The Meyer function on the handlebody group

TL;DR

This paper provides an explicit formula for the signature of handlebody bundles over the circle, linking homological monodromy to Meyer's signature cocycle, and offers topological insights into the hyperelliptic handlebody group.

Contribution

It introduces a new explicit formula for the signature of handlebody bundles and interprets the generator of the first cohomology of the hyperelliptic handlebody group.

Findings

Explicit signature formula in terms of homological monodromy

Cobounding function for Meyer's signature cocycle on handlebody group

Topological interpretation of the hyperelliptic handlebody group's cohomology

Abstract

We give an explicit formula for the signature of handlebody bundles over the circle in terms of the homological monodromy. This gives a cobounding function of Meyer's signature cocycle on the mapping class group of a $3$-dimensional handlebody, i.e., the handlebody group. As an application, we give a topological interpretation for the generator of the first cohomology group of the hyperelliptic handlebody group.

Equations

$$H^2(\mathrm{Mod}({\Sigma }_g))\cong \mathbb{Z}\text{for g≥3.}$$

$${\tau }_g:\mathrm{Sp}(2g;\mathbb{Z})\times \mathrm{Sp}(2g;\mathbb{Z})\to \mathbb{Z}$$

$${\tau }_g(\rho ({\phi }_1),\rho ({\phi }_2))={\varphi }_g({\phi }_1)+{\varphi }_g({\phi }_2)-{\varphi }_g({\phi }_1{\phi }_2)\text{for any φ1,φ2∈Mod(Σg).}$$

$${\varphi }_g^V(\phi ):=\mathrm{Sign}{M}_{\phi }\in \mathbb{Z}.$$

$$⟨x,y{⟩}_{\phi }:={}^{t}x{}^{t}Qy,\text{for x,y∈Uφ}.$$

$${\varphi }_g^{\mathcal{H}}-{\varphi }_g^V\in \mathrm{Hom}(\mathcal{H}(V_g),\mathbb{Q})=H^1(\mathcal{H}(V_g);\mathbb{Q})$$

$${\varphi }_g^{\mathcal{H}}-{\varphi }_g^V=\{\begin{array}{ll}\frac{2}{2g+1}\mu & \text{if }g\text{ is even},\\ \frac{1}{2g+1}\mu & \text{if }g\text{ is odd}.\end{array}$$

$$\begin{array}{ccc}\mathcal{H}(V_g)& \stackrel{i_2}{\to }& \mathrm{Mod}(V_g)\\ i_1\downarrow & & \downarrow j_2\\ \mathcal{H}({\Sigma }_g)& \underset{j_1}{\to }& \mathrm{Mod}({\Sigma }_g).\end{array}$$

$${\rm Y}:H^2(\mathrm{Mod}({\Sigma }_g);\mathbb{Q})\to H^1(\mathcal{H}(V_g);\mathbb{Q})$$

$$\rho :\mathrm{Mod}({\Sigma }_g)\to \mathrm{Sp}(2g;\mathbb{Z}),\phi \mapsto {\phi }_{\ast }.$$

$$\mathrm{Sp}(2g;\mathbb{Z})=\{A\in \mathrm{GL}(2g;\mathbb{Z})\mid {}^{t}AJA=J\},$$

$$V_{A,B}:=\{(x,y)\in {\mathbb{R}}^{2g}\oplus {\mathbb{R}}^{2g}\mid (A^{-1}-I_{2g})x+(B-I_{2g})y=0\}$$

$$⟨(x,y),(x^{\prime },y^{\prime }){⟩}_{A,B}:={}^t(x+y)J(I_{2g}-B)y^{\prime }.$$

$${\tau }_g(A,B):=\mathrm{Sign}(V_{A,B},⟨\cdot ,\cdot {⟩}_{A,B}).$$

$$0⟶H_2(V_g,{\Sigma }_g)\stackrel{{\partial }_{\ast }}{⟶}{H}_1({\Sigma }_g)\stackrel{i_{\ast }}{⟶}{H}_1(V_g)⟶0$$

$$\mathrm{urSp}(2g;\mathbb{Z}):=\left\{A\in \mathrm{Sp}(2g;\mathbb{Z})\mid A=\left(\begin{array}{cc}P& Q\\ O_g& S\end{array}\right)\right\},$$

$${}^{t}PS=I_g,{}^{t}QS={}^{t}SQ.$$

$$\mathcal{H}({\Sigma }_g):=\{\phi \in \mathrm{Mod}({\Sigma }_g)\mid \phi \iota =\iota \phi \}.$$

$$\mathcal{H}(V_g):=\mathcal{H}({\Sigma }_g)\cap \mathrm{Mod}(V_g).$$

$$\mathcal{H}(V_g{)}^{\mathrm{abel}}\cong \mathbb{Z}\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\text{for g≥2}.$$

$$\mathcal{H}(V_g{)}^{\mathrm{abel}}\cong \{\begin{array}{ll}\mathbb{Z}[s_1]\oplus {\mathbb{Z}}_2[t_1{s}_1^{\frac{g}{2}}]\oplus {\mathbb{Z}}_2[r_1]& \text{if g is even},\\ \mathbb{Z}[t_1{s}_1^{\frac{g+1}{2}}]\oplus {\mathbb{Z}}_2[t_1^2{s}_1^g]\oplus {\mathbb{Z}}_2[r_1]& \text{if g is odd}.\end{array}$$

$$2[r_1]=0,4[t_1]+2g[s_1]=0,2(g+1)[t_1]+g(g+1)[s_1]=0.$$

$$M_{\phi }:=(I\times V_g)/(0,x)\sim (1,\phi (x)).$$

$$\Phi :I\times V_g\to M_{\phi },(t,x)\mapsto [t,x].$$

$$V_g\stackrel{0\times \mathrm{id}}{\cong }\{0\}\times V_g\stackrel{\Phi (0,\cdot )}{⟶}{\pi }^{-1}([0])={\pi }^{-1}([1])\stackrel{\Phi (1,\cdot {)}^{-1}}{⟶}\{1\}\times V_g\stackrel{1\times \mathrm{id}}{\cong }{V}_g.$$

$$H_2(M_{\phi })\cong E_{1,1}^{\infty }\cong E_{1,1}^2=H_1(S^1;{\mathcal{H}}_1(V_g)).$$

$$H_1(S^1;{\mathcal{H}}_1(V_g))$$

$$=\mathrm{Ker}(\partial :\mathbb{Z}{e}^1\otimes H_1(V_g)\to \mathbb{Z}{e}^0\otimes H_1(V_g)=H_1(V_g)),$$

$$\partial (e^1\otimes \alpha )={\ell }_{\ast }(\alpha )-\alpha =(\Phi (0,\cdot {)}^{-1}\circ \Phi (1,\cdot ){)}_{\ast }(\alpha )-\alpha ={\phi }_{\ast }^{-1}(\alpha )-\alpha .$$

$$H_2(M_{\phi },\partial M_{\phi })\cong E_{0,2}^{\infty }\cong E_{0,2}^2=H_0(S^1;{\mathcal{H}}_2(V_g,{\Sigma }_g)).$$

$$H_1(V_g{)}^{{\pi }_1(S^1)}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1. Introduction

