2.1 The mapping class group and the symplectic representation

In this subsection, we recall some basic facts about the mapping class group; see [Reference Farb and Margalit19] for a detailed treatment.

Throughout the paper, let $\Sigma _g$ denote a closed genus g surface and $\Sigma _g^1$ denote a genus g surface with one puncture. Let $\Gamma _g$ (resp. $\Gamma _g^1$ ) be the mapping class group of $\Sigma _g$ (resp. $\Sigma _g^1$ ), that is, the group of isotopy classes of orientation-preserving diffeomorphisms of the surface, and $\Pi _g = \pi _1(\Sigma _g)$ . These groups fit into the Birman exact sequence:

(2.1) $$ \begin{align} 1 \to \Pi_g \to \Gamma_g^1 \to \Gamma_g \to 1. \end{align} $$

Given a simple closed curve a in $\Sigma _{g}$ (resp. $\Sigma _g^1$ ), denote by $T_a$ the left-handed Dehn twist of a. A separating twist is a Dehn twist $T_a$ , where a is a separating curve, and a bounding pair map is $T_{a}T_{b}^{-1}$ , where a and b are homologous nonseparating, nonintersecting, simple closed curves.

The singular homology group $H_1(\Sigma _g^1,\mathbb {Z}) \cong H_1(\Sigma _g,\mathbb {Z})$ , which we denote by H, has a symplectic structure given by the the algebraic intersection pairing $\hat {i}: H \wedge H \to \mathbb {Z}$ . The action of $\Gamma _g$ (resp. $\Gamma _g^1$ ) on H respects this pairing. This yields the symplectic representation of $\Gamma _g$ (resp. $\Gamma _g^1$ ), and we have the short exact sequence

(2.2) $$ \begin{align} 1 \to \mathcal{I}_g \text{ (resp. }\mathcal{I}_g^1 \text{)} \to \Gamma_g \text{ (resp. }\Gamma_g^1 \text{)} \to \operatorname{\mathrm{Sp}}(2g,\mathbb{Z}) \to 1, \end{align} $$

where $\mathcal {I}_g$ (resp. $\mathcal {I}_g^1$ ) is called the Torelli group. By [Reference Birman9, Reference Powell35], the Torelli group is generated by separating twists and bounding pair maps.

The Johnson homomorphism was introduced by Johnson in [Reference Johnson27] to study the action of the Torelli group on the third nilpotent quotient of a surface group. We provide the following characterization. Recall that for any symplectic free $\mathbb {Z}$ -module V with symplectic basis $\alpha _i,\beta _i$ ( $i=1,\ldots ,g$ ), the form

(2.3) $$ \begin{align} \omega_V= \left(\sum_{i=1}^g \alpha_i \wedge \beta_i\right) \end{align} $$

does not depend on the choice of symplectic basis. When $V=H$ , we simply write $\omega $ for this form. Set $L=\wedge ^3H$ , and view H as a subgroup of L via the embedding $h\mapsto \omega \wedge h$ . The Johnson homomorphism for a once-punctured surface is a group homomorphism $J:\mathcal {I}_{g}^1 \to L$ ; by the previous paragraph, it suffices to describe how J operates on separating twists and bounding pair maps. If $T_a$ is a separating twist, then $J(T_a)=0$ . Suppose $T_aT_{b}^{-1}$ is a bounding pair map. The curves a and b separate $\Sigma _{g}^1$ into two subsurfaces; let S be the subsurface which does not contain the puncture. The inclusion $S\hookrightarrow \Sigma _{g}^1$ induces an injective map $H_1(S,\mathbb {Z}) \to H$ which respects the symplectic forms on these spaces. Denote the image of this map by W. Then $\omega $ restricts to $\omega _W$ on W, and

(2.4) $$ \begin{align} J(T_{a}T_{b}^{-1}) = \omega_{W}\wedge [a]. \end{align} $$

The Johnson homomorphism for $\Sigma _{g}$ is a homomorphism $J:\mathcal {I}_g\to L/H$ and operates on separating twists and bounding pair maps as above, except that S may be either of the two subsurfaces cut off by a and b.

