Weakly framed surface configurations, Heisenberg homology and Mapping Class Group action

It is shown in [8] that the braid group of an oriented surface with one boundary component has a natural quotient isomorphic to the Heisenberg group. From this one obtains homologies with coefficients in any representation of the Heisenberg group. The Mapping Class Group acts on the local coefficients and in general there is a twisted action of the Mapping Class Groups on Heisenberg homologies. For specific representations, including the famous Shrödinger one, the Mapping Class Group action can be untwisted, producing a native representation of a central extension.

In the closed case a similar quotient exists in genus $1$, but produces a version of the Heisenberg group with finite center in higher genus [9], or more involved metabelian quotients [1]. Here we will recover an homomorphism to the full Heisenberg group by replacing the surface braid group by a central extension realized using an $S^{1}$-bundle over the configuration space. Elements of this bundle will be called weakly framed configurations, and its fundamental group named the weakly framed braid group. We obtain a presentation for this newly defined group and a quotient homomorphism to an Heisenberg group with infinite cyclic center. We then define homologies of weakly framed configurations with coefficients in any representation of the Heisenberg group. Finally we construct a twisted action of what we call the $\mathtt{f}$-based Mapping Class Group, a central extension of the Mapping Class Group of the punctured surface whose elements are represented by diffeomorphisms fixing the weakly framed set of punctures.

The Heisenberg group in genus $g$ is usually realized as a group of $(g+2)\times(g+2)$ matrices. In Section 4 we introduce the linearised regular representation which achieves it as a group of $(2g+2)\times(2g+2)$ matrices. Using this representation as local coefficients, we obtain a native representation (no twisting) of the $\mathtt{f}$-based Mapping Class Group.

A famous result of Bigelow [6] and Krammer [11] states that the classical braid groups which are Mapping Class Groups in genus zero are linear. Bigelow’s proof uses an homological action on the $2$-points configuration space in the punctured disc. Then Bigelow and Budney [7] deduced that the mapping class group of the closed orientable surface of genus $2$ is also linear. We speculate that our representations can be used for the linearity problem in higher genus.

Let $\Sigma_{g}$, $g\geqslant 1$, be a closed oriented genus $g$ surface. For $n\geqslant 2$, the unordered configuration space of $n$ points in $\Sigma_{g}$ is

The surface braid group is then defined as $\mathbb{B}_{n}(\Sigma_{g})=\pi_{1}(\mathcal{C}_{n}(\Sigma_{g}),*)$. Here $*=\{*_{1},\dots,*_{n}\}$ is a base configuration. A presentation for this group was first obtained by G. P. Scott [13] and revisited by Gonzáles-Meneses [10], Bellingeri [2]. The braid group of a bounded surface has a natural quotient isomorphic to the Heisenberg group of the surface. This is proved in [8] for a surface with one boundary component. In the closed case with genus $g>1$ a similar quotient produces a version of the Heisenberg group with finite center; see [5, Section 5, Example 1] in case $n\geqslant 3$. We will recover the full Heisenberg group by using an $S^{1}$-bundle over the configuration space. Let us equip $\Sigma_{g}$ with a riemannian metric (the choice is irrelevant). This determines a conformal structure on $\Sigma_{g}$, which is equivalent to a complex structure. Then the configuration space $\mathcal{C}_{n}(\Sigma_{g})$ inherits a complex structure with hermitian metric and a symplectic structure.

Using the complex structure we may define various bundles over the configuration space $\mathcal{C}_{n}(\Sigma_{g})$. We have the complex tangent bundle $T_{\mathbb{C}}(\mathcal{C}_{n}(\Sigma_{g}))$, its determinant $\Delta(\mathcal{C}_{n}(\Sigma_{g}))=\Lambda^{n}(T_{\mathbb{C}}(\mathcal{C}_{n}(\Sigma_{g}))$, the square determinant $\Delta^{2}(\mathcal{C}_{n}(\Sigma_{g}))=\Delta(\mathcal{C}_{n}(\Sigma_{g}))^{\otimes 2}$.