Let $\Sigma_{g}$ be a closed connected oriented surface of genus $g$ and $\operatorname{Mod}(\Sigma_{g})$ the mapping class group of $\Sigma_{g}$, namely the group of isotopy classes of orientation-preserving self-diffeomorphisms of $\Sigma_{g}$. Unless otherwise stated, we assume that (co)homology groups have coefficients in ${\mathbb{Z}}$. The second cohomology of $\operatorname{Mod}(\Sigma_{g})$ has been determined for all $g\geq 1$ by works of many people, in particular by the seminal work of Harer [6, 7] for $g\geq 3$. We have $H^{2}(\operatorname{Mod}(\Sigma_{1}))\cong{\mathbb{Z}}/12{\mathbb{Z}}$, $H^{2}(\operatorname{Mod}(\Sigma_{2}))\cong{\mathbb{Z}}/10{\mathbb{Z}}$, and

[TABLE]

There are various interesting constructions of non-trivial second cohomology class of $\operatorname{Mod}(\Sigma_{g})$; the reader is referred to the survey article [13]. Among others, the remarkable approach of Meyer [16, 17] was to consider the signature of $\Sigma_{g}$-bundles over surfaces. The central object that Meyer used was a normalized 2-cocycle

[TABLE]

on the integral symplectic group of degree $2g$.

Meyer showed that for $g\geq 3$ the pullback of the cohomology class of $\tau_{g}$ by the homology representation $\rho\colon\operatorname{Mod}(\Sigma_{g})\to\operatorname{Sp}(2g;{\mathbb{Z}})$ is of infinite order in $H^{2}(\operatorname{Mod}(\Sigma_{g}))$. On the other hand, if $g=1,2$ then $[\rho^{*}\tau_{g}]$ is torsion and there exists a (unique) rational valued cobounding function $\phi_{g}\colon\operatorname{Mod}(\Sigma_{g})\to{\mathbb{Q}}$ of $\rho^{*}\tau_{g}$. This means that

[TABLE]

Since the case $g=1$ was extensively studied by Meyer, such a cobounding function is called a Meyer function. Some number-theoretic and differential geometric aspects of the function $\phi_{1}$ were studied by Atiyah [2]. The case $g=2$ was studied by Matsumoto [15], Morifuji [18] and Iida [11]. For $g\geq 3$, there is no cobounding function of $\rho^{*}\tau_{g}$ on the whole mapping class group $\operatorname{Mod}(\Sigma_{g})$. However, if we restrict $\rho^{*}\tau_{g}$ to a subgroup called the hyperelliptic mapping class group $\mathcal{H}(\Sigma_{g})$, then it is known that there is a (unique) cobounding function $\phi^{\mathcal{H}}_{g}\colon\mathcal{H}(\Sigma_{g})\to{\mathbb{Q}}$ of $\rho^{*}\tau_{g}$. Note that $\mathcal{H}(\Sigma_{g})=\operatorname{Mod}(\Sigma_{g})$ for $g=1,2$. This function $\phi^{\mathcal{H}}_{g}$ was studied by Endo [4] and Morifuji [18]. One motivation for studying Meyer functions comes from the localization phenomenon of the signature of fibered $4$-manifolds. See, e.g., [1, 14].

In this paper, we study a new example of Meyer functions: the Meyer function on the handlebody group. The handlebody group of genus $g$, which we denote by $\operatorname{Mod}(V_{g})$, is defined as the group of isotopy classes of orientation-preserving self-diffeomorphisms of the 3-dimensional handlebody $V_{g}$ of genus $g$. It is well known that the natural homomorphism $\operatorname{Mod}(V_{g})\to\operatorname{Mod}(\Sigma_{g}),\varphi\mapsto\varphi|_{\Sigma_{g}}$ is injective since $V_{g}$ is an irreducible 3-manifold. Therefore, we can think of $\operatorname{Mod}(V_{g})$ as a subgroup of $\operatorname{Mod}(\Sigma_{g})$. For a mapping class $\varphi\in\operatorname{Mod}(V_{g})$, we denote by $M_{\varphi}$ the mapping torus of $\varphi$. It is a compact oriented $4$-manifold. We define

[TABLE]

We show in Lemma 4.2 that $\phi_{g}^{V}$ is a cobounding function of the cocycle $\rho^{*}\tau_{g}$ on the handlebody group $\operatorname{Mod}(V_{g})$. If $g\geq 3$, this is the unique cobounding function since $H_{1}(\operatorname{Mod}(V_{g}))$ is torsion (see [21, Theorem 20] and [12, Remark 3.5]).

The value $\phi_{g}^{V}(\varphi)$ can be computed from the action of $\varphi$ on the first homology $H_{1}(\Sigma_{g})$, and our first result gives its explicit description. To state it, we take a suitable basis of $H_{1}(\Sigma_{g})$ so that the homology representation $\rho$ restricted to $\operatorname{Mod}(V_{g})$ takes values in a subgroup $\operatorname{urSp}(2g;{\mathbb{Z}})\subset\operatorname{Sp}(2g;{\mathbb{Z}})$. (See Section 2.3 for details.) Then, $\rho(\varphi)$ is of the form $\rho(\varphi)=\begin{pmatrix}P&Q\\ O_{g}&S\end{pmatrix}$, where $P$, $Q$ and $S$ are $g\times g$ matrices. We consider a ${\mathbb{Q}}$-linear space $U_{\varphi}:=\operatorname{Ker}(S-I_{g})\subset{\mathbb{Q}}^{g}$, and define a bilinear form $\langle\ ,\ \rangle_{\varphi}$ on it by

[TABLE]

It turns out that $\langle\ ,\ \rangle_{\varphi}$ is symmetric, and we have the following:

Theorem 1.1**.**

The value $\phi_{g}^{V}(\varphi)$ coincides with the signature of the symmetric bilinear form $\langle\ ,\ \rangle_{\varphi}$ on $U_{\varphi}$.

In fact, we will show in Section 3.5 that the intersection form on $H_{2}(M_{\varphi})$ is equivalent to the bilinear form $\langle\ ,\ \rangle_{\varphi}$.

