\Sigma\times I$
	$\displaystyle e^{0}=e^{0}_{1}\sqcup\cdots\sqcup e^{0}_{N-1}\sqcup e^{0}_{0}$	$\displaystyle:\Sigma_{1}\times I\sqcup\cdots\sqcup\Sigma_{N-1}\times I\sqcup\Sigma_{0}\times I\hookrightarrow\Sigma\times I$

be embeddings satisfying the three conditions.

•

The images of $e^{+},e^{-},e^{0}$ equal to each other except for a closed ball $D^{3}$ .
•

In $D^{3}$ , the images of $e^{+},e^{-},e^{0}$ are submanifolds shown in the figure.

•

We have

	$\displaystyle e_{i}^{+}=e_{i}^{-}=e_{i}^{0}\mathrm{\ for\ }i=1,\cdots,N-1,$
	$\displaystyle(\Sigma_{i}\times I)\backslash(e^{+})^{-1}(D^{3})=(\Sigma_{i}\times I)\backslash(e^{-})^{-1}(D^{3})\mathrm{\ for\ }i=N,N+1,$
	$\displaystyle e^{+}_{i\|(\Sigma_{i}\times I)\backslash(e^{+})^{-1}(D^{3})}=e^{-}_{i\|(\Sigma_{i}\times I)\backslash(e^{-})^{-1}(D^{3})}\mathrm{\ for\ }i=N,N+1,$

For non-increasing sequences $\mathbf{b}_{1},\cdots\mathbf{b}_{N+1}$ of non-negative integers, integers $m_{1}$ , $\cdots$ , $m_{N+1}\in\mathbb{Z}_{\geq 0}$ , and

\displaystyle x_{1}\in F^{(e_{1},\mathbf{b}_{1})m_{1}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{1}),\cdots,x_{N+1}\in F^{(e_{N+1},\mathbf{b}_{N+1})m_{N+1}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{N+1}),

the two statements hold.

(1)

There exists $y\in F^{(e_{0},\Delta(\mathbf{b}_{N},\mathbf{b}_{N+1}))+m_{N}+m_{N+1}-2}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{0})$ satisfying

	$\displaystyle e^{+}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))-e^{-}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle=he^{0}_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\Psi_{\mathcal{U}_{h}}^{\mathcal{A}^{\prime}}(x_{N-1})\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y)).$

(2)

If Statement $(N)$ holds, we have

	$\displaystyle e^{+}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))-e^{-}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle\in F^{\sum_{i=1}^{k}m_{i}+\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N+1})(0)}\mathcal{A}^{\prime}(\Sigma).$

Proof.

(1)We set embeddings $e_{\mathrm{over}},e_{\mathrm{under}}:\Sigma_{0}\times I\to\Sigma_{0}\times I$ as $e_{\mathrm{over}}(p,t)=(p,\frac{t+1}{2})$ and $e_{\mathrm{under}}(p,t)=(p,\frac{t}{2})$ . Let $e_{N,0}:\Sigma_{N}\times I\to\Sigma_{0}\times I$ and $e_{N+1,0}:\Sigma_{N+1}\times I\to\Sigma_{0}\times I$ be embeddings satisfying the condition.

•

The embedding $e^{+}=e^{+}_{1}\sqcup\cdots\sqcup e^{+}_{N+1}$ is isotopic to $e^{+}_{1}\sqcup\cdots\sqcup e^{+}_{N-1}\sqcup e^{0}_{0}\circ e_{\mathrm{under}}\circ e_{N,0}\sqcup e^{0}_{0}\circ e_{\mathrm{over}}\circ e_{N+1,0}$ .
•

The embedding $e^{-}=e^{-}_{1}\sqcup\cdots\sqcup e^{-}_{N+1}$ is isotopic to $e^{-}_{1}\sqcup\cdots\sqcup e^{-}_{N-1}\sqcup e^{0}_{0}\circ e_{\mathrm{over}}\circ e_{N,0}\sqcup e^{0}_{0}\circ e_{\mathrm{under}}\circ e_{N+1,0}$ .

Then we obtain

	$\displaystyle e^{+}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))-e^{-}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle=e^{0}_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N-1})\otimes$
	$\displaystyle(e_{N+1,0}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N}+1))e_{N,0}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N}))-e_{N,0}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N}))e_{N+1,0}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1})))).$

By Lemma 7.3, we have

	$\displaystyle e^{+}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))-e^{-}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle=he^{0}_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N-1})\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y))$

where we set

y\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}[e_{N+1*}(x_{N+1}),e_{N*}(x_{N})]\in F^{(e_{0},\Delta(\mathbf{b}_{N},\mathbf{b}_{N+1}))(0)+m_{N}+m_{N+1}-2}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{N+1}).

It proves the statement.

(2)Using the statement (1) and Statement $(N)$ , we obtain

	$\displaystyle e^{0}_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N-1})\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y))$
	$\displaystyle\in F^{\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N-1},\Delta(\mathbf{b}_{N},\mathbf{b}_{N+1}))(0)+(\sum_{i=1}^{N+1}m_{i})-2}\mathcal{A}^{\prime}(\Sigma).$

By definition, we have

\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N-1},\Delta(\mathbf{b}_{N},\mathbf{b}_{N+1}))=\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N+1}).

So we have

	$\displaystyle e^{+}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))-e^{-}_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle=he^{0}_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N-1})\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(y))$
	$\displaystyle\in F^{\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N+1})(0)+(\sum_{i=1}^{N+1}m_{i})}\mathcal{A}^{\prime}(\Sigma).$

It proves the statement.

∎

Lemma 7.5.

We assume Statement $(N)$ . Let

\displaystyle e=e_{1}\sqcup\cdots\sqcup e_{N+1}

\displaystyle:\Sigma_{1}\times I\sqcup\cdots\sqcup\Sigma_{N+1}\times I\hookrightarrow\Sigma\times I

be an embedding. For non-increasing sequences $\mathbf{b}_{1},\cdots,\mathbf{b}_{N+1}$ of non-negative integers, integers $m_{1},\cdots,m_{N+1}$ , and elements

\displaystyle x_{1}\in F^{(e_{1},\mathbf{b}_{1})m_{1}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{1}),\cdots,x_{N+1}\in F^{(e_{N+1},\mathbf{b}_{N+1})m_{N+1}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{N+1}),

we have

(1)		$\displaystyle e_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle-e_{1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1}))\cdots e_{N+1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle\in F^{\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N+1})(0)+\sum_{i=1}^{N+1}m_{i}}\mathcal{A}^{\prime}(\Sigma),$
(2)		$\displaystyle e_{1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1}))\cdots e_{N+1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle\in F^{\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N+1})(0)+\sum_{i=1}^{N+1}m_{i}}\mathcal{A}^{\prime}(\Sigma),$
(3)		$\displaystyle e_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1})\otimes\cdots\otimes\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle\in F^{\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N+1})(0)+\sum_{i=1}^{N+1}m_{i}}\mathcal{A}^{\prime}(\Sigma).$

Proof.

(1)Using Lemma 7.4 (2) repeatedly, we obtain the equation (1).

(2)By definition, we have

e_{i*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{i}))\in F^{m_{i}+\mathbf{b}_{i}(0)}\mathcal{A}^{\prime}(\Sigma).

So we obtain

	$\displaystyle e_{1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{1}))\cdots e_{N+1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(x_{N+1}))$
	$\displaystyle\in F^{\sum_{i=1}^{N+1}(m_{i}+\mathbf{b}_{i}(0))}\mathcal{A}^{\prime}(\Sigma)\subset F^{\sum_{i=1}^{N+1}m_{i}*+\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N+1})(0)}\mathcal{A}^{\prime}(\Sigma).$

(3)Using the equations (1) and (2), we obtain the equation(3).

∎

Theorem 7.6.

For any $N\in\mathbb{Z}_{\geq 1}$ , Statement $(N)$ holds. In other words, for any $N\in\mathbb{Z}_{\geq 1}$ , we have the following. Let $\Sigma_{1},\cdots,\Sigma_{N},\Sigma$ be compact connected oriented surfaces and $e=e_{1}\sqcup\cdots\sqcup e_{N}:\Sigma_{1}\times I\sqcup\cdots\sqcup\Sigma_{N}\times I\hookrightarrow\Sigma\times I$ be an embedding that induces homomorphism $e_{*}:\mathcal{A}^{\prime}(\Sigma_{1})\otimes\cdots\otimes\mathcal{A}^{\prime}(\Sigma_{N})\to\mathcal{A}^{\prime}(\Sigma)$ . For any integers $m_{1},\cdots,m_{N}$ and non-increasing sequences $\mathbf{b}_{1}=\{b_{(1)n}\}_{n\geq 0},\cdots,\mathbf{b}_{N}=\{b_{(N)i}\}_{n\geq 0}$ of non-negative integers, we have

	$\displaystyle e_{*}(\otimes_{i=1}^{j}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(F^{(e_{i},\mathbf{b}_{i})m_{i}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma_{i})))\subset F^{(\sum_{i=1}^{j}m_{i})+\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N})(0)}\mathcal{A}^{\prime}(\Sigma)$
	$\displaystyle=F^{(\sum_{i=1}^{N}m_{i})+\min\{b_{(1)k_{1}}+\cdots b_{(N)k_{N}}\|k_{1}+\cdots+k_{N}\geq 2N-2\}}\mathcal{A}^{\prime}(\Sigma).$

Proof.