Using the symplectic structure we also have the lagrangian grassmannian bundle $\mathcal{L}(\mathcal{C}_{n}(\Sigma_{g}))$, whose fiber is the grassmanian of lagrangian $n$-spaces in $\mathbb{C}^{n}=\mathbb{R}^{2n}$ which can be identified with $U(n)/O(n)$. We have a square determinant map $\det^{2}:\mathcal{L}(\mathcal{C}_{n}(\Sigma_{g}))\rightarrow\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g})$, which allows to consider $\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g})$ as a quotient of the lagrangian grassmannian bundle $\mathcal{L}(\mathcal{C}_{n}(\Sigma_{g}))$.

The framed braid group of surfaces $FB_{n}(\Sigma_{g})$ is studed in [3]. It is defined as the fundamental group of the space $F_{n}(\Sigma_{g})$ of $n$-points configurations with a unit tangent vector at each point. A framing generates a lagrangian subspace which gives a map $FB_{n}(\Sigma_{g})\rightarrow\mathcal{L}(\mathcal{C}_{n}(\Sigma_{g}))$. Composing with the projection we obtain a fibration $FB_{n}(\Sigma_{g})\rightarrow\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g})$ whose fiber is the kernel of $\det^{2}:(S^{1})^{n}\rightarrow S^{1}$. We can deduce an homomorphism $FB_{n}(\Sigma_{g})\rightarrow\mathbb{B}^{\mathtt{f}}_{n}(\Sigma_{g})$ whose image is an index $2$ subgroup which identifies each framing generator with $F^{2}$, where $F$ is the weak framing generator (see below for a definition). A presentation of our weakly framed braid group $\mathbb{B}^{\mathtt{f}}_{n}(\Sigma_{g})$ can be then deduced from [3, Theorem 13]. We will give below a short proof which will clarify our conventions and choice of generators.

We fix a decomposition of $\Sigma_{g}$ as a disc with $2g$ handles of index $1$, which gives $\Sigma_{g,1}$, completed by a final handle of index $2$. The based loops, $\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g}$ are depicted in Figure 1.

The boundary of the $2$-handle gives a loop homotopic to $\delta^{-1}$ with $\delta=\beta_{g}\overline{\alpha}_{g}\overline{\beta}_{g}\alpha_{g}\dots\beta_{1}\overline{\alpha}_{1}\overline{\beta}_{1}\alpha_{1}$, which gives the relation in $\pi_{1}(\Sigma_{g},*_{1})$. Here we write the composition of loops from right to left. We fix a unit vector field $X$ on $\Sigma_{g,1}$. The loops $\alpha_{i}$, $\beta_{i}$, $1\leqslant i\leqslant g$, represent free generators for $\pi_{1}(\Sigma_{g,1},*_{1})$. Here the base point $*_{1}$ belongs to the base configuration $*$. We will use the same notation $\alpha_{i}$, $\beta_{i}$, $1\leqslant i\leqslant g$, for the corresponding braids where the weak framing is given by the square determinant of the framing obtained using $X$ at each point in the configuration. We have classical (positive) generators $\sigma_{1}$, …,$\sigma_{n-1}$ where the weak framing is also given by $X$. We have a weak framing generator $F$, which rotate counterclockwise the framing vector by $\pi$ around $*_{1}$. The choice of the vector field $X$ is irrelevant in the presentation, but will be needed when acting with mapping classes. For comparing with Bellingeri and al. relations, note that they involve the negative classical generators; see e.g. [2, Fig. 1].