As a corollary, we see that the function $\phi_{g}^{V}$ is bounded by $g=\operatorname{rank}H_{1}(V_{g})$. We also give sample calculations of $\phi_{g}^{V}$ in Lemmas 4.4 and 4.5. Walker also constructed a function $j\colon\operatorname{Mod}(\Sigma_{g})\to{\mathbb{Q}}$ whose restriction to $\operatorname{Mod}(V_{g})$ coincides with $\phi_{g}^{V}$. Our description of $\phi_{g}^{V}$ in Theorem 1.1 is similar to but different from a description of $j$ given by Gilmer and Masbaum [5, Proposition 6.9]. See, for details, Remark 3.6.

As an application of the function $\phi_{g}^{V}$, we obtain a non-trivial first cohomology class in the intersection $\mathcal{H}(\Sigma_{g})\cap\operatorname{Mod}(V_{g})$ called the hyperelliptic handlebody group, denoted by $\mathcal{H}(V_{g})$. The group $\mathcal{H}(V_{g})$ is an extension by $\mathbb{Z}/2\mathbb{Z}$ of a subgroup of the mapping class group of a 2-sphere with $(2g+2)$-punctures, called the Hilden group. The Hilden group was introduced in [8], and it is related to the study of links in $3$-manifolds. In [10], Hirose and Kin studied the minimal dilatation of pseudo-Anosov elements in $\mathcal{H}(V_{g})$, and gave a presentation of $\mathcal{H}(V_{g})$.

We consider the difference

[TABLE]

of the Meyer functions on $\mathcal{H}(\Sigma_{g})$ and on $\operatorname{Mod}(V_{g})$. From the abelianization of $\mathcal{H}(V_{g})$ obtained in [10, Corollary A.9], we see that the rank of $H^{1}(\mathcal{H}(V_{g}))$ is one. Let us denote a generator of $H^{1}(\mathcal{H}(V_{g}))$ by $\mu$. Our second result is:

Theorem 1.2**.**

Let $g\geq 1$. We have

[TABLE]

When $g=1,2$, we have $\mathcal{H}(V_{g})=\operatorname{Mod}(V_{g})$, and $\phi_{g}^{\mathcal{H}}-\phi_{g}^{V}$ gives an abelian quotient of $\operatorname{Mod}(V_{g})$.

There is an interpretation of the cohomology class $\phi_{g}^{\mathcal{H}}-\phi_{g}^{V}$ in terms of a kind of connecting homomorphism. We assume that $g\geq 3$. From the diagram

[TABLE]

of groups and their inclusions, we have a natural homomorphism

[TABLE]

defined as follows. For $[c]\in H^{2}(\operatorname{Mod}(\Sigma_{g});\mathbb{Q})$, there are cobounding functions $f^{\mathcal{H}}\colon\mathcal{H}(\Sigma_{g})\to\mathbb{Q}$ of $j_{1}^{*}c$ and $f^{V}\colon\operatorname{Mod}(V_{g})\to\mathbb{Q}$ of $j_{2}^{*}c$, respectively. The cochain $i_{1}^{*}f^{\mathcal{H}}-i_{2}^{*}f^{V}$ is actually a homomorphism on $\mathcal{H}(V_{g})$. It does not depend on the choices of the representatives $c$, $f^{\mathcal{H}}$, and $f^{V}$ since $H^{1}(\operatorname{Mod}(V_{g});\mathbb{Q})=H^{1}(\mathcal{H}(\Sigma_{g});\mathbb{Q})=0$ when $g\geq 3$. Then $\Upsilon([c])$ is defined to be $i_{1}^{*}f^{\mathcal{H}}-i_{2}^{*}f^{V}$. In this setting, our cohomology class is written as $\Upsilon([\tau_{g}])=\phi_{g}^{\mathcal{H}}-\phi_{g}^{V}\in H^{1}(\mathcal{H}(V_{g});{\mathbb{Q}})$.

The outline of this paper is as follows. In Section 2, we review the definition of Meyer’s signature cocycle and the handlebody group $\operatorname{Mod}(V_{g})$. We also review the abelianization of the hyperelliptic handlebody group obtained in [10], and describe a generator of the cohomology group $H^{1}(\mathcal{H}(V_{g}))$ in Corollary 2.6. In Section 3, we investigate the intersection form of the mapping torus of $\varphi\in\operatorname{Mod}(V_{g})$, and prove Theorem 1.1. As it turns out, we can explicitly describe $\phi^{V}_{g}$ as a function on a subgroup $\operatorname{urSp}(2g;\mathbb{Z})$ of the integral symplectic group. In Section 4, we prove Theorem 1.2 by using explicit calculations of the Meyer function $\phi_{g}^{V}\colon\operatorname{Mod}(V_{g})\to\mathbb{Z}$ in Lemmas 4.4 and 4.5.

2. Preliminaries on mapping class groups

Fix a non-negative integer $g$.

2.1. Mapping class group of a surface

Let $\Sigma_{g}$ be a closed connected oriented surface of genus $g$. The mapping class group of $\Sigma_{g}$, denoted by $\operatorname{Mod}(\Sigma_{g})$, is the group of isotopy classes of orientation-preserving self-diffeomorphisms of $\Sigma_{g}$. To simplify notation, we will use the same letter for a self-diffeomorphism of $\Sigma_{g}$ and its isotopy class.

The first homology group $H_{1}(\Sigma_{g})$ is equipped with a non-degenerate skew-symmetric pairing $\langle\cdot,\cdot\rangle$, namely the intersection form. Thus we can take a symplectic basis $\alpha_{1},\ldots,\alpha_{g},\beta_{1},\ldots,\beta_{g}$ for $H_{1}(\Sigma_{g})$. This means that $\langle\alpha_{i},\beta_{j}\rangle=\delta_{ij}$ and $\langle\alpha_{i},\alpha_{j}\rangle=\langle\beta_{i},\beta_{j}\rangle=0$ for any $i,j\in\{1,\ldots,g\}$, where $\delta_{ij}$ is the Kronecker symbol.

Once a symplectic basis for $H_{1}(\Sigma_{g})$ is fixed, we obtain the homology representation

[TABLE]

Here, the target is the integral symplectic group

[TABLE]

where $J=\begin{pmatrix}O_{g}&I_{g}\\ -I_{g}&O_{g}\end{pmatrix}$, and $\rho(\varphi)=\varphi_{*}$ is the matrix presentation of the action of $\varphi$ on $H_{1}(\Sigma_{g})$ with respect to the fixed symplectic basis. We use block matrices to denote elements in $\operatorname{Sp}(2g;{\mathbb{Z}})$, e.g., $A=\begin{pmatrix}P&Q\\ R&S\end{pmatrix}$ with $g\times g$ integral matrices $P$, $Q$, $R$, and $S$.