2.2 Construction of the Ceresa class

In this section, we construct a class in $H^1(\Gamma _g,L/H)$ whose restriction to $\mathcal {I}_g$ equals twice the Johnson homomorphism. By the work of Hain and Matsumoto [Reference Hain and Matsumoto22], this class with $\ell $ -adic coefficients agrees with the universal Ceresa class over $\mathcal {M}_g$ . We discuss this further in §3.2.

Let $F_{2g} = \pi _1(\Sigma _g^1)$ , which is the rank- $2g$ free group, and

$$ \begin{align*} F_{2g} = L^1F_{2g} \supseteq L^2F_{2g} \supseteq L^3F_{2g} \supseteq \ldots \end{align*} $$

be the lower central series of $F_{2g}$ , that is, $L^{k+1}F_{2g} = [F_{2g},L^kF_{2g}]$ . The k-th nilpotent quotient of $F_{2g}$ is $N_k = F_{2g} / L^{k} F_{2g}$ . Note that $N_2\cong H$ . Set $\operatorname {\mathrm {gr}}_L^k F_{2g} = L^kF_{2g}/L^{k+1}F_{2g}$ , which is a central subgroup of $N_{k+1}$ . We note that the $N_k$ and the $\operatorname {\mathrm {gr}}_L^k F_{2g}$ are all characteristic quotients of $F_{2g}$ and thus carry natural actions of $\operatorname {\mathrm {Aut}}(F_{2g})$ ; in particular, they carry actions of the mapping class group $\Gamma _g^1$ . What’s more, the action of $\operatorname {\mathrm {Aut}}(F_{2g})$ on $\operatorname {\mathrm {gr}}_L^kF_{2g}$ factors through the natural homomorphism $\operatorname {\mathrm {Aut}}(F_{2g}) \rightarrow \operatorname {\mathrm {GL}}(H)$ .

By [Reference Morita31, Proposition 2.3], $\operatorname {\mathrm {Aut}}(N_3)$ fits into an exact sequence of groups

(2.5) $$ \begin{align} 1 \to \operatorname{\mathrm{Hom}}(H, \operatorname{\mathrm{gr}}_L^2 F_{2g}) \xrightarrow{\phi} \operatorname{\mathrm{Aut}}(N_3) \xrightarrow{p} \operatorname{\mathrm{GL}}(H) \to 1. \end{align} $$

Here, for any $f \in \operatorname {\mathrm {Hom}}(H,\operatorname {\mathrm {gr}}_L^2 F_{2g})$ the action of $\phi (f)$ on $N_3$ is to send $\gamma \in N_3$ to $\gamma f([\gamma ])$ , where $[\gamma ]$ is the image of $\gamma $ under the natural projection to H. Because $\operatorname {\mathrm {Hom}}(H, \operatorname {\mathrm {gr}}_L^2 F_{2g})$ is abelian, we write the group operation additively. The group $\operatorname {\mathrm {Aut}}(N_3)$ acts on $\operatorname {\mathrm {Hom}}(H, \operatorname {\mathrm {gr}}_L^2F_{2g})$ by conjugation.

Let $\tau $ be an element of $\operatorname {\mathrm {Aut}}(N_3)$ such that $p(\tau ) = -I$ . Since $p(\tau )$ is central in $\operatorname {\mathrm {GL}}(H)$ , any commutator in $\operatorname {\mathrm {Aut}}(N_3)$ of the form $[x,\tau ]$ lies in $\operatorname {\mathrm {Hom}}(H, \operatorname {\mathrm {gr}}_L^2F_{2g})$ . Define

(2.6) $$ \begin{align} \mu_{\tau}:\operatorname{\mathrm{Aut}}(N_3) \to \operatorname{\mathrm{Hom}}(H, \operatorname{\mathrm{gr}}_L^2 F_{2g}) \hspace{30 pt} x \mapsto [x,\tau]. \end{align} $$

Proposition 2.1. The map $\mu _{\tau }$ is a crossed homomorphism, and its cohomology class

$$ \begin{align*} \mu \in H^1(\operatorname{\mathrm{Aut}}(N_3), \operatorname{\mathrm{Hom}}(H, \operatorname{\mathrm{gr}}_L^2 F_{2g})) \end{align*} $$

is independent of the choice of $\tau $ .