By Lemma 7.1, we have Statement $(1)$ . Using Lemma 7.5(3), by induction, Statement $(N)$ holds for any $N\in\mathbb{Z}_{\geq 1}$ . By definition, we can check $\Delta(\mathbf{b}_{1},\cdots,\mathbf{b}_{N})(k)=\min\{b_{(1)k_{1}}+\cdots b_{(N)k_{N}}|k_{1}+\cdots+k_{N}\geq 2N-2+k\}$ .

∎

Using the lemma, we can prove the corollary.

Corollary 7.7.

Let $\Sigma^{\prime}$ and $\Sigma$ be compact connected oriented surfaces and $e:\Sigma^{\prime}\times I\to\Sigma\times I$ an embedding. For an element $\gamma\in[\pi_{1}(\Sigma^{\prime}),\pi_{1}(\Sigma^{\prime})]$ satisfying $e_{*}(\gamma)\in 1+I_{\mathbb{Q}\pi_{1}(\Sigma\times I)}^{m+2}$ , we have

e_{*}(\prod_{i=1}^{k}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma-1)^{m_{i}}\rvert))\in F^{2(\sum_{i=1}^{k}m_{i})+m(\sum_{i=1}^{k}m_{i}-(2k-2))}\mathcal{A}^{\prime}(\Sigma).

Proof.

For $i=1,\cdots,k$ , we set an embedding $e_{i}:\Sigma^{\prime}\times I\hookrightarrow\Sigma\times I$ as $e_{i}(p,t)=e(p,\frac{2k-2i+t}{2k})$ and consider the embedding $\widetilde{e}=e_{1}\sqcup\cdots\sqcup e_{k}:\coprod\Sigma^{\prime}\times I\hookrightarrow\Sigma\times I$ . Then we have

e_{*}(\prod_{i=1}^{k}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma-1)^{m_{i}}\rvert))=\widetilde{e}_{*}(\otimes_{i=1}^{k}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma-1)^{m_{i}}\rvert)).

Setting a sequence $\mathbf{b}^{(i)}=\{b^{(i)}_{n}\}_{n\geq 0}$ as

b^{(i)}_{n}=\max(m(m_{i}-n),0)

for any $i=1,\cdots,k$ , we obtain

\lvert(\gamma-1)^{m_{i}}\rvert\in F^{(e_{i},\mathbf{b}^{(i)})2m_{i}}\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma).

Using Lemma 7.6, we have

\displaystyle\widetilde{e}_{*}(\otimes_{i=1}^{k}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma-1)^{m_{i}}\rvert))\in F^{2(\sum_{i=1}^{k}m_{i})+\min\{b_{(1)j_{1}}+\cdots b_{(k)j_{k}}|j_{1}+\cdots+j_{k}\geq 2k-2\}}\mathcal{A}^{\prime}(\Sigma).

Since

m(\sum_{i=1}^{k}m_{i}-2k+2)\leq\min\{b_{(1)j_{1}}+\cdots b_{(k)j_{k}}|j_{1}+\cdots+j_{k}\geq 2k-2\}

we have

\displaystyle\widetilde{e}_{*}(\otimes_{i=1}^{k}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma-1)^{m_{i}}\rvert))\in F^{2(\sum_{i=1}^{k}m_{i})+m(\sum_{i=1}^{k}m_{i}-2k+2)}\mathcal{A}^{\prime}(\Sigma).

It proves the corollary.

∎

8. Some formulas of the action of some homology cylinders

In this section, we will give another proof of the theorem (Theorem 8.1 in this paper) of Kuno and Massuyeau [7] and clarify it. Kuno and Massuyeau obtain this theorem by observing how paths behave in the neighborhood of a Seifert surface. On the other hand, we will prove it by using Theorem 5.5 in this paper.

We apply our knowledge of the skein algebra $\mathcal{A}^{\prime}$ to compute the map $\widetilde{\zeta}:\mathcal{IC}(\Sigma)\to(F^{3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma),\mathrm{bch})$ in the situation of the formula of Kuno and Massuyeau. Let $\Sigma$ be a compact connected oriented surface. For a knot $K$ in $\Sigma\times I$ , the two properties are equivalent.

(1)

The homology class of $K$ equals 0, and the framing number $w(K)$ of $K$ $0$ .
(2)

The knot $K$ is a boundary link, which means a boundary knot.

Theorem 8.1 ([7]).

Let $\Sigma$ be a compact connected oriented surface, $\star$ a base point in $\partial\Sigma$ , and $K$ a boundary knot in $\Sigma\times I$ . We denote by $\lambda_{\epsilon},\pi_{0}(K)\to\{\pm 1\},[K]\mapsto\epsilon$ the label of a Seifert surface of $K$ and by $(\Sigma\times I)(K(\epsilon))$ the element $(\Sigma\times I)(K(\lambda_{\epsilon}))$ of $\mathcal{IC}(\Sigma)$ . If there exists a path $\gamma\in\pi_{1}(\Sigma)\cap(1+I_{\mathbb{Q}\pi_{1}(\Sigma)}^{m})$ whose conjugacy class equals the homotopy class of $K$ , we have

	$\displaystyle\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K(-\epsilon)))=\epsilon\lvert\frac{1}{2}(\log(\gamma))^{2}\rvert\mod F^{2m+2}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$
	$\displaystyle\epsilon\lvert\frac{1}{2}(\log(\gamma))^{2}\rvert\in F^{2m}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma).$

In other words, for $\star_{1},\star_{2}\in\partial\Sigma$ , we have

	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\mathrm{id}$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m-1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star)/F^{2m-1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\exp(\sigma(\epsilon\frac{1}{2}\lvert(\log\gamma)^{2}\rvert))$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star 1,\star_{2})/F^{2m+1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star)/F^{2m+1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}).$

Remark 8.2.

In the above theorem, Kuno and Massuyeau proved that $\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K(-\epsilon)))\mod F^{2m+2}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ depends only on the homotopy type of the knot $K$ . In this paper §9, we will verify that the number $2m+2$ is the best possible.

Proof.

Let $\Sigma^{\prime}$ be a compact connected oriented surface and $e:\Sigma^{\prime}\times I\to\Sigma\times I$ an embedding such that the image $e(\partial\Sigma^{\prime}\times I)$ represents $K$ . Using Theorem 5.5, we have

\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K(-\epsilon))))=h\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(\log(e_{*}(\exp(\frac{\epsilon}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\frac{1}{2}(\lvert(\log(\gamma_{\partial}))^{2}\rvert)))))),

where $\gamma_{\partial}$ is an element of $\pi_{1}(\Sigma^{\prime})$ whose conjugacy class equals the boundary $\partial\Sigma^{\prime}$ . We remark $e_{*}(\lvert\gamma_{\partial}^{n}\rvert)=\lvert\gamma^{n}\rvert$ and $e_{*}(\gamma_{\partial})\in 1+I_{\mathbb{Q}\pi_{1}(\Sigma)}^{m}$ . So we have $\epsilon\lvert\frac{1}{2}(\log(\gamma))^{2}\rvert\in F^{2m}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ .

By Corollary 7.7, since $4N+(m-2)(2N-2(N-1))=4(N-1)+2m$ , we obtain

	$\displaystyle e_{*}(\exp(\frac{\epsilon}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\frac{1}{2}(\lvert(\log(\gamma_{\partial}))^{2}\rvert))))$
	$\displaystyle=\sum_{i=0}^{\infty}\frac{\epsilon^{i}}{i!h^{i}}e_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)^{i}))$
	$\displaystyle=1+\frac{\epsilon}{h}e_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)))\mod F^{2m}\widehat{(\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3})}\mathcal{A}^{\prime}(\Sigma).$

Using

\displaystyle e_{*}(\exp(\frac{\epsilon}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\frac{1}{2}(\lvert(\log(\gamma_{\partial}))^{2}\rvert))))-1\in F^{2m-2}\widehat{(\sum_{*\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{*}}F^{3*})}\mathcal{A}^{\prime}(\Sigma),

we have

	$\displaystyle h(\log(e_{*}(\exp(\frac{\epsilon}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\frac{1}{2}(\lvert(\log(\gamma_{\partial}))^{2}\rvert))))))$
	$\displaystyle=\sum_{i=1}^{\infty}\frac{(-1)^{i-1}}{i}(e_{*}(\exp(\frac{\epsilon}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\frac{1}{2}(\lvert(\log(\gamma_{\partial}))^{2}\rvert))))-1)^{i}$
	$\displaystyle=\epsilon e_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)))\mod F^{2m+2}\widehat{(\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3})}\mathcal{A}^{\prime}(\Sigma).$

It proves

	$\displaystyle\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K(-\epsilon)))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\epsilon e_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert))))\mod F^{2m+2}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$
	$\displaystyle=\epsilon e_{*}(\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)=\epsilon\frac{1}{2}\lvert(\log(\gamma))^{2}\rvert$

as desired.