In genus $1$, the braid group $\mathbb{B}_{n}(\Sigma_{1})$ quotiented by $\sigma_{1}$ made central is isomorphic to the standard discrete Heisenberg group, and the construction from [8] applies. In this section we will suppose $g>1$ and consider a version of the discrete Heisenberg group designed for our situation. The Heisenberg group $\mathcal{H}_{g}$ is $\mathbb{Z}\nu\times H_{1}(\Sigma_{g},\mathbb{Z})$ with $\nu=\frac{\gcd(2g-2,g+n-1)}{2g-2}\in\mathbb{Q}$, and operation

It will be convenient to further embed this group in the rational or real Heisenberg groups $\mathcal{H}_{\mathbb{Q}}$, $\mathcal{H}_{\mathbb{R}}$ which motivate a formulation where the center is identified with $\mathbb{Z}\nu$ rather than $\mathbb{Z}$. For $g>1$, we will use the notation $a_{i}$, $b_{i}$ for the homology classes of $\alpha_{i}$, $\beta_{i}$, $1\leqslant i\leqslant g$.

Using the homomorphism $\phi$ we define a regular covering $\widetilde{\mathcal{C}}^{\mathtt{f}}_{n}(\Sigma_{g})$ of the weakly framed configuration space ${\mathcal{C}}^{\mathtt{f}}_{n}(\Sigma_{g})$. The homology of this cover is what we call the Heisenberg homology. Deck transformations endow Heisenberg homology with a right module structure over the group ring $\mathbb{Z}[\mathcal{H}_{g}]$. We may specialise to local coefficients as follows. Let us denote by $S_{*}(\widetilde{\mathcal{C}}^{\mathtt{f}}_{n}(\Sigma_{g}))$ the singular chain complex of the Heisenberg cover, which is a right $\mathbb{Z}[\mathcal{H}_{g}]$-module. Given a representation $\rho:\mathcal{H}_{g}\rightarrow GL(V)$, the corresponding local homology is that of the complex $S_{*}(\mathcal{C}^{\mathtt{f}}_{n}(\Sigma),V):=S_{*}(\widetilde{\mathcal{C}}^{\mathtt{f}}_{n}(\Sigma))\otimes_{\mathbb{Z}[\mathcal{H}_{g}]}V$. It will be called the Heisenberg homology of weakly framed surface configurations with coefficients in $V$.

It is convenient to also consider Borel-Moore homology

the inverse limit is taken over all compact subsets $T\subset\mathcal{C}_{n}(\Sigma_{g})$, and $T^{\mathtt{f}}\subset\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g})$ denotes the corresponding weakly framed configurations.

Recall that $*=\{*_{1},\dots,*_{n}\}\in\mathcal{C}_{n}(\Sigma_{g})$, $g\geqslant 2$, $n\geqslant 2$, is the base $n$-points configuration. We denote by $\mathfrak{M}(\Sigma_{g},*)$ the Mapping Class Group of the punctured surface. By a theorem of Moser [12], we may work with representatives of mapping classes which are area preserving, equivalently in this dimension with symplectomorphisms. Let us denote by $\mathcal{C}_{n}(f)$ the diffeomorphism of $\mathcal{C}_{n}(\Sigma_{g})$ corresponding to a symplectomorphism $f$ which fixes the base configuration. It is a symplectomorphism giving an action on the lagrangian bundle $\mathcal{L}(\mathcal{C}_{n}(\Sigma_{g}))$. We obtain an induced action $\mathcal{C}^{\mathtt{f}}_{n}(f)$ on the square determinant quotient $\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g})$. We denote by $*^{\mathtt{f}}$ the base configuration with a choice of weak framing, i.e. an inverse image of $*$ in $\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g})$. We will consider here an extension of the Mapping Class Group obtained with isotopy classes of symplectomorphism fixing the base configuration with weak framing $*^{\mathtt{f}}$.

We fix a lift $\tilde{*}^{\mathtt{f}}\in\widetilde{\mathcal{C}}^{\mathtt{f}}_{n}(\Sigma_{g})$ of the weakly framed base configuration $*^{\mathtt{f}}$.

The above argument also proves the following, which can be seen as an extension of similar results on surface braid groups [1, 4, 8].