Proof. That $\mu _\tau $ is a crossed homomorphism follows from the fact that $[xy,\tau ] = [y,\tau ]^x[x,\tau ]$ , and hence

$$ \begin{align*} \mu_\tau(xy) = [xy,\tau] = \mu_\tau(x) + x\cdot \mu_\tau(y). \end{align*} $$

Now, suppose we had made a different choice $\tau '$ ; then $\tau ' = t\tau $ for some $t\in \ker p$ . One checks, using the fact that $\ker p$ is abelian, that

$$ \begin{align*} [x,t\tau] = t^x t^{-1} [x,\tau] \end{align*} $$

which is to say that

$$ \begin{align*} \mu_{\tau'}(x) = \mu_{\tau}(x) - t + x \cdot t \end{align*} $$

so $\mu _{\tau '}$ is cohomologous to $\mu _\tau $ , as claimed.

The preimage of $\ker p$ under the natural morphism $\Gamma _g^1 \to \operatorname {\mathrm {Aut}}(N_3)$ is the Torelli group $\mathcal {I}_g^1$ , and the restriction of this morphism to $\mathcal {I}_g^1$ is the Johnson homomorphism. By the work of Johnson [Reference Johnson27], the homomorphism J is not surjective onto $\operatorname {\mathrm {Hom}}(H, \operatorname {\mathrm {gr}}_L^2 F_{2g})$ ; rather, its image is the natural $\operatorname {\mathrm {GSp}}(H)$ -equivariant inclusion of $L = \wedge ^3 H$ into

$$ \begin{align*} \operatorname{\mathrm{Hom}}(H, \operatorname{\mathrm{gr}}_L^2 F_{2g}) = \operatorname{\mathrm{Hom}}(H, \wedge^2 H). \end{align*} $$

We can thus inflate $\mu $ to $\Gamma _g^1$ to get a cohomology class $\mu \in H^1(\Gamma ^1_g, L)$ represented by the cocycle

(2.7) $$ \begin{align} \mu_{\tau}(\gamma) = J([\gamma,\tau]), \end{align} $$

where $\tau \in \Gamma _g^1$ acts on H as $-I$ . We say that $\tau $ is a hyperelliptic quasi-involution; a hyperelliptic quasi-involution that is an honest involution is called a hyperelliptic involution. Following Proposition 2.1, the class $\mu $ is defined independent of the choice of $\tau $ .

Proof. Pick an element $\gamma \in \mathcal {I}_g^1$ and fix a hyperelliptic quasi-involution $\tau \in \Gamma _g^1$ . Because $\tau $ acts as $-I$ on H, it also acts as $-I$ on $\wedge ^3H$ . Therefore, we have

$$ \begin{align*} J([\gamma,\tau]) = J(\gamma)+ J(\tau\gamma^{-1} \tau^{-1}) = J(\gamma)+ \tau \cdot J(\gamma^{-1}) = 2J(\gamma). \end{align*} $$

The uniqueness of $\mu $ follows from [Reference Hain and Matsumoto22, Proposition 5.5].

Let $\nu \in H^1(\Gamma _g, L/H)$ denote the image of $\mu $ under the composition

$$ \begin{align*} H^1(\Gamma_g^1,L) \rightarrow H^1(\Gamma_g^1,L/H) \cong H^1(\Gamma_g,L/H). \end{align*} $$

The map $H^1(\Gamma _g,L/H) \to H^1(\Gamma _g^1,L/H)$ is induced by restriction, and is an isomorphism by [Reference Hain and Matsumoto22, Proposition 10.3]. Similar to the once-punctured case, the class $\nu $ is represented by the cocycle $\gamma \mapsto J([\gamma ,\tau ])$ , where $\tau \in \Gamma _g$ is a hyperelliptic quasi-involution.