∎

Using the computation in the above proof, we obtain a more accurate formula. Specifically, we have

	$\displaystyle e_{*}(\exp(\frac{\epsilon}{h}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\frac{1}{2}(\lvert(\log(\gamma_{\partial}))^{2}\rvert))))$
	$\displaystyle=\sum_{i=0}^{\infty}\frac{\epsilon^{i}}{i!h^{i}}e_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)^{i}))$
	$\displaystyle=1+\frac{\epsilon}{h}e_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)))+\frac{\epsilon^{2}}{2h^{2}}e_{}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)))^{2})$
	$\displaystyle+\frac{\epsilon^{3}}{6h^{3}}e_{}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)))^{3})\mod F^{2m+4}\widehat{(\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3})}\mathcal{A}^{\prime}(\Sigma).$

Using the notation

L^{(i)}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}e_{*}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}((\frac{1}{2}\lvert(\log(\gamma_{\partial}))^{2}\rvert)))^{i}),

we get

	$\displaystyle\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K(-\epsilon))))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(h\log(1+\frac{\epsilon}{h}L^{(1)}+\frac{\epsilon^{2}}{2h^{2}}L^{(2)}+\frac{\epsilon^{3}}{6h^{3}}L^{(3)}))\mod F^{2m+6}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(\epsilon L^{(1)}+\frac{1}{2h}(L^{(2)}-L^{(1)2})+\frac{\epsilon}{12h^{2}}(2L^{(3)}-3L^{(2)}L^{(1)}-3L^{(1)}L^{(2)}+4L^{(1)3}))$
	$\displaystyle\mod\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(F^{2m+6}\widehat{(\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3*})}\mathcal{A}^{\prime}(\Sigma)).$

Here we can prove $L^{(2)}-L^{(1)2}\in h\widehat{\mathcal{A}}^{\prime}(\Sigma)\subset\widehat{\mathcal{A}}^{\prime}(\Sigma)$ and $2L^{(3)}-3L^{(2)}L^{(1)}-3L^{(1)}L^{(2)}+4L^{(1)3}\in h^{2}\widehat{\mathcal{A}}^{\prime}(\Sigma)\subset\widehat{\mathcal{A}}^{\prime}(\Sigma)$ using the following lemma, where $x=y=z=\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\frac{1}{2}\lvert\log(\gamma_{\partial})^{2}\rvert)$ .

Lemma 8.3.

Let $e:\Sigma^{\prime}\times I\hookrightarrow\Sigma\times I$ be an embedding. Then we have

	$\displaystyle e_{}(xy)-e_{}(x)e_{*}(y)\in h\mathcal{A}^{\prime}(\Sigma),$
	$\displaystyle e_{}(xy)-e_{}(y)e_{*}(x)\in h\mathcal{A}^{\prime}(\Sigma),$
	$\displaystyle 2e_{}(xyz)-(e_{}(x)e_{}(yz)+2e_{}(y)e_{}(xz))-(e_{}(yz)e_{}(x)+2e_{}(xy)e_{*}(z))$
	$\displaystyle+e_{}(x)e_{}(y)e_{}(z)+e_{}(y)e_{}(z)e_{}(x)+2e_{}(y)e_{}(x)e_{*}(z)\in h^{2}\mathcal{A}^{\prime}(\Sigma)$

for any $x,y,z\in\mathcal{A}^{\prime}(\Sigma^{\prime})$ .

Proof.

We will prove the equations by the three steps. We assume $e$ is a tubular neighborhood $e_{\partial}:\partial(\Sigma\times I)\times I\to\Sigma\times I$ , an embedding $e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}:\partial(\Sigma_{\mathrm{st}}\times I)\times I\to\Sigma\times I$ defined by a standard embedding $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ , or a general embedding in (Step 1), (Step 2), or (Step 3), respectively.

(Step 1) For enough small $\epsilon>0$ , we set embeddings $\kappa_{0}$ and $\kappa_{1}$ as

	$\displaystyle\kappa_{0}:\Sigma\times I\to\partial(\Sigma\times I)\times I,(p,t)\mapsto e_{\partial}^{-1}(p,\epsilon t),$
	$\displaystyle\kappa_{1}:\Sigma\times I\to\partial(\Sigma\times I)\times I,(p,t)\mapsto e_{\partial}^{-1}(p,1-\epsilon+\epsilon t),$

which induce

	$\displaystyle\kappa_{0*}:\mathcal{A}^{\prime}(\Sigma)\to\mathcal{A}^{\prime}(\partial(\Sigma\times I)),$
	$\displaystyle\kappa_{1*}:\mathcal{A}^{\prime}(\Sigma)\to\mathcal{A}^{\prime}(\partial(\Sigma\times I)).$

Since

	$\displaystyle\mathrm{id}\simeq e_{\partial}\circ\kappa_{0}\simeq e_{\partial}\circ\kappa_{1},$
	$\displaystyle\kappa_{1}\circ e^{\Sigma}_{\mathrm{over}}\sqcup\kappa_{1}\circ e^{\Sigma}_{\mathrm{under}}\simeq e^{\partial(\Sigma\times I)}_{\mathrm{over}}\circ\kappa_{1}\sqcup e^{\partial(\Sigma\times I)}_{\mathrm{under}}\circ\kappa_{1},$
	$\displaystyle\kappa_{0}\circ e^{\Sigma}_{\mathrm{over}}\sqcup\kappa_{0}\circ e^{\Sigma}_{\mathrm{under}}\simeq e^{\partial(\Sigma\times I)}_{\mathrm{under}}\circ\kappa_{0}\sqcup e^{\partial(\Sigma\times I)}_{\mathrm{over}}\circ\kappa_{0},$
	$\displaystyle e_{\partial}\circ e_{\mathrm{over}}^{\partial(\Sigma\times I)}\sqcup e_{\partial}\circ e_{\mathrm{under}}^{\partial(\Sigma\times I)}$
	$\displaystyle\simeq e_{\partial}\circ e_{\mathrm{over}}^{\partial(\Sigma\times I)}\sqcup e_{\partial}\circ e_{\mathrm{under}}^{\partial(\Sigma\times I)}\circ\kappa_{1}\circ e_{\partial}$
	$\displaystyle\simeq e_{\partial}\circ e_{\mathrm{over}}^{\partial(\Sigma\times I)}\sqcup e_{\partial}\circ e_{\mathrm{under}}^{\partial(\Sigma\times I)}\circ\kappa_{0}\circ e_{\partial},$
	$\displaystyle e_{\mathrm{over}}^{\Sigma}\circ e_{\partial}\sqcup e_{\mathrm{under}}^{\Sigma}\circ e_{\partial}$
	$\displaystyle\simeq e_{\partial}\circ e_{\mathrm{over}}^{\partial(\Sigma\times I)}\circ\kappa_{1}\circ e_{\partial}\sqcup e_{\partial}\circ e_{\mathrm{under}}^{\partial(\Sigma\times I)}$
	$\displaystyle\simeq e_{\partial}\circ e_{\mathrm{under}}^{\partial(\Sigma\times I)}\sqcup e_{\partial}\circ e_{\mathrm{over}}^{\partial(\Sigma\times I)}\circ\kappa_{0}\circ e_{\partial},$

we have

	$\displaystyle\mathrm{id}=e_{\partial}\circ\kappa_{0}=e_{\partial}\circ\kappa_{1},$
	$\displaystyle\kappa_{1}(z^{\prime}_{1}z^{\prime}_{2})=\kappa_{1}(z^{\prime}_{1})\kappa_{1*}(z^{\prime}_{2}),$
	$\displaystyle\kappa_{0}(z^{\prime}_{1}z^{\prime}_{2})=\kappa_{0}(z^{\prime}_{2})\kappa_{0*}(z^{\prime}_{1}),$
	$\displaystyle e_{\partial}(z_{1}z_{2})=e_{\partial}(z_{1}\kappa_{0}\circ e_{\partial}(z_{2}))=e_{\partial}(z_{1}\kappa_{1}\circ e_{\partial*}(z_{2}))$
	$\displaystyle e_{\partial}(z_{1})e_{\partial}(z_{2})=e_{\partial}(\kappa_{1}\circ e_{\partial}(z_{1})z_{2})=e_{\partial}(\kappa_{0}\circ e_{\partial}(z_{2})z_{1})$

for $z_{1},z_{2}\in\mathcal{A}^{\prime}(\partial(\Sigma\times I)),z^{\prime}_{1},z^{\prime}_{2}\in\mathcal{A}^{\prime}(\Sigma)$ . We obtain

	$\displaystyle e_{\partial}(xy)-e_{\partial}(x)e_{\partial}(y)=e_{\partial}(x\kappa_{0}\circ e_{\partial}(y)-\kappa_{0}\circ e_{\partial}(y)x)\in h\mathcal{A}^{\prime}(\Sigma),$
	$\displaystyle e_{\partial}(xy)-e_{\partial}(y)e_{\partial*}(x)$
	$\displaystyle=(e_{\partial}(xy)-e_{\partial}(x)e_{\partial}(y))+(e_{\partial}(x)e_{\partial}(y)-e_{\partial}(y)e_{\partial*}(x))\in h\mathcal{A}^{\prime}(\Sigma),$

which proves the first equation and the second equation.

