Mapping class groups of circle bundles over a surface

By an oriented circle bundle we mean a fiber bundle

with structure group $\operatorname{Homeo}^{+}(S^{1})$, the group of orientation preserving homeomorphisms of $S^{1}$. The inclusion of the rotation group $\operatorname{SO}(2)$ in $\operatorname{Homeo}^{+}(S^{1})$ is a homotopy equivalence, so circle bundles are in bijection with rank-2 real vector bundles. The classifying space $B\operatorname{SO}(2)$ is homotopy equivalent to $\mathbb{C}P^{\infty}$, which is an Eilenberg–Maclane space $K(\mathbb{Z},2)$. Thus each circle bundle is uniquely determined up to isomorphism by its Euler class $eu(E)\in H^{2}(B;\mathbb{Z})$, which is the primary obstruction to a section of the bundle.

When $B=S_{g}$ is a closed, oriented surface, $H^{2}(S_{g};\mathbb{Z})\cong\mathbb{Z}$, we can speak of the Euler number. We use $X_{g}^{k}$ to denote the total space of the circle bundle

with Euler number $k$. For example, for the unit tangent bundle $eu(US_{g})=2-2g$ (the Euler characteristic), so $US_{g}\cong X_{g}^{2-2g}$. We also note that $X_{g}^{k}$ and $X_{g}^{-k}$ are homeomorphic 3-manifolds, since the sign of the Euler number of a circle bundle over $S_{g}$ depends on the choice of orientation on $S_{g}$.

From the long exact sequence of a fibration, we have an exact sequence

The group $\pi_{1}(X_{g}^{k})$ has a presentation

Using this, one finds $\langle z\rangle\cong\mathbb{Z}$ is the center of $\pi_{1}(X_{g}^{k})$ as long as $(g,k)\neq(1,0)$. When $g\geq 2$ follows from the fact that the group $\pi_{1}(S_{g})$ has trivial center; the case $g=1$ can be treated directly.

Given this computation of the center, any automorphism of $\pi_{1}(X_{g}^{k})$ induces an automorphism of $\langle z\rangle\cong\mathbb{Z}$ and descends to an automorphism of $\pi_{1}(S_{g})$. The latter gives a homomorphism

that restricts to an isomorphism between the inner automorphism groups

and hence descends to a homomorphism

These orientation-preserving subgroups contain the (respective) inner automorphism groups, and we denote the quotients $\operatorname{\mathcal{OUT}}\big{(}\pi_{1}(X_{g}^{k})\big{)}$ and $\operatorname{\mathcal{OUT}}\big{(}\pi_{1}(S_{g})\big{)}$.

Fix $g\geq 1$ and $k\in\mathbb{Z}$, and assume $(g,k)\neq(1,0)$. Let $\operatorname{Homeo}^{+}(X_{g}^{k})$ denote the group of homeomorphisms whose image in $\operatorname{Out}\big{(}\pi_{1}(X_{g}^{k})\big{)}$ is contained in $\operatorname{\mathcal{OUT}}\big{(}\pi_{1}(S_{g})\big{)}$. Define

Waldhausen [Wal68, Cor. 7.5] proved that the natural homomorphism

is an isomorphism. Then, by the definitions, this homomorphism restricts to an isomorphism $\operatorname{Mod}(X_{g}^{k})\cong\operatorname{\mathcal{OUT}}\big{(}\pi_{1}(X_{g}^{k})\big{)}$. Waldhausen also proved that $\pi_{0}\operatorname{Homeo}(X_{g}^{k})$ is isomorphic to the group of fiber-preserving homeomorphisms modulo homeomorphisms that are isotopic to the identity through fiber-preserving isotopies; see [Wal68, Rmk. following Cor. 7.5]. Consequently, there is a homomorphism

Altogether, we have the following commutative diagram relating the maps (5) and (6).

The right vertical map is an isomorphism by the Dehn–Nielsen–Baer theorem [FM12, Thm. 8.1]. Furthermore, by Conner–Raymond [CR77, Thm. 8] that there is a short exact sequence

This establishes the short exact sequence (1) in the introduction. We will give a concrete derivation of this exact sequence in Corollary 3.1 below.

Fix a basepoint $*\in S_{g}$, and set $S_{g,1}=S_{g}\setminus\{*\}$. By the Dehn–Nielsen–Baer theorem, $\operatorname{Mod}(S_{g,1})$ is isomorphic to $\operatorname{Out}^{*}(F_{2g})$, where $F_{2g}$ is the free group of rank $2g$ and $\operatorname{Out}^{*}(F_{2g})<\operatorname{Out}(F_{2g})$ is the subgroup that preserves the conjugacy class corresponding to the free homotopy class of the curve around the puncture in $S_{g,1}$. We construct $\Psi$ as a composition