2.2. Meyer’s signature cocycle

Let $A,B\in\operatorname{Sp}(2g;{\mathbb{Z}})$. We consider an ${\mathbb{R}}$-linear space

[TABLE]

and a bilinear form on $V_{A,B}$ given by

[TABLE]

The form $\langle\cdot,\cdot\rangle_{A,B}$ turns out to be symmetric, and thus its signature is defined; we set

[TABLE]

The map $\tau_{g}\colon\operatorname{Sp}(2g;{\mathbb{Z}})\times\operatorname{Sp}(2g;{\mathbb{Z}})\to{\mathbb{Z}}$ is called Meyer’s signature cocycle [16, 17]. It is a normalized 2-cocycle of the group $\operatorname{Sp}(2g;{\mathbb{Z}})$.

Let $P$ be a compact oriented surface of genus [math] with three boundary components, i.e., a pair of pants. We denote by $C_{1}$, $C_{2}$ and $C_{3}$ the boundary components of $P$. Choose a base point in $P$, and let $\ell_{1}$, $\ell_{2}$ and $\ell_{3}$ be based loops in $P$ such that $\ell_{i}$ is parallel to the negatively oriented boundary component $C_{i}$ for any $i\in\{1,2,3\}$ and $\ell_{1}\ell_{2}\ell_{3}=1$ holds in the fundamental group $\pi_{1}(P)$.

For given two mapping classes $\varphi_{1},\varphi_{2}\in\operatorname{Mod}(\Sigma_{g})$, there is an oriented $\Sigma_{g}$-bundle $E(\varphi_{1},\varphi_{2})\to P$ such that the monodromy along $\ell_{i}$ is $\varphi_{i}$ for $i=1,2$. It is unique up to bundle isomorphisms. The total space $E(\varphi_{1},\varphi_{2})$ is a compact 4-manifold equipped with a natural orientation, and hence its signature is defined.

Proposition 2.1** (Meyer [16, 17]).**

$\operatorname{Sign}(E(\varphi_{1},\varphi_{2}))=\tau_{g}(\rho(\varphi_{1}),\rho(\varphi_{2}))$.

Remark 2.2*.*

Turaev [20] independently found the signature cocycle. He also studied its relation to the Maslov index.

2.3. Handlebody group

Let $V_{g}$ be a handlebody of genus $g$. That is, $V_{g}$ is obtained by attaching $g$ one-handles to the $3$-ball $D^{3}$. We identify $\Sigma_{g}$ and the boundary of $V_{g}$ by choosing an orientation-preserving diffeomorphism between them. We have the following short exact sequence

[TABLE]

which is a part of the homology exact sequence of the pair $(V_{g},\Sigma_{g})$. There are properly embedded, oriented and pairwise disjoint disks $D_{1},\ldots,D_{g}$ in $V_{g}$ whose homology classes (denoted by the same letters) constitute a basis for $H_{2}(V_{g},\Sigma_{g})$. We set $\alpha_{i}:=\partial_{*}(D_{i})\in H_{1}(\Sigma_{g})$ for $i\in\{1,\ldots,g\}$. Then $\alpha_{i}$’s extend to a symplectic basis $\alpha_{1},\ldots,\alpha_{g},\beta_{1},\ldots,\beta_{g}$ for $H_{1}(\Sigma_{g})$. In what follows, we fix a symplectic basis obtained in this way. The image of the homology classes $\beta_{1},\ldots,\beta_{g}$ by the map $i_{*}$ constitute a basis for $H_{1}(V_{g})$. For simplicity, we denote them by the same letters $\beta_{1},\ldots,\beta_{g}$.

We denote by $\operatorname{Mod}(V_{g})$ the handlebody group of genus $g$. It can be considered as a subgroup of $\operatorname{Mod}(\Sigma_{g})$. For any $\varphi\in\operatorname{Mod}(V_{g})$, the matrix $\rho(\varphi)$ lies in the subgroup of $\operatorname{Sp}(2g;{\mathbb{Z}})$ defined by

[TABLE]

cf. [3, 9] for details. The matrices $P$, $Q$ and $S$ satisfy the following relations:

[TABLE]

Remark 2.3*.*

The group $\operatorname{Mod}(V_{g})$ acts naturally on the groups in (2.1), and the maps $\partial_{*}$ and $i_{*}$ are $\operatorname{Mod}(V_{g})$-module homomorphisms. The matrix presentation of the action $\varphi_{*}$ on $H_{1}(V_{g})$ is $S$.

2.4. Hyperelliptic handlebody group

An involution of $\Sigma_{g}$ is called hyperelliptic if it acts on $H_{1}(\Sigma_{g})$ as $-\operatorname{id}$. We fix an hyperelliptic involution $\iota$ which extends to an involution of $V_{g}$, as in Figure 1.

The hyperelliptic mapping class group $\mathcal{H}(\Sigma_{g})$ is the centralizer of $\iota$ in $\operatorname{Mod}(\Sigma_{g})$:

[TABLE]

Definition 2.4** ([10]).**

The hyperelliptic handlebody group $\mathcal{H}(V_{g})$ is defined by

[TABLE]

Hirose and Kin [10, Appendix A] gave a finite presentation of the group $\mathcal{H}(V_{g})$. Moreover they determined the abelianization of $\mathcal{H}(V_{g})$ as

[TABLE]

In fact, using their presentation, it is easy to make this result more explicit. Let $C_{1}$, $C_{2}$ and $C_{3}$ be simple closed curves on $\Sigma_{g}$ as in Figure 1. For each $i\in\{1,2,3\}$ denote by $t_{i}$ the right handed Dehn twist along $C_{i}$. Following [10], set $r_{1}=t_{2}^{-1}t_{3}^{-1}t_{1}t_{2}$ and $s_{1}=t_{2}t_{3}t_{1}t_{2}$. (Note that in [10], $t_{C}$ denotes the left handed Dehn twist along $C$.)

Lemma 2.5**.**

When $g=1$, one has $\mathcal{H}(V_{1})\cong{\mathbb{Z}}\,[t_{1}s_{1}]\oplus{\mathbb{Z}}_{2}\,[t_{1}^{2}s_{1}]$. If $g\geq 2$, then

[TABLE]

Here, $[s_{1}]$ is the class of $s_{1}$ in $\mathcal{H}(V_{g})^{\rm abel}$, and ${\mathbb{Z}}\,[s_{1}]$ is the infinite cyclic group generated by $[s_{1}]$, etc.

Proof.

The case $g=1$ follows from the fact that $\mathcal{H}(V_{1})\cong\operatorname{Mod}(V_{1})$ and a result of Wajnryb [21, Theorem 14].