Using the first equation and the second equation, we will prove

(a)		$\displaystyle e_{\partial}(xyz)-e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial*}(z)))$
	$\displaystyle-e_{\partial}(x)e_{\partial}(yz)+e_{\partial}(x)e_{\partial}(y)e_{\partial*}(z)\in h^{2}\mathcal{A}^{\prime}(\Sigma),$
(b)		$\displaystyle e_{\partial}(xyz)-e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial*}(z)))$
	$\displaystyle-e_{\partial}(yz)e_{\partial}(x)+e_{\partial}(y)e_{\partial}(z)e_{\partial*}(x)\in h^{2}\mathcal{A}^{\prime}(\Sigma),$
(c)		$\displaystyle e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(y)e_{\partial}(xz)$
	$\displaystyle-e_{\partial}(xy)e_{\partial}(z)+e_{\partial}(y)e_{\partial}(x)e_{\partial*}(z)\in h^{2}\mathcal{A}^{\prime}(\Sigma)$

for $x,y,z\in\mathcal{A}^{\prime}(\partial(\Sigma\times I))$ .

We will prove the equations (a) and (b). Using the first and the second equation, we obtain

	$\displaystyle e_{\partial}(xyz)-e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(x)e_{\partial}(yz)+e_{\partial}(x)e_{\partial}(y)e_{\partial}(z)$
	$\displaystyle=e_{\partial}(x\kappa_{0}(e_{\partial}(yz)-e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(x)e_{\partial}(\kappa_{0}(e_{\partial}(yz)-e_{\partial}(y)e_{\partial*}(z)))$
	$\displaystyle\in h^{2}\mathcal{A}^{\prime}(\Sigma)$
	$\displaystyle e_{\partial}(xyz)-e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(yz)e_{\partial}(x)+e_{\partial}(y)e_{\partial}(z)e_{\partial}(x)$
	$\displaystyle=e_{\partial}(xyz)-e_{\partial}(x\kappa_{1}(e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(yz)e_{\partial}(x)+e_{\partial}(y)e_{\partial}(z)e_{\partial}(x)$
	$\displaystyle=e_{\partial}(x\kappa_{1}(e_{\partial}(yz)-e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(\kappa_{1}(e_{\partial}(yz)-e_{\partial}(y)e_{\partial}(z)))e_{\partial*}(x)$
	$\displaystyle\in h^{2}\mathcal{A}^{\prime}(\Sigma),$

which verify the equations (a) and (b).

We will prove the equation (c). Since

\displaystyle z^{\prime}_{1}z^{\prime}_{2}=e_{\partial*}(\kappa_{1*}(z^{\prime}_{1}z^{\prime}_{2}))=e_{\partial*}(\kappa_{1*}(z^{\prime}_{1})\kappa_{1*}(z^{\prime}_{2}))=e_{\partial*}(\kappa_{1*}(z^{\prime}_{1})\kappa_{0*}(z^{\prime}_{2}))

for $z^{\prime}_{1},z^{\prime}_{2}\in\mathcal{A}^{\prime}(\Sigma)$ , we obtain

	$\displaystyle e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial}(z)))$
	$\displaystyle=e_{\partial}(x\kappa_{1}(e_{\partial}(y)e_{\partial}(z)))$
	$\displaystyle=e_{\partial}(x\kappa_{1}\circ e_{\partial}\circ\kappa_{1}(e_{\partial}(y)e_{\partial*}(z)))$
	$\displaystyle=e_{\partial}(x\kappa_{1}\circ e_{\partial}((\kappa_{1}\circ e_{\partial}(y))(\kappa_{0}\circ e_{\partial}(z))))$
	$\displaystyle=e_{\partial}(x((\kappa_{1}\circ e_{\partial}(y))(\kappa_{0}\circ e_{\partial*}(z)))).$

By definition, we have

	$\displaystyle e_{\partial}(y)e_{\partial}(xz)=e_{\partial}((\kappa_{1}\circ e_{\partial}(y))x(\kappa_{0}\circ e_{\partial*}(z))),$
	$\displaystyle e_{\partial}(xy)e_{\partial}(z)=e_{\partial}((\kappa_{0}\circ e_{\partial}(z))x(\kappa_{1}\circ e_{\partial*}(y))).$

We remark that these elements $\kappa_{0*}(z_{1})$ and $\kappa_{1*}(z_{2})$ are commutative for $z_{1},z_{2}\in\mathcal{A}^{\prime}(\Sigma)$ . Using it, we have

\displaystyle e_{\partial*}(y)e_{\partial*}(x)e_{\partial*}(z)=e_{\partial*}((\kappa_{0*}\circ e_{\partial*}(z))(\kappa_{1*}\circ e_{\partial*}(y))x).

Hence, we obtain

	$\displaystyle e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(y)e_{\partial}(xz)-e_{\partial}(xy)e_{\partial}(z)+e_{\partial}(y)e_{\partial}(x)e_{\partial*}(z)$
	$\displaystyle=e_{\partial}(x(\kappa_{1}\circ e_{\partial}(y))(\kappa_{1}\circ_{\partial}(z))-(\kappa_{1}\circ e_{\partial}(y))x(\kappa_{1}\circ e_{\partial*}(z))$
	$\displaystyle-(\kappa_{1}\circ e_{\partial}(z))x(\kappa_{1}\circ e_{\partial}(y))+(\kappa_{1}\circ e_{\partial}(y))(\kappa_{1}\circ e_{\partial}(z))x)$
	$\displaystyle=e_{\partial}(\kappa_{1}\circ e_{\partial}(y)((\kappa_{1}\circ e_{\partial}(z))x-x(\kappa_{1}\circ e_{\partial*}(z)))$
	$\displaystyle-((\kappa_{1}\circ e_{\partial}(z))x-x(\kappa_{1}\circ e_{\partial}(z))\kappa_{1}\circ e_{\partial}(y)))$
	$\displaystyle\in h^{2}\mathcal{A}^{\prime}(\Sigma),$

which verifies the equation (c).

Using the equations (a), (b), and (c), we have

	$\displaystyle e_{\partial}(xyz)-e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(x)e_{\partial}(yz)+e_{\partial}(x)e_{\partial}(y)e_{\partial}(z)$
	$\displaystyle+e_{\partial}(xyz)-e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(yz)e_{\partial}(x)+e_{\partial}(y)e_{\partial}(z)e_{\partial}(x)$
	$\displaystyle+2(e_{\partial}(x\kappa_{0}(e_{\partial}(y)e_{\partial}(z)))-e_{\partial}(y)e_{\partial}(xz)-e_{\partial}(xy)e_{\partial}(z)+e_{\partial}(y)e_{\partial}(x)e_{\partial*}(z))$
	$\displaystyle=2e_{\partial}(xyz)-(e_{\partial}(x)e_{\partial}(yz)+2e_{\partial}(y)e_{\partial}(xz))-(e_{\partial}(yz)e_{\partial}(x)+2e_{\partial}(xy)e_{\partial*}(z))$
	$\displaystyle+e_{\partial}(x)e_{\partial}(y)e_{\partial}(z)+e_{\partial}(y)e_{\partial}(z)e_{\partial}(x)+2e_{\partial}(y)e_{\partial}(x)e_{\partial*}(z)\in h^{2}\mathcal{A}^{\prime}(\Sigma).$

In other words, we obtain the third equation under the assumption of (Step 1).

(Step 2) Let $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ be a standard embedding, $e_{\mathrm{st}\partial}:\partial(\Sigma_{\mathrm{st}}\times I)\to\Sigma_{\mathrm{st}}\times I$ a tubular neighborhood. Using (Step 1), we obtain

	$\displaystyle e_{\mathrm{st}\partial}(xy)-e_{\mathrm{st}\partial}(x)e_{\mathrm{st}\partial*}(y)\in h\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}}),$
	$\displaystyle e_{\mathrm{st}\partial}(xy)-e_{\mathrm{st}\partial}(y)e_{\mathrm{st}\partial*}(x)\in h\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}}),$
	$\displaystyle 2e_{\mathrm{st}\partial}(xyz)-(e_{\mathrm{st}\partial}(x)e_{\mathrm{st}\partial}(yz)+2e_{\mathrm{st}\partial}(y)e_{\mathrm{st}\partial*}(xz))$
	$\displaystyle-(e_{\mathrm{st}\partial}(yz)e_{\mathrm{st}\partial}(x)+2e_{\mathrm{st}\partial}(xy)e_{\mathrm{st}\partial}(z))$
	$\displaystyle+e_{\mathrm{st}\partial}(x)e_{\mathrm{st}\partial}(y)e_{\mathrm{st}\partial}(z)+e_{\mathrm{st}\partial}(y)e_{\mathrm{st}\partial}(z)e_{\mathrm{st}\partial}(x)$
	$\displaystyle+2e_{\mathrm{st}\partial}(y)e_{\mathrm{st}\partial}(x)e_{\mathrm{st}\partial*}(z)\in h^{2}\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}}).$

We can prove the equations under the assumption $e=e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}$ of (Step 2) using that $e_{\mathrm{st}*}:\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}})\to\mathcal{A}^{\prime}(\Sigma)$ is an algebra isomorphism.