To define $\sigma$, fix a generating set $\alpha_{1},\beta_{1},\ldots,\alpha_{g},\beta_{g}$ for $F_{2g}$ such that $c=\prod_{i=1}^{g}[\alpha_{i},\beta_{i}]$ represents the conjugacy class of the curve around the puncture. Let

be the homomorphism defined by $\alpha_{i}\mapsto A_{i}$ and $\beta_{i}\mapsto B_{i}$. Given $f\in\operatorname{Out}^{*}(F_{2g})$, fix an automorphism $\tilde{f}:F_{2g}\to F_{2g}$ that represents $f$, and assume that $\tilde{f}(c)=c$ (this can always be achieved by composing any lift with an inner automorphism of $F_{2g}$). Next we define $\sigma(f)$ on generators of $\pi_{1}(X_{g}^{k})$ by

To show that $\sigma(f)$ extends to a homomorphism of $\pi_{1}(X_{g}^{k})$, we check that the relation $[A_{1},B_{1}]\cdots[A_{g},B_{g}]=z^{k}$ is preserved under $\sigma(f)$:

The second equality uses the the fact that $\tilde{f}(c)=c$. The map $\sigma(f)$ is independent of the choice of $\tilde{f}$ because different choices of $\tilde{f}$ differ by conjugation by powers of $c$ (because the centralizer of $c$ in $F_{2g}$ is the cyclic subgroup $\langle c\rangle$)¹¹1Note that the centralizer is isomorphic to $\mathbb{Z}$ and contains $\langle c\rangle$. It is only bigger if $c=u^{i}$ for some $u\in F_{2g}$ and $i\geq 2$. By contradiction, if $c=u^{i}$ for $i\geq 2$, then $u$ is cyclically reduced because $c$ is. This implies that $u$ is a subword of $c=\prod[\alpha_{i},\beta_{i}]$, which is absurd. and $\iota(c)=z^{k}$ is central in $\pi_{1}(X_{g}^{k})$. The homomorphism $\sigma(f):\pi_{1}(X_{g}^{k})\to\pi_{1}(X_{g}^{k})$ is an automorphism and belongs to $\operatorname{\mathcal{AUT}}\big{(}\pi_{1}(X_{g}^{k})\big{)}$ by definition. Furthermore, $f\mapsto\sigma(f)$ is a homomorphism, which is easy to check using the observation that if $w=\iota w^{\prime}$, then $\sigma(f)(w)=\iota\tilde{f}(w^{\prime})$.

Composing $\sigma$ with $\operatorname{\mathcal{AUT}}\to\operatorname{\mathcal{OUT}}$ gives the desired homomorphism $\Psi$. As a corollary of this construction, we have proved the following.

Corollary 3.1 and equation (4) combine to give the short exact sequence of outer automorphism groups (8).

Observe that the kernel of $\Psi$ is contained in the point-pushing subgroup $\pi_{1}(S_{g})<\operatorname{Mod}(S_{g,1})$. This is because $\Psi$ composed with the natural map $\operatorname{Mod}(X_{g}^{k})\to\operatorname{Mod}(S_{g})$ is the natural map $\operatorname{Mod}(S_{g,1})\to\operatorname{Mod}(S_{g})$, whose kernel is the point-pushing subgroup. Thus we want to understand the image of the point-pushing subgroup under the section $\sigma$ used to define $\Psi$. What we find is a simple relationship between three surface group representations:

The main results are Proposition 3.4 and Corollary 3.5 below. In order to state Proposition 3.4, we need the following notation. Let

be the Poincaré duality map, given explicitly by $\gamma\mapsto\langle-,\gamma\rangle$, where

is the algebraic intersection form. We use $\hat{\delta}$ denote the composition

This map is given explicitly by $\hat{\delta}(\gamma)(w)=w\cdot z^{\langle[\bar{w}],\gamma\rangle}$, where $\bar{w}$ is the image of $w$ under $\pi_{1}(X_{g}^{k})\to\pi_{1}(S_{g})$ and $[\bar{w}]\in H_{1}(S_{g};\mathbb{Z})$ is the corresponding homology class.

Fix a basepoint $\star\in S_{g,1}$. Recall that we have fixed a standard generating set $\{\alpha_{i},\beta_{i}\}$ of $\pi_{1}(S_{g,1},\star)\cong F_{2g}$ so that $c:=\prod_{i}[\alpha_{i},\beta_{i}]$ is a loop around the puncture $*$ of $S_{g,1}=S_{g}\setminus\{*\}$. Define

by $\gamma\mapsto\epsilon.\gamma.\bar{\epsilon}$, where $\epsilon$ is a fixed arc from $*$ to $\star$.

As a sanity check, observe that $C_{\iota\tilde{t}}$ does not depend on the choice of lift $\tilde{t}$ because any two lifts differ by an element of the normal closure of $c$ in $\pi_{1}(S_{g,1},\star)=F_{2g}$, and conjugation by any such element is trivial on $\pi_{1}(X_{g}^{k})$.

The following corollary is an immediate consequence of Proposition 3.4.

Using the isomorphisms between mapping class groups and automorphism groups, the desired diagram is equivalent to the following one.

The map $\Psi$ in (15) descends to the middle vertical map and restricts to the left vertical map by Corollary 3.5. The fact that $\sigma$ is a section (Corollary 3.1) implies that the middle vertical map descends to the identity map on $\operatorname{\mathcal{OUT}}\big{(}\pi_{1}(S_{g})\big{)}$. When $k=1$, the middle vertical map is an isomorphism by the five lemma. This concludes the proof of Theorem A.