The isotopy between the identity and the half twist around $*_{1}$ rotates the framing at $*_{1}$ by $\pi$, which generates $\pi_{1}$ of the fiber. This identifies the kernel generator. This half twist commutes with symplectomorphisms which are identity on a disc neighbourhood of the base configuration $*$. One can check that it also commutes up to isotopy fixing $*^{\mathtt{f}}$ with symplectomorphisms supported in a disc containing $*$, which are classical braids. Composing with classical braids any symplectomorphism fixing $*^{\mathtt{f}}$ is isotopic to one which is the identity on a disc neighbourhood of $*$. Centrality follows.   

We denote by $\mathrm{Aut}^{+}(\mathcal{H}_{g})$ the group of oriented automorphisms of $\mathcal{H}_{g}$ which means automorphisms which are identity on the center. We have an action of the $\mathtt{f}$-based Mapping Class Group on the Heisenberg group $\mathcal{H}_{g}$, $\Psi\colon\mathfrak{M}^{\mathtt{f}}(\Sigma_{g},*^{\mathtt{f}})\longrightarrow\mathrm{Aut}^{+}(\mathcal{H}_{g})$, $f\mapsto f_{\mathcal{H}}$. The quotient of $\mathcal{H}_{g}$ by its center is equal to $H_{1}(\Sigma_{g},\mathbb{Z})$ hence every oriented automorphism $\tau\in\mathrm{Aut}^{+}(\mathcal{H}_{g})$ induces an automorhism of $H_{1}(\Sigma_{g},\mathbb{Z})$. The triviality of the action on the center implies that the induced map $\overline{\tau}$ is symplectic, so we have an homomorphism $\mathrm{Aut}^{+}(\mathcal{H}_{g})\to Sp(H_{1}(\Sigma_{g},\mathbb{Z}))$. This homomorphism has a section and its kernel is isomorphic to $\mathrm{Hom}(H_{1}(\Sigma_{g},\mathbb{Z}),\mathbb{Z}\nu)\cong H^{1}(\Sigma_{g},\mathbb{Z}\nu)$, see [8, Lemma 16]. This identifies the group $\mathrm{Aut}^{+}(\mathcal{H}_{g})$ as a semidirect product

The action of a $\mathtt{f}$-based mapping class $f\in\mathfrak{M}^{\mathtt{f}}(\Sigma_{g},*^{\mathtt{f}})$ on $\mathcal{H}_{g}$ writes down

where $\delta:\mathfrak{M}^{\mathtt{f}}(\Sigma_{g},*^{\mathtt{f}})\rightarrow H^{1}(\Sigma_{g},{\mathbb{Z}}\nu)$ is a crossed homomorphism, i.e. for all $f,g\in\mathfrak{M}(\Sigma_{g})$ we have $\delta_{g\circ f}=\delta_{f}+f^{*}(\delta_{g})$, see [8, Section 3.3].

From proposition 4 we obtain for $f\in\mathfrak{M}^{\mathtt{f}}(\Sigma_{g},*^{\mathtt{f}})$ an homology isomorphism

which is $\mathbb{Z}$-linear. This provides a representation of the $\mathtt{f}$-based Mapping Class Group

This representation is twisted with respect to the right $\mathbb{Z}[\mathcal{H}_{g}]$-module structure, which means that for $x\in H_{*}(\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g}),\mathbb{Z})$, $h\in\mathcal{H}_{g}$, we have

For a representation $\rho:\mathcal{H}_{g}\rightarrow GL(V)$ and automorphism $\tau\in\mathrm{Aut}^{+}(\mathcal{H}_{g})$, we denote by ${}_{\tau}\!V$ the twisted representation $\rho\circ\tau$. Recall that the homology with coefficient in $V$ is computed from the complex $S_{*}(\mathcal{C}^{\mathtt{f}}_{n}(\Sigma),V):=S_{*}(\widetilde{\mathcal{C}}^{\mathtt{f}}_{n}(\Sigma_{g}))\otimes_{\mathbb{Z}[\mathcal{H}_{g}]}V$.