(Step 3) By Lemma 4.7, there exists a standard embedding $e_{\mathrm{st}}:\Sigma_{\mathrm{st}}\times I\to\Sigma\times I$ and exists an embedding $e^{\prime}:\Sigma^{\prime}\to\partial(\Sigma_{\mathrm{st}}\times I)$ satisfying $e_{\mathrm{st}}\circ e^{\prime}\simeq e(\cdot,1)$ . We can extend $e^{\prime}:\Sigma^{\prime}\to\partial(\Sigma_{\mathrm{st}})$ to $e^{\prime\prime}=e^{\prime}\times\mathrm{id}_{I}:\Sigma^{\prime}\times I\to\partial(\Sigma_{\mathrm{st}})\times I$ . By definition, $e_{\mathrm{st}}\circ e_{\partial\mathrm{st}}\circ e^{\prime\prime}$ is also isotopic to $e$ . By the second step, we obtain

	$\displaystyle e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime}y^{\prime})-e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(y^{\prime})\in h\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}}),$
	$\displaystyle e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime}y^{\prime})-e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(y^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime})\in h\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}}),$
	$\displaystyle 2e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime}y^{\prime}z^{\prime})$
	$\displaystyle-(e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(y^{\prime}z^{\prime})+2e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(y^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime}z^{\prime}))$
	$\displaystyle-(e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(y^{\prime}z^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime})+2e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime}y^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(z^{\prime}))$
	$\displaystyle+e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(y^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(z^{\prime})$
	$\displaystyle+e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(y^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(z^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime})$
	$\displaystyle+2e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(y^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(x^{\prime})e_{\mathrm{st}}\circ e_{\mathrm{st}\partial}(z^{\prime})\in h^{2}\mathcal{A}^{\prime}(\Sigma_{\mathrm{st}})$

for $x^{\prime},y^{\prime},z^{\prime}\in\mathcal{A}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})$ . We can prove the equations under the assumption of the (Step 3) using that $e^{\prime\prime}_{*}:\mathcal{A}^{\prime}(\Sigma^{\prime})\to\mathcal{A}^{\prime}(\widetilde{\Sigma}_{\mathrm{st}})$ is an algebra isomorphism.

∎

To state an accurate formula, we use the following notation.

Definition 8.4.

Let $K$ be a null homologous unoriented framed knot in $\Sigma\times I$ satisfying $w(K)=0$ and $e^{\prime}_{K}:S_{K}\to\Sigma\times I$ a Seifert surface of $K$ . We choose an element $\gamma_{\partial}\in\pi_{1}(S_{K})$ whose conjugacy class is $\partial S_{K}$ . We can extend $e^{\prime}_{K}$ to $e_{K}:S_{K}\times I\to\Sigma\times I$ such that $e_{K}(\cdot,0)=e^{\prime}_{K}(\cdot)$ . We use the notation $L_{\mathcal{A}^{\prime}}(c_{\partial})\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(L_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(c_{\partial}))=\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert)$ . Then we set $L_{1}(K),L_{2}(K),L_{3}(K)\in\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ as

	$\displaystyle L_{1}(K)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}$	$\displaystyle\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(e_{K*}(L_{\mathcal{A}^{\prime}}(c_{\partial}))),$
	$\displaystyle L_{2}(K)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}$	$\displaystyle\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\frac{1}{2h}(e_{K}((L_{\mathcal{A}^{\prime}}(c_{\partial}))^{2})-(e_{K}(L_{\mathcal{A}^{\prime}}(c_{\partial})))^{2})),$
	$\displaystyle L_{3}(K)\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}$	$\displaystyle\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\frac{1}{12h^{2}}(2e_{K}((L_{\mathcal{A}^{\prime}}(c_{\partial}))^{3})-3e_{K}((L_{\mathcal{A}^{\prime}}(c_{\partial}))^{2})e_{K*}(L_{\mathcal{A}^{\prime}}(c_{\partial}))$
		$\displaystyle-3e_{K}(L_{\mathcal{A}^{\prime}}(c_{\partial}))e_{K}((L_{\mathcal{A}^{\prime}}(c_{\partial}))^{2})+4(e_{K*}((L_{\mathcal{A}^{\prime}}(c_{\partial}))))^{3})).$

Using the notation, we state the theorem.

Theorem 8.5.

In the situation of Theorem 8.1, we have

	$\displaystyle\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K(-\epsilon)))$
	$\displaystyle=\epsilon L_{1}(K)+L_{2}(K)+\epsilon L_{3}(k)\mod F^{2m+6}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma),$

where $L_{1}(K)$ , $L_{2}(K)$ , and $L_{3}(K)$ are elements of $F^{2m}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ , $F^{2m+2}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ , and $F^{2m+4}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ , respectively. In other words, for $\star_{1},\star_{2}\in\partial\Sigma$ , we have

	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\mathrm{id}$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m-1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m-1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\exp(\sigma(\epsilon L_{1}(K)))$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\exp(\sigma(\epsilon L_{1}(K)+L_{2}(K)))$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+3}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+3}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\exp(\sigma(\epsilon L_{1}(K)+L_{2}(K)+\epsilon L_{3}(K)))$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+5}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+5}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}).$

Proof.

Using the notation $L^{(N)}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}e_{K*}((L_{\mathcal{A}^{\prime}}(c_{\partial}))^{N})$ , by the above computation, we obtain

	$\displaystyle\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K(-\epsilon))))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(h\log(1+\frac{\epsilon}{h}L^{(1)}+\frac{\epsilon^{2}}{2h^{2}}L^{(2)}+\frac{\epsilon^{3}}{6h^{3}}L^{(3)}))\mod F^{2m+6}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(\epsilon L^{(1)}+\frac{1}{2h}(L^{(2)}-L^{(1)2})+\frac{\epsilon}{12h^{2}}(2L^{(3)}-3L^{(2)}L^{(1)}-3L^{(1)}L^{(2)}+4L^{(1)3}))$
	$\displaystyle\mod\Psi_{\mathcal{A}^{\prime}}^{\mathcal{U}_{h}}(F^{2m+6}\widehat{(\sum_{\in\mathbb{Z}_{\geq 0}}\frac{1}{h^{}}F^{3*})}\mathcal{A}^{\prime}(\Sigma))$
	$\displaystyle=\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{U}_{h}}(\epsilon L_{1}(K)+L_{2}(K)+\epsilon L_{3}(k))\mod F^{2m+6}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma).$

By Corollary 7.7, we have $L_{1}(K)\in F^{2m}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ , $L_{2}(K)\in F^{2m+2}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ , and $L_{3}(K)\in F^{2m+4}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$ . It proves the first formula. Since $\sigma(F^{N}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma))(\mathbb{Q}\pi_{1}(\Sigma,\star))\subset F^{N-1}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star)$ for any $N\in\mathbb{Z}_{\geq 0}$ , we have

	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\mathrm{id}$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m-1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m-1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\exp(\sigma(\epsilon L_{1}(K)))$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+1}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\exp(\sigma(\epsilon L_{1}(K)+L_{2}(K)))$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+3}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+3}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle((\Sigma\times I)(K(-\epsilon))_{*}=\exp(\sigma(\epsilon L_{1}(K)+L_{2}(K)+\epsilon L_{3}(K)))$
	$\displaystyle:\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+5}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})\to\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})/F^{2m+5}\mathbb{Q}\pi_{1}(\Sigma,\star_{1},\star_{2})$

The above equations prove the theorem.

∎

9. An application

In this section, we will use Theorem 8.5 in a concrete example. Let $\Sigma_{1,1}$ be a compact connected oriented surface of genus one having a connected boundary. We fix a generating set $\{\gamma_{\alpha},\gamma_{\beta}\}$ of $\pi_{1}(\Sigma_{1,1})$ as the figure and denote the element $\gamma_{\alpha}\gamma_{\beta}\gamma_{\alpha}^{-1}\gamma_{\beta}^{-1}$ by $\gamma_{\partial}$ .

We choose an embedding $e^{(1)}_{\Sigma_{1,1}}:\Sigma_{1,1}\times I\to\Sigma_{1,1}\times I$ satisfying the two conditions.

•

It is an embedding shown in the figure.
•

The induced map $\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)\to\mathbb{Q}\lvert{\pi}_{1}\rvert(\Sigma)$ is an isomorphism.

To apply Theorem 8.5, we need the lemma.

Lemma 9.1.

Using the above notation, we have

	$\displaystyle e^{(1)}_{\Sigma_{1,1}}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2})-(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2}$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1){\gamma^{\prime}_{\beta}}$
	$\displaystyle-(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1){\gamma^{\prime}_{\beta}}$
	$\displaystyle-(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1){\gamma^{\prime}_{\beta}}^{-1}(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1){\gamma^{\prime}_{\beta}}^{-1}({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)))\rvert)$
	$\displaystyle+4h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)\rvert).$

Proof.