The Ceresa class $\nu (\gamma )$ lies in $H^1(\langle \gamma \rangle , L/H) \cong L / ((\gamma -1)L+H)$ . It is certainly trivial for any $\gamma $ which commutes with $\tau $ , which is to say it is trivial for any $\gamma $ in the hyperelliptic mapping class group. But the converse is not true; for instance, if $\gamma $ is in the kernel of the Johnson homomorphism then so is $[\gamma ,\tau ]$ , so $\mu (\gamma ) = J([\gamma ,\tau ]) = 0$ , but such a $\gamma $ certainly need not be hyperelliptic. More generally, any mapping class whose commutator with $\tau $ lands in the Johnson kernel has trivial Ceresa class.

When the action of $\gamma $ on H has no eigenvalues that are roots of unity, the group $H^1(\langle \gamma \rangle , L/H)$ is finite and the Ceresa class is torsion. At the other extreme, if $\gamma $ lies in the Torelli group, $H^1(\langle \gamma \rangle , L/H) = L/H$ and the Ceresa class is an element of this free $\mathbb {Z}$ -module of positive rank and can be of infinite order.

The case of primary interest in the present paper is that where $\gamma $ is a product of commuting Dehn twists, or a positive multitwist. In this case, the action of $\gamma $ on H is unipotent, and $H^1(\langle \gamma \rangle , L/H)$ is infinite; however, in this case, as we shall prove in §6, the Ceresa class is still of finite order.

In fact, we will show how the order of the Ceresa class can be explicitly computed, though the computation is somewhat onerous. We will, along the way, prove the analogous statement for multitwists in the punctured mapping class group $\Gamma _g^1$ as well.

Our method for proving Theorem 2.2 will be to show that the Ceresa class lies in a canonical finite subgroup of $H^1(\langle \gamma \rangle , L/H)$ , which we might think of as the tropical intermediate Jacobian; see Remark 6.11. A notion of tropical intermediate Jacobian was proposed by Mikhalkin and Zharkov in [Reference Mikhalkin and Zharkov30]; it would be interesting to know whether the two notions agree in the context considered here.

The next two sections will explain the relationship between the topological, tropical and local-algebraic pictures; the reader whose interest is solely in the mapping class group can skip ahead to §5.

3.1 Monodromy

Our discussion on the monodromy of a degenerating family of stable curves mainly follows [Reference Amini, Bloch, Burgos Gil and Fresán2, §3.2] and [Reference Asada, Matsumoto and Oda4, §1.1]. For details on the construction of a local universal family of a stable curve, see [Reference Tsuchiya, Ueno and Yamada41, §3]. Our goal is to recall the nonabelian Picard–Lefschetz formula [Reference Asada, Matsumoto and Oda4, Theorem 2.2] which says that the monodromy action on the fundamental group of a smooth fiber is given by a multitwist.

Let $X_0$ be a stable complex curve of genus $g\geq 2$ . Let $\mathcal {Y} \to \mathcal {D}$ be the local universal family for $X_0$ as was constructed in [Reference Tsuchiya, Ueno and Yamada41, Theorem 3.1.5]. The base $\mathcal {D}$ is homeomorphic to $D^{3g-3}$ , where D denotes a small open complex disc centered at $0$ . Let $\mathcal {B}\subset \mathcal {D}$ be the discriminant locus of $\mathcal {Y}\to \mathcal {D}$ and $\mathcal {D}^* = \mathcal {D}\setminus \mathcal {B}$ ; each fiber $\mathcal {Y}_p$ for $p\in \mathcal {D}^*$ is diffeomeorphic to a closed surface $\Sigma _g$ . Choose a point $p_0\in \mathcal {D}^*$ sufficiently close to $0$ at which all loops in $\mathcal {D}^*$ will be based when we consider its fundamental group.

The combinatorial data of a stable curve are recorded in its dual graph, which is a connected vertex-weighted graph defined in the following way. Recall that a vertex-weighted graph $\mathbf {G}$ is a connected graph G, possibly with loops or multiple edges, together with a nonnegative integer $w_v$ for each vertex v of G. The dual graph $\mathbf {G}$ of $X_0$ consists of

1. - a vertex $v_{C}$ for each irreducible component C of $X_{0}$ whose weight is the arithmetic genus of C, and

2. - an edge e between $v_C$ and $v_{C'}$ for each node n in the intersection of C and $C'$ .