In this section we obtain finite dimensional representations of the Mapping Class Groups from the left regular action on the Heisenberg group $\mathcal{H}_{g}^{\mathbb{Q}}$. The group $\mathcal{H}_{g}$ is a subgroup in $\mathcal{H}_{g}^{\mathbb{Q}}$. We endow $\mathcal{H}_{g}^{\mathbb{Q}}=\mathbb{Q}\times H_{1}(\Sigma_{g},\mathbb{Q})$ with affine structure isomorphic to $\mathbb{Q}^{2g+1}$. The left regular action $l_{(k_{0},x_{0})}$ is then an affine automorphism. We decompose $x_{0}=p_{0}+q_{0}$, $p_{0}\in\Lambda_{a}=\mathrm{Span}(a_{i},1\leqslant i\leqslant g)$, $q_{0}\in\Lambda_{b}=\mathrm{Span}(b_{i},1\leqslant i\leqslant g)$, then the action is written

We consider the linearisation $\rho_{L}$ of this affine action on $L=\mathcal{H}_{g}^{\mathbb{Q}}\oplus\mathbb{Q}$. The linear action of $\rho_{L}(k_{0},x_{0})$ is as follows.

The nice feature of this representation is that the twisted representation ${}_{\tau}\!L$ is canonically isomorphic to $L$.

Composing the homology isomorphism induced by the intertwinning of representations with the twisted action from Theorem 8, we obtain a natural homological action of $\mathtt{f}$-based mapping classes by automorphisms.

In [8] it is proved that a relative Borel-Moore Heisenberg homology of configurations in $\Sigma_{g,1}$ is free of finite dimension over the group ring of the Heisenberg group. The argument does not work for closed surfaces. A more careful analysis of a cell decomposition of weakly framed configurations is likely to be needed. We first quote that $\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g})$ has the homotopy type of a finite CW-complex. Indeed, we get the same homotopy type if we consider weakly framed configurations where points cannot be $\epsilon$-closed with $\epsilon$ small enough, i.e we replace the condition $x_{i}\neq x_{j}$ by $d(x_{i},x_{j})\geqslant\epsilon$, $i\neq j$, and get a compact manifold with boundary $\mathcal{C}^{\epsilon,\mathtt{f}}_{n}(\Sigma_{g})$. It follows that for finite dimensional representations of the Heisenberg group, the obtained homologies are finite dimensional.

It is exciting to analyse submanifolds representing cycles in Heisenberg homologies, expecting that certain family could generate a subspace invariant under Mapping Class Group action. Let us denote by $\mathcal{C}^{\mathtt{f}}_{*_{1},n-1}(\Sigma_{g})$ the subspace of weakly framed configurations containing the point $*_{1}$. For a partition $n=n_{1}+n_{2}+\dots+n_{2g-1}+n_{2g}$, we obtain a cell formed with configurations having $n_{1}$, $n_{2}$, …,$n_{2g-1}$, $n_{2g}$ points respectively on $\alpha_{1}$, $\beta_{1}$, …, $\alpha_{g}$, $\beta_{g}$, weakly framed by the vector field $X$. This gives a properly embedded cell representing an homology class in $H^{BM}_{*}\bigl{(}\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g}),\mathcal{C}^{\mathtt{f}}_{*_{1},n-1}(\Sigma_{g}),V\bigr{)}$, where $\mathcal{C}^{\mathtt{f}}_{*_{1},n-1}(\Sigma_{g})$ denotes the subspace of $n$-points configurations containing $*_{1}$. We will obtain classes in $H_{n}\bigl{(}\mathcal{C}^{\mathtt{f}}_{n}(\Sigma_{g}),V)$ by studying the kernel of the boundary map

The case $n=2$ already looks promising.