Using the notation $l\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}\rvert)$ , $l^{\prime}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}^{-1}\rvert)$ , we have

	$\displaystyle e^{(1)}_{\Sigma_{1,1}}((l+l^{\prime}+2)^{2})-((l+l^{\prime}+2)^{2})$
	$\displaystyle=(e^{(1)}_{\Sigma_{1,1}}(l^{2})-l^{2})+2(e^{(1)}_{\Sigma_{1,1}}(ll^{\prime})-ll^{\prime})+(e^{(1)}_{\Sigma_{1,1}}(l^{\prime 2})-l^{\prime}2).$

We will compute the three elements $(e^{(1)}_{\Sigma_{1,1}}(l^{2})-l^{2})$ , $(e^{(1)}_{\Sigma_{1,1}}(ll^{\prime})-ll^{\prime})$ , $(e^{(1)}_{\Sigma_{1,1}}(l^{\prime 2})-l^{\prime}2)$ . In the proof, we denote by $L(X)$ the link presented by the link diagram.

Here $X$ is one of the following.

Using the notation $x=\gamma_{\alpha},y=\gamma_{\beta}$ , we have

	$\displaystyle(e^{(1)}_{\Sigma_{1,1}}(l^{2})-l^{2})$
	$\displaystyle=L(d(1,1))-L(d(1,2))=h(-L(d(1,3))+L(d(1,4))-L(d(1,5))+L(d(1,6)))$
	$\displaystyle=h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert-(xyx^{-1})^{2}y^{-2}+xyx^{-1}y^{-1}xyx^{-1}y^{-1}-(xyx^{-1})^{2}y^{-2}+xyx^{-1}y^{-1}xyx^{-1}y^{-1}\rvert)$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert-(xyx^{-1})^{2}y^{-2}+xyx^{-1}y^{-1}xyx^{-1}y^{-1}\rvert),$

	$\displaystyle(e^{(1)}_{\Sigma_{1,1}}(ll^{\prime})-ll^{\prime})$
	$\displaystyle=L(d(2,1))-L(d(2,2))=h(L(d(2,3))-L(d(2,4))+L(d(2,5))-L(d(2,6)))$
	$\displaystyle=h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert xyx^{-1}y^{-1}xy^{-1}x^{-1}y-1+xyx^{-1}y^{-1}xy^{-1}x^{-1}y-1\rvert)$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert xyx^{-1}y^{-1}xy^{-1}x^{-1}y-1\rvert),$

	$\displaystyle(e^{(1)}_{\Sigma_{1,1}}(l^{\prime 2})-l^{\prime 2})$
	$\displaystyle=L(d(3,1))-L(d(3,2))=h(-L(d(3,3))+L(d(3,4))-L(d(3,5))+L(d(3,6)))$
	$\displaystyle=h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert-(xyx^{-1})^{-2}y^{2}+yxy^{-1}x^{-1}yxy^{-1}x^{-1}-(xyx^{-1})^{-2}y^{2}+yxy^{-1}x^{-1}yxy^{-1}x^{-1}\rvert)$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert-(xyx^{-1})^{-2}y^{2}+yxy^{-1}x^{-1}yxy^{-1}x^{-1}\rvert).$

So we obtain

	$\displaystyle e^{(1)}_{\Sigma_{1,1}}((l+l^{\prime}+2)^{2})-((l+l^{\prime}+2)^{2})$
	$\displaystyle=(e^{(1)}_{\Sigma_{1,1}}(l^{2})-l^{2})+2(e^{(1)}_{\Sigma_{1,1}}(ll^{\prime})-ll^{\prime})+(e^{(1)}_{\Sigma_{1,1}}(l^{\prime 2})-l^{\prime}2)$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert y^{\prime}y^{-1}(y^{\prime}y^{-1}-y^{-1}y^{\prime}+y^{\prime-1}y-yy^{\prime-1})$
	$\displaystyle+yy^{\prime-1}(yy^{\prime-1}-y^{\prime-1}y+y^{-1}y^{\prime}-y^{\prime}y^{-1})\rvert)$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-yy^{\prime-1})(y^{\prime}y^{-1}-y^{-1}y^{\prime}+y^{\prime-1}y-yy^{\prime-1})\rvert),$

where $y^{\prime}=xyx^{-1}$ . Furthermore, using

	$\displaystyle y^{\prime}y^{-1}-y^{-1}y^{\prime}+y^{\prime-1}y-yy^{\prime-1}$
	$\displaystyle=(y^{\prime}y^{-1}y^{\prime-1}-y^{\prime-1}y^{\prime}y^{-1})y^{\prime}+y^{\prime-1}(yy^{\prime-1}y^{\prime}-y^{\prime}yy^{\prime-1})$
	$\displaystyle=((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))y^{\prime}$
	$\displaystyle+y^{\prime-1}((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1)),$

	$\displaystyle y^{\prime}y^{-1}-yy^{\prime-1}$
	$\displaystyle=(y^{\prime}y^{-1}-1)(yy^{\prime-1}-1)+2y^{\prime}y^{-1}-2$
	$\displaystyle=-(y^{\prime}y^{-1}-1)(yy^{\prime-1}-1)-2yy^{\prime-1}+2,$

we obtain

	$\displaystyle 2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-yy^{\prime-1})(y^{\prime}y^{-1}-y^{-1}y^{\prime}+y^{\prime-1}y-yy^{\prime-1})\rvert)$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-yy^{\prime-1})(((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))y^{\prime}$
	$\displaystyle+y^{\prime-1}((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1)))\rvert)$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-1)(yy^{\prime-1}-1)(((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))y^{\prime}$
	$\displaystyle-y^{\prime-1}((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1)))\rvert)$
	$\displaystyle+4h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-1)((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))y^{\prime}$
	$\displaystyle-(yy^{\prime-1}-1)y^{\prime-1}((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1))\rvert)$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-1)(yy^{\prime-1}-1)(((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))y^{\prime}$
	$\displaystyle-y^{\prime-1}((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1)))\rvert)$
	$\displaystyle+4h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-1)((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))$
	$\displaystyle-(yy^{\prime-1}-1)((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1))\rvert)$
	$\displaystyle+4h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-1)((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))(y^{\prime}-1)$
	$\displaystyle-(yy^{\prime-1}-1)(y^{\prime-1}-1)((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1))\rvert)$

Using $\lvert z_{1}z_{2}\rvert=\lvert z_{2}z_{1}\rvert$ for $z_{1},z_{2}\in\mathbb{Q}\pi_{1}(\Sigma)$ , we have

	$\displaystyle e^{(1)}_{\Sigma_{1,1}}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2})-(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2}$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-1)(yy^{\prime-1}-1)(((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))y^{\prime}$
	$\displaystyle-y^{\prime-1}((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1)))\rvert)$
	$\displaystyle+4h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(y^{\prime}y^{-1}-1)((y^{\prime}y^{-1}-1)(y^{\prime-1}-1)-(y^{\prime-1}-1)(y^{\prime}y^{-1}-1))(y^{\prime}-1)$
	$\displaystyle-(yy^{\prime-1}-1)(y^{\prime-1}-1)((yy^{\prime-1}-1)(y^{\prime}-1)-(y^{\prime}-1)(yy^{\prime-1}-1))\rvert).$

Recalling the notations

	$\displaystyle y^{\prime}y^{-1}=\gamma_{\alpha}\gamma_{\beta}\gamma_{\alpha}^{-1}\gamma_{\beta}^{-1}=\gamma_{\partial},$
	$\displaystyle yy^{\prime-1}=\gamma_{\partial}^{-1},$
	$\displaystyle y=\gamma_{\beta},$
	$\displaystyle y^{\prime}=\gamma_{\alpha}\gamma_{\beta}\gamma_{\alpha}^{-1}={\gamma^{\prime}_{\beta}},$

we get

	$\displaystyle e^{(1)}_{\Sigma_{1,1}}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2})-(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2}$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)(((\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)-({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1)){\gamma^{\prime}_{\beta}}$
	$\displaystyle-{\gamma^{\prime}_{\beta}}^{-1}((\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)-({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)))\rvert)$
	$\displaystyle+4h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma_{\partial}-1)((\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)-({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1))({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)((\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)-({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1))\rvert)$

Finally, we can expand it into

	$\displaystyle e^{(1)}_{\Sigma_{1,1}}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2})-(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2}$
	$\displaystyle=2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1){\gamma^{\prime}_{\beta}}$
	$\displaystyle-(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1){\gamma^{\prime}_{\beta}}$
	$\displaystyle-(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1){\gamma^{\prime}_{\beta}}^{-1}(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1){\gamma^{\prime}_{\beta}}^{-1}({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)))\rvert)$
	$\displaystyle+4h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)\rvert),$

as desired.

∎

Using the lemma, we will state the theorem.

Theorem 9.2.