For each edge $e_i$ of $\mathbf {G}$ , choose a small loop $\ell _i'$ in the smooth locus of $X_0$ that goes around $n_i$ . Shrinking $\mathcal {D}$ if necessary, the inclusion $X_0\hookrightarrow \mathcal {Y}$ admits a retraction $\mathcal {Y} \to X_0$ so that $\mathcal {Y}\to X_0 \to \mathcal {Y}$ is homotopic to the identity. This gives rise to a specialization map $r_p:\mathcal {Y}_p \to X_0$ for each fiber $\mathcal {Y}_p$ over $p \in \mathcal {D}$ . Then each $\ell _i = r_p^{-1}(\ell _i')$ defines a closed curve in $\mathcal {Y}_p$ .

The discriminant locus $\mathcal {B}$ is a normal crossings divisor [Reference Asada, Matsumoto and Oda4, Proposition 1.1] (see also [Reference Knudsen28, Theorem 2.7]). Following [Reference Asada, Matsumoto and Oda4, Proposition 1.1(3)], choose coordinates $z_1,\ldots ,z_{3g-3}$ on $\mathcal {D}$ so that $B_i$ , the irreducible component of $\mathcal {B}$ consisting of those $p\in \mathcal {D}$ such that $\ell _i$ is contractible in $\mathcal {Y}_p$ , has the form $B_i = \{z_i=0\} \cap \mathcal {D}$ . In particular, $\mathcal {Y}_0 = X_0$ . Then $\mathcal {D}^*$ is homeomorphic to $(D^*)^N\times (D)^{3g-3-N}$ , where N is the number of edges of G and $D^* = D\setminus 0$ . Thus, $\pi _1(\mathcal {D}^*)$ is isomorphic to $\oplus _{i=1}^N\mathbb {Z}\cdot \lambda _i$ , where $\lambda _i$ is a loop in $\mathcal {D}^*$ based at $p_0$ that goes around $B_i$ . The monodromy action on $\Pi _g$ is given by a nonabelian Picard–Lefschetz formula [Reference Asada, Matsumoto and Oda4, Theorem 2.2]:

(3.1) $$ \begin{align} \rho_{\mathcal{Y}}:\pi_1(\mathcal{D}^*) \to \operatorname{\mathrm{Out}}(\Pi_g) \hspace{30 pt} \lambda_i \mapsto T_{\ell_i}^{-1}. \end{align} $$

Let $\mathfrak {Y} \to D$ be a one-parameter degeneration such that $\mathfrak {Y}_0 = X_0$ is the only singular fiber. Suppose that the local equation in $\mathfrak {Y}$ near the node $n_i$ corresponding to the edge $e_i$ of the dual graph is $xy=t^{c_i}$ , where t is the parameter on D and $c_i \in \mathbb {Z}_{>0},i=1,\ldots ,N$ . The following proposition is a variant of [Reference Oda32, Main Lemma].

Proposition 3.1. The restriction of the monodromy map $\rho _{\mathcal {Y}}$ to $\pi _1(D^*) = \mathbb {Z}\cdot \gamma $ is given by

$$ \begin{align*} \rho_{\mathfrak{Y}}:\pi_1(D^*) \to \operatorname{\mathrm{Out}}(\Pi_g) \hspace{30 pt} \gamma \mapsto \prod_{i=1}^N T_{\ell_i}^{-c_i}. \end{align*} $$

Proof. The map $\mathfrak {Y} \to D$ is given by the pullback of $\mathcal {Y}\to \mathcal {D}$ under a map $j:D\to \mathcal {D}$ , and $c_i$ is the multiplicity at which $j(D)$ intersects the divisor $B_i$ . Explicitly, with t being the local coordinate on D, the map j is given by

$$ \begin{align*} j(t) = (a_1t^{c_1},\ldots,a_{N}t^{c_N},0,\ldots,0) + \text{higher order terms}, \end{align*} $$

where $a_i\in \mathbb {C}^*$ . Composing with the orthogonal projection to $B_i^{\perp } = \{z_k=0\, | \, k\neq i\}$ yields the map $D\to B_i^{\perp }$ given by $t\mapsto t^{c_i}$ , and therefore $\pi _{1}(D^*) \to \pi _1(B_i^{\perp }\setminus 0)$ is given by ${\gamma \mapsto c_i\cdot \lambda _i}$ . The proposition now follows from Equation (3.1).