Assume that $g\geq 2$. Using [10, Theorem A.8], one sees that $\mathcal{H}(V_{g})^{\rm abel}$ is generated by $[r_{1}]$, $[s_{1}]$ and $[t_{1}]$ with the relations

[TABLE]

The assertion follows from these relations by a direct computation. ∎

The following corollary to Lemma 2.5 will be used in Section 4.4 to prove Theorem 1.2.

Corollary 2.6**.**

Let $g\geq 1$. There is a unique homomorphism $\mu\colon\mathcal{H}(V_{g})\to{\mathbb{Z}}$ satisfying the following property:

1. $(1)$

If $g$ is even, $\mu(s_{1})=1$ and $\mu(t_{1})=-g/2$; 2. $(2)$

If $g$ is odd, $\mu(t_{1})=-g$, $\mu(s_{1})=2$, and thus $\mu(t_{1}s_{1}^{\frac{g+1}{2}})=1$.

Moreover, the first cohomology group $H^{1}(\mathcal{H}(V_{g}))=\operatorname{Hom}(\mathcal{H}(V_{g}),{\mathbb{Z}})$ is an infinite cyclic group generated by $\mu$.

3. Handlebody bundles over $S^{1}$

3.1. Mapping torus

Let $I=[0,1]$ be the unit interval. By identifying the endpoints of $I$, we obtain the circle $S^{1}=[0,1]/0\sim 1$. Let $\ell\colon I\to S^{1}$ be the natural projection. For $t\in I$, we set $[t]:=\ell(t)$. Choose $[0]$ as a base point of $S^{1}$. Then the fundamental group $\pi_{1}(S^{1})$ is an infinite cyclic group generated by the homotopy class of $\ell$.

In what follows, we use the following cell decomposition of $S^{1}$: the [math]-cell is $e^{0}=[0]$ and the 1-cell is $e^{1}=S^{1}\setminus e^{0}$. The map $\ell$ induces an orientation of $e^{1}$.

Let $\varphi\in\operatorname{Mod}(V_{g})$. The mapping torus of $\varphi$ is the quotient space

[TABLE]

For $(t,x)\in I\times V_{g}$, its class in $M_{\varphi}$ is denoted by $[t,x]$. The natural projection $\pi\colon M_{\varphi}\to S^{1},[t,x]\mapsto[t]$ is an oriented $V_{g}$-bundle, and the total space $M_{\varphi}$ is a compact 4-manifold with boundary equipped with a natural orientation. The pullback of $M_{\varphi}\to S^{1}$ by $\ell$ is a trivial $V_{g}$-bundle over $I$, and its trivialization is given by the map

[TABLE]

The following composition of maps coincides with $\varphi$:

[TABLE]

Therefore, the monodromy of $M_{\varphi}\to S^{1}$ along $\ell$ is equal to the mapping class $\varphi$. As was mentioned in Remark 2.3, the groups $H_{2}(V_{g},\Sigma_{g})$, $H_{1}(\Sigma_{g})$, and $H_{1}(V_{g})$ are $\operatorname{Mod}(V_{g})$-modules. Thus, these groups become $\pi_{1}(S^{1})$-modules; the homotopy class of $\ell$, which is a generator of $\pi_{1}(S^{1})$, acts as the monodromy $\varphi\in\operatorname{Mod}(V_{g})$.

3.2. Second homology of the mapping torus

For a non-negative integer $q\geq 0$, let $\mathscr{H}_{q}(V_{g})$ be the local system on $S^{1}$ which comes from the $V_{g}$-bundle $\pi\colon M_{\varphi}\to S^{1}$, and whose fiber at $x\in S^{1}$ is the $q$-th homology group $H_{q}(\pi^{-1}(x))$. Similarly, we consider the local system $\mathscr{H}_{q}(V_{g},\Sigma_{g})$ whose fiber at $x\in S^{1}$ is the $q$-th relative homology group $H_{q}(\pi^{-1}(x),\partial\pi^{-1}(x))$.

Consider the Serre homology spectral sequence of the $V_{g}$-bundle $M_{\varphi}\to S^{1}$. It degenerates at the $E^{2}$ page, which is given by $E^{2}_{p,q}=H_{p}(S^{1};\mathscr{H}_{q}(V_{g}))$. Since $H_{2}(V_{g})=0$ and the base space $S^{1}$ is $1$-dimensional, we obtain

[TABLE]

Moreover, using the cellular homology of $S^{1}$ with coefficients in $\mathscr{H}_{1}(V_{g})$, we have

[TABLE]

where the boundary map is given by

[TABLE]

In summary, we have proved the following lemma. In the statement, $H_{1}(V_{g})^{\pi_{1}(S^{1})}$ is the space of invariants under the action of $\pi_{1}(S^{1})$, i.e., $H_{1}(V_{g})^{\pi_{1}(S^{1})}=\{\alpha\in H_{1}(V_{g})\mid\varphi_{*}(\alpha)=\alpha\}$.

Lemma 3.1**.**

We have $H_{2}(M_{\varphi})\cong H_{1}(S^{1};\mathscr{H}_{1}(V_{g}))\cong H_{1}(V_{g})^{\pi_{1}(S^{1})}$.

Similarly, for the relative homology of the pair $(M_{\varphi},\partial M_{\varphi})$, there is a spectral sequence converging to $H_{*}(M_{\varphi},\partial M_{\varphi})$ such that $E^{2}_{p,q}=H_{p}(S^{1};\mathscr{H}_{q}(V_{g},\Sigma_{g}))$. This degenerates at the $E^{2}$ page, too. Since $H_{1}(V_{g},\Sigma_{g})=0$, we obtain

[TABLE]

By the same argument as above, we obtain the following lemma. In the statement, $H_{2}(V_{g},\Sigma_{g})_{\pi_{1}(S^{1})}$ is the space of coinvariants under the action of $\pi_{1}(S^{1})$, i.e., the quotient of $H_{2}(V_{g},\Sigma_{g})$ by the subgroup generated by the set $\{\varphi_{*}(\delta)-\delta\mid\delta\in H_{2}(V_{g},\Sigma_{g})$}.

Lemma 3.2**.**

We have $H_{2}(M_{\varphi},\partial M_{\varphi})\cong H_{0}(S^{1};\mathscr{H}_{2}(V_{g},\Sigma_{g}))\cong H_{2}(V_{g},\Sigma_{g})_{\pi_{1}(S^{1})}$.

3.3. Description of the inclusion homomorphism

Recall that the short exact sequence (2.1) is $\operatorname{Mod}(V_{g})$-equivariant. Let $\alpha\in H_{1}(V_{g})^{\pi_{1}(S^{1})}$ be a $\varphi_{*}$-invariant homology class. Pick an element $\tilde{\alpha}\in H_{1}(\Sigma_{g})$ such that $i_{*}(\tilde{\alpha})=\alpha$. Then $\varphi_{*}(\tilde{\alpha})-\tilde{\alpha}\in\operatorname{Ker}(i_{*})=\operatorname{Im}(\partial_{*})$.