We use the notation in Theorem 8.1. Let $e:\Sigma_{1,1}\times I\to\Sigma\times I$ be an embedding satisfying $e_{*}(\gamma_{\alpha}\gamma_{\beta}\gamma_{\alpha}^{-1}\gamma_{\beta}^{-1})\in 1+I_{\mathbb{Q}\pi_{1}(\Sigma\times I)}^{N}$ , and $K_{\partial}$ the boundary knot in $\Sigma_{1,1}\times I$ whose Seifert surface is $\Sigma_{1,1}\times\{\frac{1}{2}\}$ , respectively. Then we have

	$\displaystyle\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))=\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e(K_{\partial})(-\epsilon)))$
	$\displaystyle=\epsilon\lvert\frac{1}{2}(\log(e_{*}(\gamma_{\partial}))^{2})\rvert\mod F^{2N+2}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma),$
	$\displaystyle\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))-\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e(K_{\partial})(-\epsilon)))$
	$\displaystyle=\lvert(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)-(e_{}(\gamma_{\partial})-1)^{2}(e_{}(\gamma_{\beta})-1)^{2}\rvert$
	$\displaystyle\mod F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma).$

In other words, for $\star_{1},\star_{2}\in\partial\Sigma$ , we have

	$\displaystyle(\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon))_{}=(\Sigma\times I)(e(K_{\partial})(-\epsilon))_{}=\exp(\sigma(\epsilon\lvert\frac{1}{2}(\log(e_{*}(\gamma_{\partial})))^{2}\rvert)):$
	$\displaystyle\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})/F^{2N+1}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})/F^{2N+1}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2}),$
	$\displaystyle(\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon))_{}\circ((\Sigma\times I)(e(K_{\partial})(-\epsilon))_{})^{-1}$
	$\displaystyle=\exp(\sigma(\lvert(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)-(e_{}(\gamma_{\partial})-1)^{2}(e_{}(\gamma_{\beta})-1)^{2}\rvert)):$
	$\displaystyle\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star)/F^{2N+2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star)\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})/F^{2N+2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2}).$

Proof.

The first formula and the third formula are special ones in Theorem 8.1. We will prove the second equation using Theorem 8.5.

By Theorem 8.5, we have

	$\displaystyle\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))-\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e(K_{\partial})(-\epsilon)))$
	$\displaystyle=(\epsilon L_{1}(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial}))+L_{2}(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})))$
	$\displaystyle-(\epsilon L_{1}(e(K_{\partial}))+L_{2}(e(K_{\partial})))\mod F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$
	$\displaystyle=L_{2}(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial}))-L_{2}(e(K_{\partial}))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\frac{1}{2h}((e_{}\circ e^{(1)}_{\Sigma_{1},1}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert))^{2}))$
	$\displaystyle-(e_{}\circ e^{(1)}_{\Sigma_{1},1}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert)))^{2})$
	$\displaystyle-\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\frac{1}{2h}((e_{}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert))^{2}))-(e_{}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert)))^{2})).$

Using

e_{*}\circ e^{(1)}_{\Sigma_{1},1*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert))=e_{*}(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert)),

we have

	$\displaystyle\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))-\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e(K_{\partial})(-\epsilon)))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\frac{1}{2h}((e_{}\circ e^{(1)}_{\Sigma_{1},1}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert))^{2})))$
	$\displaystyle-(e_{*}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\frac{1}{2}(\log\gamma_{\partial})^{2}\rvert))^{2})))\mod F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma).$

Since $\frac{1}{2}(\log(X))^{2}=\frac{1}{2}(X-2+X^{-1})$ , by Corollary 7.7, we obtain

	$\displaystyle\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))-\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e(K_{\partial})(-\epsilon)))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\frac{1}{8h}e_{*}(e^{(1)}_{\Sigma_{1,1}}((\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2})-(\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert\gamma_{\partial}-2+\gamma_{\partial}^{-1}\rvert))^{2}))$
	$\displaystyle\mod F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma).$

Using Lemma 9.1, we continue the computation. By the lemma, we have

	$\displaystyle\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))-\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e(K_{\partial})(-\epsilon)))$
	$\displaystyle=\Psi_{\mathcal{A}^{\prime}}^{\mathbb{Q}\lvert{\pi}_{1}\rvert}(\frac{1}{8}e_{*}($
	$\displaystyle 2h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1){\gamma^{\prime}_{\beta}}$
	$\displaystyle-(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1){\gamma^{\prime}_{\beta}}$
	$\displaystyle-(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1){\gamma^{\prime}_{\beta}}^{-1}(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1){\gamma^{\prime}_{\beta}}^{-1}({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)))\rvert)$
	$\displaystyle+4h\Psi_{\mathbb{Q}\lvert{\pi}_{1}\rvert}^{\mathcal{A}^{\prime}}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)\rvert))\mod F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$
	$\displaystyle=\frac{1}{4}e_{*}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1){\gamma^{\prime}_{\beta}}$
	$\displaystyle-(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1){\gamma^{\prime}_{\beta}}$
	$\displaystyle-(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1){\gamma^{\prime}_{\beta}}^{-1}(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}-1)(\gamma_{\partial}^{-1}-1){\gamma^{\prime}_{\beta}}^{-1}({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)))\rvert)$
	$\displaystyle-\frac{1}{2}e_{*}(\lvert(\gamma_{\partial}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}^{-1}-1)({\gamma^{\prime}_{\beta}}^{-1}-1)({\gamma^{\prime}_{\beta}}-1)(\gamma_{\partial}^{-1}-1)\rvert))\mod F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$

where $\gamma^{\prime}_{\beta}\stackrel{{\scriptstyle\mathrm{def.}}}{{=}}\gamma_{\alpha}\gamma_{\beta}\gamma_{\alpha}^{-1}$ . Furthermore, since

	$\displaystyle e_{*}(\gamma_{\partial}-1)\in F^{N}\mathbb{Q}\pi_{1}(\Sigma),$
	$\displaystyle e_{}(\gamma_{\partial}-1)=e_{}(\gamma_{\partial}^{-1}-1)\mod F^{2N}\mathbb{Q}\pi_{1}(\Sigma),$
	$\displaystyle e_{}({\gamma^{\prime}_{\beta}}-1)=-e_{}({\gamma^{\prime}_{\beta}}^{-1}-1)=e_{}(\gamma_{\beta}-1)=-e_{}(\gamma_{\beta}^{-1}-1)\mod F^{2}\mathbb{Q}\pi_{1}(\Sigma),$

we obtain

	$\displaystyle\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))-\zeta_{\mathbb{Q}}\lvert{\pi}_{1}\rvert((\Sigma\times I)(e(K_{\partial})(-\epsilon)))$
	$\displaystyle=\frac{1}{2}e_{*}(\lvert-(\gamma_{\partial}-1)(\gamma_{\partial}-1)({\gamma_{\beta}}-1)({\gamma_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}-1)({\gamma_{\beta}}-1)(\gamma_{\partial}-1)({\gamma_{\beta}}-1)$
	$\displaystyle+(\gamma_{\partial}-1)({\gamma_{\beta}}-1)(\gamma_{\partial}-1)({\gamma_{\beta}}-1)$
	$\displaystyle-(\gamma_{\partial}-1)({\gamma_{\beta}}-1)({\gamma_{\beta}}-1)(\gamma_{\partial}-1)\rvert))\mod F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$
	$\displaystyle=\lvert(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)-(e_{}(\gamma_{\partial})-1)^{2}(e_{}(\gamma_{\beta})-1)^{2}\rvert.$

It proves the second formula.

Finally, using the second formula, we verify the fourth formula. By definition of the Baker-Campbell-Hausdorff series, we obtain

	$\displaystyle(\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon))_{}\circ((\Sigma\times I)(e(K_{\partial})(-\epsilon))_{})^{-1}$
	$\displaystyle=\exp(\sigma(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))))\circ\exp(\sigma(-\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e(K_{\partial})(-\epsilon)))))$
	$\displaystyle=\exp(\sigma(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))))\circ\exp(\sigma(-\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e(K_{\partial})(-\epsilon)))))$
	$\displaystyle=\exp(\sigma(\mathrm{bch}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))),-\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e(K_{\partial})(-\epsilon)))))).$

By the second formula, we have

	$\displaystyle\mathrm{bch}(\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon)))),-\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e(K_{\partial})(-\epsilon))))$
	$\displaystyle=\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}(((\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon))))-\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(e(K_{\partial})(-\epsilon)))\mod F^{4N-2}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma)$
	$\displaystyle=\lvert(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)-(e_{}(\gamma_{\partial})-1)^{2}(e_{}(\gamma_{\beta})-1)^{2}\rvert$
	$\displaystyle\mod F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma).$

Since $\sigma(F^{2N+3}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma))\subset F^{2N+2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma)$ for any $N\in\mathbb{Z}_{\geq 0}$ , we get

	$\displaystyle(\Sigma\times I)(e\circ e^{(1)}_{\Sigma_{1},1}(K_{\partial})(-\epsilon))_{}\circ((\Sigma\times I)(e(K_{\partial})(-\epsilon))_{})^{-1}$
	$\displaystyle=\exp(\sigma(\lvert(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)(e_{}(\gamma_{\partial})-1)(e_{}(\gamma_{\beta})-1)-(e_{}(\gamma_{\partial})-1)^{2}(e_{}(\gamma_{\beta})-1)^{2}\rvert)):$
	$\displaystyle\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star)/F^{2N+2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star)\to\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})/F^{2N+2}\widehat{\mathbb{Q}\pi_{1}}(\Sigma,\star_{1},\star_{2})$

as desired.

∎

Remark 9.3.

The theorem verifies that the number $2m+2$ in Theorem 8.1 is the best possible. In other words, there exist two boundary knots $K,K^{\prime}$ having the same homotopy type in $\lvert\pi_{1}\rvert(\Sigma)\cap\lvert 1+I_{\mathbb{Q}\pi_{1}(\Sigma)}^{m}\rvert$ such that

\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K(-\epsilon))-\widetilde{\zeta}_{\mathbb{Q}\lvert{\pi}_{1}\rvert}((\Sigma\times I)(K^{\prime}(-\epsilon))\neq 0\mod F^{2m+3}\widehat{\mathbb{Q}\pi_{1}}(\Sigma).