Definition 3.3**.**

$d(\alpha):=[\partial_{*}^{-1}(\varphi_{*}(\tilde{\alpha})-\tilde{\alpha})]\in H_{2}(V_{g},\Sigma_{g})_{\pi_{1}(S^{1})}$.

It is easy to see that $d(\alpha)$ is independent of the choice of $\tilde{\alpha}$. Thus we obtain a well-defined map $d\colon H_{1}(V_{g})^{\pi_{1}(S^{1})}\to H_{2}(V_{g},\Sigma_{g})_{\pi_{1}(S^{1})}$.

Proposition 3.4**.**

The following diagram is commutative:

[TABLE]

where the bottom horizontal arrow is the inclusion homomorphism, and the vertical arrows are the isomorphisms in Lemmas 3.1 and 3.2.

3.4. Proof of Proposition 3.4

In this section, for a topological space $X$, we denote by $S_{n}(X)$ and $Z_{n}(X)$ the groups of singular $n$-chains and singular $n$-cycles, respectively.

Let $\alpha\in H_{1}(V_{g})^{\pi_{1}(S^{1})}$. Pick its lift $\tilde{\alpha}\in H_{1}(\Sigma_{g})$ such that $i_{*}(\tilde{\alpha})=\alpha$. Take a singular $1$-cycle $\tilde{a}\in Z_{1}(\Sigma_{g})$ representing the homology class $\tilde{\alpha}$. Then, $\varphi^{-1}_{\sharp}(\tilde{a})-\tilde{a}$ is a singular $1$-boundary in $V_{g}$ since $\varphi_{*}^{-1}(\tilde{\alpha})-\tilde{\alpha}\in\operatorname{Ker}(i_{*})$. Therefore, there exists $\sigma_{\varphi,\alpha}\in S_{2}(V_{g})$ such that $\partial\sigma_{\varphi,\alpha}=\varphi^{-1}_{\sharp}(\tilde{a})-\tilde{a}$.

First we compute the composition of $d$ and the right vertical map. We claim that $d(\alpha)$ is represented by the relative $2$-cycle $-\sigma_{\varphi,\alpha}\in Z_{2}(V_{g},\Sigma_{g})$. This follows from the equality $\varphi_{*}(\tilde{\alpha})-\tilde{\alpha}=-(\varphi^{-1}_{*}(\tilde{\alpha})-\tilde{\alpha})$ in $H_{1}(\Sigma_{g})_{\pi_{1}(S^{1})}$ and the relation $\partial\sigma_{\varphi,\alpha}=\varphi^{-1}_{\sharp}(\tilde{a})-\tilde{a}$. Hence, the right vertical map sends $d(\alpha)$ to the homology class represented by the relative $2$-cycle $-e^{0}\times\sigma_{\varphi,\alpha}\in Z_{2}(M_{\varphi},\partial M_{\varphi})$, where the symbol $\times$ means the cross product.

Next we compute the composition of the left vertical map and $i_{*}$. For this purpose, we set

[TABLE]

Here, $\Phi$ is the map defined in (3.1), and the unit interval is regarded as a singular $1$-chain in the obvious way. Actually, $\mathcal{Z}_{\alpha}$ is a $2$-cycle in $M_{\varphi}$.

Lemma 3.5**.**

The isomorphism in Lemma 3.1 sends $\alpha$ to the homology class of $\mathcal{Z}_{\alpha}$.

Proof.

We need to inspect the spectral sequence involved in Lemma 3.1. For simplicity we denote $M=M_{\varphi}$, and for every non-negative integer $q\geq 0$ let $M^{(q)}$ be the inverse image of the $q$-skeleton of $S^{1}$ by the projection map $\pi$. Thus we have $\emptyset\subset M^{(0)}=\pi^{-1}([0])\subset M^{(1)}=M$. Accordingly, the singular chain complex $S_{*}(M)$ has an increasing filtration: $\{0\}\subset S_{*}(M^{(0)})\subset S_{*}(M^{(1)})=S_{*}(M)$. The associated spectral sequence is the one that we consider.

Now let $\alpha\in H_{1}(V_{g})^{\pi_{1}(S^{1})}$. There is an isomorphism

[TABLE]

under which the homology class $[e^{1}\otimes\alpha]$ is mapped to the homology class of the relative $2$-cycle $\Phi_{\sharp}(I\times\tilde{a})$. However, since $e^{0}\times\sigma_{\varphi,\alpha}\in S_{2}(M^{(0)})$, it holds that

[TABLE]

Thus the homology class under consideration is now represented by a genuine $2$-cycle in $M$. Finally, we observe that the natural map

[TABLE]

coincides with the inclusion homomorphism. This completes the proof. ∎

By Lemma 3.5, it is enough to compute $i_{*}([\mathcal{Z}_{\alpha}])$. Since $\tilde{a}$ is a $1$-cycle in $\Sigma_{g}=\partial V_{g}$, the $2$-chain $\Phi_{\sharp}(I\times\tilde{a})$ lies in $\partial M_{\varphi}$. Hence

[TABLE]

This shows that $i_{*}([\mathcal{Z}_{\alpha}])$ is represented by the relative $2$-cycle $-e^{0}\times\sigma_{\varphi,\alpha}$. This completes the proof of Proposition 3.4.

3.5. Proof of Theorem 1.1

We describe the intersection form of $M_{\varphi}$ and prove Theorem 1.1.

First we claim that the second homology group $H_{2}(M_{\varphi})$ is naturally isomorphic to $U_{\varphi}^{{\mathbb{Z}}}:=\operatorname{Ker}(S-I_{g})\subset{\mathbb{Z}}^{g}$. In fact, by Lemma 3.1 we have $H_{2}(M_{\varphi})\cong H_{1}(V_{g})^{\pi_{1}(S^{1})}$, and the action of $\varphi$ on $H_{1}(V_{g})\cong\mathbb{Z}^{g}$ is given by the matrix $S$. Thus the claim follows.

We next claim that under the isomorphism $H_{2}(M_{\varphi})\cong U_{\varphi}^{{\mathbb{Z}}}$, the intersection form on $H_{2}(M_{\varphi})$ is transferred to the bilinear form $\langle\ ,\ \rangle_{\varphi}$. Since $\phi_{g}^{V}(\varphi)=\operatorname{Sign}M_{\varphi}$, this will complete the proof of Theorem 1.1. The proof of this claim consists of two steps.

Step 1. We give a description of the bilinear form on $H_{1}(V_{g})^{\pi_{1}(S^{1})}$ that is obtained by transferring the intersection form on $H_{2}(M_{\varphi})$. Let $\langle\cdot,\cdot\rangle_{V}\colon H_{2}(V_{g},\Sigma_{g})\times H_{1}(V_{g})\to\mathbb{Z}$ be the intersection product of the compact oriented $3$-manifold $V_{g}$. We have

[TABLE]

Let

[TABLE]

be the intersection product of $H_{0}(S^{1};\mathscr{H}_{2}(V_{g},\Sigma_{g}))$ and $H_{1}(S^{1};\mathscr{H}_{1}(V_{g}))$ followed by the contraction of the coefficients by the form $\langle\cdot,\cdot\rangle_{V}$. Under the isomorphisms in Lemmas 3.1 and 3.2, this is equivalent to the intersection product $H_{2}(M_{\varphi})\times H_{2}(M_{\varphi},\partial M_{\varphi})\to\mathbb{Z}$. By composing (3.3) and the homomorphism

[TABLE]

we obtain a bilinear form on $H_{1}(V_{g})^{\pi_{1}(S^{1})}$. Proposition 3.4 implies that this is equivalent to the intersection form on $H_{2}(M_{\varphi})$.

Step 2. We prove that the bilinear form on $H_{1}(V_{g})^{\pi_{1}(S^{1})}$ described in the previous paragraph is equivalent to $\langle\ ,\ \rangle_{\varphi}$ under the identification $H_{1}(V_{g})^{\pi_{1}(S^{1})}\cong U_{\varphi}^{{\mathbb{Z}}}$. Let $x=(x_{1},\ldots,x_{g})$, $y=(y_{1},\ldots,y_{g})\in U_{\varphi}^{{\mathbb{Z}}}\subset{\mathbb{Z}}^{g}$. We regard $x$ as an element of $H_{1}(V_{g})^{\pi_{1}(S^{1})}$. Then, we can take $\tilde{x}=\sum_{i=1}^{g}x_{i}\beta_{i}\in H_{1}(\Sigma_{g})$ as a lift of $x$ which we need to compute $d(x)$. Thus we have

[TABLE]

and hence $d(x)=(x_{1},\ldots,x_{g})\,{}^{t}Q\,{}^{t}(D_{1},\ldots,D_{g}).$ Therefore, the pairing of $x$ and $y$ by the bilinear form on $H_{1}(V_{g})^{\pi_{1}(S^{1})}$ described above is equal to

[TABLE]

Here we used the equality (3.2). This completes the proof of Theorem 1.1.

Remark 3.6*.*

There is a $2$-cocycle $m_{\lambda}$ on $\operatorname{Sp}(2g;\mathbb{Z})$ constructed by Turaev [20] which satisfies $[m_{\lambda}]=-[\tau_{g}]\in H^{2}(\operatorname{Sp}(2g;{\mathbb{Z}}))$, and Walker, in page 124 of his note111K. Walker (1991). On Witten’s $3$-manifold invariants, Preliminary Version [online]. Website https://canyon23.net/math/1991TQFTNotes.pdf [accessed 1 May 2020]., constructed a (unique) cobounding function $j\colon\operatorname{Mod}(\Sigma_{g})\to{\mathbb{Q}}$ of the sum $\rho^{*}\tau_{g}+\rho^{*}m_{\lambda}$ of $2$-cocycles. The $2$-cocycle $m_{\lambda}$ and the function $j$ depend on the choice of a lagrangian $\lambda\subset H_{1}(\Sigma_{g};{\mathbb{Q}})$. If we choose a suitable lagrangian $\lambda$, the restriction of $j$ to $\operatorname{Mod}(V_{g})$ is known to be a cobounding function of $\rho^{*}\tau_{g}$, and coincides with our function $\phi_{g}^{V}$. Gilmer and Masbaum [5, Proposition 6.9] described $j$ explicitly in a way which is similar to but different from ours.

Remark 3.7*.*

Since $Sy=y$ for any $y\in U_{\varphi}$, we have $\langle x,y\rangle_{\varphi}={}^{t}x\,{}^{t}QS\,y$ for any $x,y\in U_{\varphi}$. Since ${}^{t}QS$ is symmetric by (2.2), this gives a purely algebraic explanation for the symmetric property of the form $\langle\ ,\ \rangle_{\varphi}$ on $U_{\varphi}$.

Remark 3.8*.*

By Theorem 1.1, one can regard $\phi^{V}_{g}$ as a $1$-cochain on $\operatorname{urSp}(2g;{\mathbb{Z}})$. For $g\geq 3$, it is the unique $1$-cochain which cobounds $\tau_{g}$ on $\operatorname{urSp}(2g;{\mathbb{Z}})$ since $H^{1}(\operatorname{urSp}(2g;{\mathbb{Z}}))=0$; see [19, Corollary 4.4].

4. Evaluation of Meyer functions

4.1. The Meyer function on the hyperelliptic mapping class group

There is a unique $1$-cochain $\phi_{g}^{\mathcal{H}}\colon\mathcal{H}(\Sigma_{g})\to{\mathbb{Q}}$ such that for any $\varphi_{1},\varphi_{2}\in\mathcal{H}(\Sigma_{g})$,

[TABLE]

The $1$-cochain $\phi_{g}^{\mathcal{H}}$ is called the Meyer function on the hyperelliptic mapping class group of genus $g$; see [4, 18].

Recall the element $s_{1}=t_{2}t_{3}t_{1}t_{2}\in\mathcal{H}(V_{g})\subset\mathcal{H}(\Sigma_{g})$ which was defined in Section 2.4.

Lemma 4.1**.**

$\phi_{g}^{\mathcal{H}}(s_{1})=(2g+3)/(2g+1)$.

Proof.

Set $T_{i}=\rho(t_{i})$ for every $i\in\{1,2,3\}$. Using (4.1), we have

[TABLE]

As was shown in [4, Lemma 3.3] and [18, Proposition 1.4], we have $\phi_{g}^{\mathcal{H}}(t_{i})=(g+1)/(2g+1)$ for all $i\in\{1,2,3\}$. Also, by a direct computation we obtain $\tau_{g}(T_{1},T_{2})=0$, $\tau_{g}(T_{3},T_{1}T_{2})=0$, and $\tau_{g}(T_{2},T_{3}T_{1}T_{2})=1$. The result follows from these equalities. ∎

4.2. The Meyer function on the handlebody group

Recall from the introduction that we defined $\phi_{g}^{V}\colon\operatorname{Mod}(V_{g})\to\mathbb{Z}$ by $\varphi\mapsto\operatorname{Sign}M_{\varphi}$, where $M_{\varphi}$ is the mapping torus of $\varphi$.