References

[1] S. Garoufalidis and J. Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, In: Graphs and patterns in mathematics and theoretical physics, volume 73 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence 2005, 173–203.
[2] W. M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface groups representations, Invent. Math. 85, 263-302(1986).
[3] N. Habegger, Milnor, Johnson, and tree level perturbative invariants, preprint.
[4] D. Johnson, An abelian quotient of the mapping class group $\mathcal{I}_{g}$ , Math. Ann. 249, 225-242(1980).
[5] N. Kawazumi and Y. Kuno, The logarithms of Dehn twists, Quantum Topology, Vol. 5(2014), Issue 3, pp. 347–423.
[6] N. Kawazumi and Y. Kuno, Groupoid-theoretical methods in the mapping class groups of surfaces, arXiv: 1109.6479 (2011), UTMS preprint: 2011–28.
[7] Y. Kuno and G. Massuyeau, Generalized Dehn twists on surfaces and homology cylinders. Preprint, arXiv:1902.02592.
[8] W. B. R. Lickorish, A representation of orientable combinatorial $3$ -manifolds, Ann. of Math (2) 76 (9152), 531-540.
[9] G. Massuyeau and V. Turaev, Fox pairings and generalized Dehn twists, Ann. Inst. Fourier 63 (2013) 2403-2456.
[10] S. Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. I, Topology 28 (1989) 305–323.
[11] S. Morita, On the structure of the Torelli group and the Casson invariant, Topology, Volume 30(1991), 603-621.
[12] S. Morita, Structures of the mapping class group of surface: a survey and a prospect, Proceedings of the Kirbyfest Geom. Topol.Monogr., 2 (1998) 349-406.
[13] J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170-181.
[14] S. Tsuji, A formula for the action of Dehn twists on HOMFLY-PT skein modules and its applications, preprint, arXiv:1801.00580.
[15] V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Ecole Norm. Sup. (4) 24 (1991), no. 6, 635-704.

	$\displaystyle v_{k+2}(x)-v_{k+1}(x)=\kappa_{}(\mathrm{bch}(-\iota_{1}(v_{k+1}(x)),x))$
	$\displaystyle=-\kappa_{}\circ\iota_{1}(v_{k+1}(x))+\kappa_{}(x)+(\kappa_{}\circ\iota_{1}(v_{k+1}(x))-\kappa_{}(x)+\kappa_{}(\mathrm{bch}(-\iota_{1}(v_{k+1}(x)),x)))$
	$\displaystyle=-\kappa_{}\circ\iota_{1}(v_{k+1}(x))+\kappa_{*}(x)$
	$\displaystyle+(\kappa_{}\circ\iota_{1}(v_{k}(x))-\kappa_{}(x)+\kappa_{}(\mathrm{bch}(-\iota_{1*}(v_{k}(x)),x)))\mod F^{3+k}\widehat{\mathbb{Q}\lvert\pi_{1}\rvert}(\Sigma_{\mathrm{st}})$
	$\displaystyle=-v_{k+1}(x)+\kappa_{}(x)+(v_{k}(x)-\kappa_{}(x)+\kappa(\mathrm{bch}(-\iota_{1*}(v_{k}(x)),x)))$
	$\displaystyle=-v_{k+1}(x)+v_{k}(x)+\kappa(\mathrm{bch}(-\iota_{1*}(v_{k}(x)),x))=0$

	$\displaystyle\kappa^{\prime}_{}(xy)=\kappa^{\prime}_{}(x\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y))=\kappa^{\prime}_{}(x\iota^{\prime}_{0}\circ\kappa^{\prime}_{*}(y)),$
	$\displaystyle\kappa^{\prime}_{}(yx)=\kappa^{\prime}_{}(y\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(x))=\kappa^{\prime}_{}(y\iota^{\prime}_{0}\circ\kappa^{\prime}_{*}(x)),$
	$\displaystyle\kappa^{\prime}_{}(x)\kappa^{\prime}_{}(y)=\kappa^{\prime}_{}(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(x)y)=\kappa^{\prime}_{}(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(y)x),$
	$\displaystyle\kappa^{\prime}_{}(y)\kappa^{\prime}_{}(x)=\kappa^{\prime}_{}(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)x)=\kappa^{\prime}_{}(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x)y)$

	$\displaystyle h\kappa^{\prime}_{*}(\sigma(x)(y))$
	$\displaystyle=\kappa^{\prime}_{*}(xy-yx)$
	$\displaystyle=\kappa^{\prime}_{}(x\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)-y\iota^{\prime}_{0}\circ\kappa^{\prime}_{*}(x))$
	$\displaystyle=(\kappa^{\prime}_{}(x\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y))-\kappa^{\prime}_{}(y)\kappa^{\prime}_{}(x))+(\kappa^{\prime}_{}(y)\kappa^{\prime}_{}(x)-\kappa^{\prime}_{}(y\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x)))$
	$\displaystyle=\kappa^{\prime}_{}(x\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)-\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)x)+\kappa^{\prime}_{}(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x)y-y\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))$
	$\displaystyle=h(\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}(y)))+\kappa^{\prime}_{}(\sigma(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))(y)))$

	$\displaystyle\kappa^{\prime}_{}((\sigma(x))^{n}(\iota^{\prime}_{0}(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}((\sigma(x))^{n-1}(\iota^{\prime}_{0}(y)))))+\kappa^{\prime}_{}(\sigma(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))(\sigma(x))^{n-1}(\iota^{\prime}_{0}(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}((\sigma(x))^{n-1}(\iota^{\prime}_{0}(y))))),$
	$\displaystyle\kappa^{\prime}_{}((\sigma(x))^{n}(\iota^{\prime}_{1}(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}((\sigma(x))^{n-1}(\iota^{\prime}_{1}(y)))))+\kappa^{\prime}_{}(\sigma(\iota^{\prime}_{0}\circ\kappa^{\prime}_{}(x))(\sigma(x))^{n-1}(\iota^{\prime}_{1}(y)))$
	$\displaystyle=\kappa^{\prime}_{}(\sigma(x)(\iota^{\prime}_{1}\circ\kappa^{\prime}_{}((\sigma(x))^{n-1}(\iota^{\prime}_{1}(y))))).$

	$\displaystyle\kappa^{\prime}_{}\circ\chi_{}^{-1}\circ\iota^{\prime}_{0*}(x)$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-\zeta))(\iota^{\prime}_{0}(x)))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-\mathrm{bch}(\iota^{\prime}_{1}(v),v^{\prime}))(\iota^{\prime}_{0*}(x)))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-v^{\prime}))\circ\exp(\sigma(-\iota^{\prime}_{1}(v)))(\iota^{\prime}_{0*}(x)))$
	$\displaystyle=\kappa^{\prime}_{}(\exp(\sigma(-v^{\prime}))(\iota^{\prime}_{0}(x))),$

The total Johnson homomorphism on the homology cylinder and the bracket-quantization HOMFLY-PT skein algebra

Abstract.

Acknowledgment

1. Introduction and the main result

1.1. Back ground

1.2. The Goldman Lie algebra

1.3. Homology cylinders

Definition 1.1.

Proposition 1.2 (Proposition 4.9).

Definition 1.3.

1.4. Main result

Theorem 1.4.

1.5. An application

2. Review of the Goldman Lie algebra

Theorem 2.1 ([5] [6] [9]).

Proposition 2.2.

Proposition 2.3.

Proof.

3. The bracket-quantization HOMFLY-PT skein algebra

Definition 3.1.

Definition 3.2.

Theorem 3.3.

Proposition 3.4.

Proof.

4. Homology cylinders

Definition 4.1.

Proposition 4.2.

Proof.

Remark 4.3.

“A construction using a boundary link”

Definition 4.4.

Proposition 4.5 ([3] p.4 Theorem 2.5, p.17 Lemma 6.1).

“A construction using a standard embedding”

Definition 4.6.

Lemma 4.7.

Proof.

Lemma 4.8.

Proposition 4.9.

Proof.

5. The main result

Remark 5.1.

“A construction using a standard embedding”

Definition 5.2.

Proposition 5.3.

Proof.

Theorem 5.4.

“A construction using a boundary link”

Theorem 5.5.

6. Proof of the main result

Proposition 6.1.

Proof.

Lemma 6.2.

Proof.

proof of Theorem 5.4.

Lemma 6.3.

Proof.

proof of Theorem 5.5.

7. A lemma of 𝒜′\mathcal{A}^{\prime}

Lemma 7.1.

Proof.

Lemma 7.2.

Proof.

Lemma 7.3.

Proof.

Lemma 7.4.

Proof.

Lemma 7.5.

Proof.

Theorem 7.6.

Proof.

Corollary 7.7.

Proof.

8. Some formulas of the action of some homology cylinders

Theorem 8.1 ([7]).

Remark 8.2.

Proof.

Lemma 8.3.

Proof.

Definition 8.4.

Theorem 8.5.

7. A lemma of $\mathcal{A}^{\prime}$