Lemma 4.2**.**

The function $\phi^{V}_{g}\colon\operatorname{Mod}(V_{g})\to\mathbb{Z}$ cobounds the cocycle $\rho^{*}\tau_{g}$ in the handlebody group $\operatorname{Mod}(V_{g})$. If $g\geq 3$, $\phi^{V}_{g}$ is the unique cobounding function of $\rho^{*}\tau_{g}$.

Proof.

The uniqueness follows from the fact that $H_{1}(\operatorname{Mod}(V_{g}))$ is torsion when $g\geq 3$.

For given two mapping classes $\varphi,\psi\in\operatorname{Mod}(V_{g})$, there is an oriented $V_{g}$-bundle $W(\varphi,\psi)\to P$ such that the monodromy along $\ell_{1}$, $\ell_{2}$ and $\ell_{3}$ are $\varphi$, $\psi$ and $(\varphi\psi)^{-1}$, respectively. The boundary of $W(\varphi,\psi)$ is written as

[TABLE]

Note that $M_{\varphi^{-1}}$ is diffeomorphic to $-M_{\varphi}$ under an orientation-preserving diffeomorphism, where $-M_{\varphi}$ denotes the mapping torus $M_{\varphi}$ with orientation reversed. Since the signature of $\partial W(\varphi,\psi)$ is zero, Novikov additivity implies that

[TABLE]

This shows that $\phi^{V}_{g}$ is a cobounding function of $\rho^{*}\tau_{g}$ restricted to $\operatorname{Mod}(V_{g})$. ∎

Since $\dim V_{A,B}\leq 4g$ for any $A,B\in\operatorname{Sp}(2g;{\mathbb{Z}})$, the signature cocycle $\tau_{g}$ is a bounded 2-cocycle. Therefore, it represents a class in the second bounded cohomology group $H_{b}^{2}(\operatorname{Mod}(\Sigma_{g}))$. The image of $[\tau_{g}]$ under the natural homomorphism $H_{b}^{2}(\operatorname{Mod}(\Sigma_{g});\mathbb{Q})\to H_{b}^{2}(\mathcal{H}(\Sigma_{g});\mathbb{Q})$ is non-trivial since the Meyer function $\phi_{g}^{\mathcal{H}}$ is unbounded. In contrast, we have:

Proposition 4.3**.**

Under the natural homomorphism $H_{b}^{2}(\operatorname{Mod}(\Sigma_{g});\mathbb{Q})\to H_{b}^{2}(\operatorname{Mod}(V_{g});\mathbb{Q})$, the image of the cohomology class $[\tau_{g}]$ vanishes.

Proof.

The restriction of the signature cocycle $\tau_{g}$ to $\operatorname{Mod}(V_{g})$ is cobounded by the function $\phi_{g}^{V}$, and $\phi_{g}^{V}$ is a bounded function since the rank of $H_{2}(M_{\varphi})$ is at most $g$. ∎

4.3. Computation of the Meyer function on the handlebody group

Theorem 1.1 shows that the bilinear form $\langle\ ,\ \rangle_{\varphi}$ on $U_{\varphi}$, whose signature coincides with $\phi_{g}^{V}(\varphi)$, can be computed from the homological monodromy $\rho(\varphi)\in\operatorname{urSp}(2g;{\mathbb{Z}})$. In more detail, if $\rho(\varphi)=\begin{pmatrix}P&Q\\ O_{g}&S\end{pmatrix}$, then $U_{\varphi}=\operatorname{Ker}(S-I_{g})\subset{\mathbb{Q}}^{g}$ and $\langle x,y\rangle_{\varphi}={}^{t}x\,{}^{t}Q\,y$ for $x,y\in U_{\varphi}$.

The $1$-cochain $\phi_{g}^{V}$, regarded as the one defined on $\operatorname{urSp}(2g;{\mathbb{Z}})$, is stable with respect to $g$ in the following sense. For every non-negative integer $g\geq 0$, there is a natural embedding $\iota\colon\operatorname{urSp}(2g;{\mathbb{Z}})\hookrightarrow\operatorname{urSp}(2(g+1);{\mathbb{Z}})$;

[TABLE]

where

[TABLE]

Then $\phi_{g+1}^{V}(\iota(A))=\phi_{g}^{V}(A)$ for any $A\in\operatorname{urSp}(2g;{\mathbb{Z}})$.

Lemma 4.4**.**

For any positive integer $m$, we have $\phi_{g}^{V}(t_{1}^{m})=1$.

Proof.

Since the action of $\rho(t_{1}^{m})$ on $H_{1}(\Sigma_{g})$ is given by

[TABLE]

we may assume that $g=1$. Then $\rho(t_{1}^{m})=\begin{pmatrix}1&m\\ 0&1\end{pmatrix}$, and $\operatorname{Ker}(S-I_{1})={\mathbb{Z}}$ on which the pairing is given by the $1\times 1$ matrix $\begin{pmatrix}m\end{pmatrix}$. Hence $\phi_{g}^{V}(t_{1}^{m})=1$, as required. ∎

Lemma 4.5**.**

$\phi^{V}_{g}(s_{1})=1$.

Proof.

The proof proceeds as in the same way as the previous lemma. In this case we may assume that $g=2$. Then

[TABLE]

The rest of computation is straightforward, so we omit it. ∎

4.4. Proof of Theorem 1.2

Since both the $1$-cochains $\phi_{g}^{\mathcal{H}}$ and $\phi_{g}^{V}$ cobound the signature cocycle, their difference becomes a ${\mathbb{Q}}$-valued homomorphism on $\mathcal{H}(V_{g})=\mathcal{H}(\Sigma_{g})\cap\operatorname{Mod}(V_{g})$.

We compare the homomorphism $\phi_{g}^{\mathcal{H}}-\phi_{g}^{V}$ with the generator $\mu\in H^{1}(\mathcal{H}(V_{g}))$ in Corollary 2.6. It is sufficient to evaluate $\phi_{g}^{\mathcal{H}}-\phi_{g}^{V}$ on $s_{1}$ if $g$ is even, and on $t_{1}s_{1}^{\frac{g+1}{2}}$ if $g$ is odd. By Lemmas 4.1 and 4.5 we immediately obtain

[TABLE]

This settles the case where $g$ is even. When $g$ is odd, we compute

[TABLE]

Here, we used the fact that $\phi_{g}^{\mathcal{H}}-\phi_{g}^{V}$ is a homomorphism on $\mathcal{H}(V_{g})$ in the first line; we used the fact that $\phi_{g}^{\mathcal{H}}(t_{1})=(g+1)/(2g+1)$ (see the proof of Lemma 4.1), Lemma 4.4 and (4.2) in the second line. This completes the proof of Theorem 1.2.

Acknowledgments. The authors would like to thank Susumu Hirose for his helpful comments. Y. K.  is supported by JSPS KAKENHI 18K03308. M. S.  is supported by JSPS KAKENHI 18K03310.