3.2 The $\ell $ -adic Ceresa class for algebraic curves over $\mathbb {C}(\!(t)\!)$

In this subsection, we recall the definition of the Ceresa cycle associated with an algebraic curve and its induced class in Galois cohomology, following [Reference Hain and Matsumoto22]. Using comparison theorems, this class agrees with the topological Ceresa class with $\ell $ -adic coefficients, justifying the definition of the Ceresa class $\nu $ in §2.

Let K be a field of characteristic $0$ , $G_K$ its absolute Galois group and $\ell $ a fixed prime number. Let X be a smooth, complete, genus $g\geq 3$ curve over K. For the moment, suppose X has a K-rational point $\xi $ , which yields an embedding $\Phi _{\xi }:X\to \operatorname {\mathrm {Jac}}(X)$ . Define algebraic cycles in $\text {CH}_1(\operatorname {\mathrm {Jac}}(X))$ given by $X_{\xi }:=(\Phi _{\xi })_*(X)$ and $X_{\xi }^{-}:=(\iota )_*(X_{\xi })$ , where $\iota $ is the inverse map on $\operatorname {\mathrm {Jac}}(X)$ . By [Reference Hain and Matsumoto22, § 4.3], the cycle $X_{\xi } - X_{\xi }^{-}$ is null-homologous. Thus, the image of $X_{\xi } - X_{\xi }^{-}$ under the $\ell $ -adic Abel–Jacobi map produces a Galois cohomology class

$$ \begin{align*} \mu^{(\ell)}(X,\xi) \in H^1(G_K, H_{\operatorname{\mathrm{\acute{e}t}}}^{2g-3}((\operatorname{\mathrm{Jac}} X)_{\overline{K}},\, \mathbb{Z}_\ell(g-1))). \end{align*} $$

Via Poincaré duality,

$$ \begin{align*} H_{\operatorname{\mathrm{\acute{e}t}}}^{2g-3}((\operatorname{\mathrm{Jac}} X)_{\overline{K}},\mathbb{Z}_{\ell}(g-1)) \cong H_{\operatorname{\mathrm{\acute{e}t}}}^{3}((\operatorname{\mathrm{Jac}} X)_{\overline{K}},\mathbb{Z}_{\ell}(1))(-1) \cong (\wedge^3H_{\operatorname{\mathrm{\acute{e}t}}}^{1}(X_{\overline{K}},\mathbb{Z}_{\ell}(1)))(-1). \end{align*} $$

Let $H_{\mathbb {Z}_\ell } = H^1_{\operatorname {\mathrm {\acute {e}t}}}(X_{\overline {K}}, \mathbb {Z}_\ell (1))$ , $L_{\mathbb {Z}_\ell } = (\wedge ^3H_{\mathbb {Z}_\ell })(-1)$ and $\omega \in \wedge ^2H_{\mathbb {Z}_\ell }$ the polarization.

The map $h\mapsto \omega \wedge h$ yields an embedding $H_{\mathbb {Z}_\ell } \hookrightarrow L_{\mathbb {Z}_\ell }$ . The $\ell $ -adic Ceresa class, denoted by $\nu ^{(\ell )}(X)$ , is the image of $\mu ^{(\ell )}(X,\xi )$ under the map $H^1(G_K, L_{\mathbb {Z}_\ell }) \to H^1(G_K, L_{\mathbb {Z}_\ell }/H_{\mathbb {Z}_\ell })$ , where we view $\mu ^{(\ell )}(X,\xi )$ as an element of $H^1(G_K, L_{\mathbb {Z}_\ell })$ . By [Reference Hain and Matsumoto22, §10.4], the class $\nu ^{(\ell )}(X)$ only depends on the curve $X/K$ and can be defined when X has no K-rational point. Hain and Matsumoto construct a universal characteristic class

$$ \begin{align*} \hat{n}^{(\ell)} \in H^1(\operatorname{\mathrm{Out}} \Pi^{(\ell)}_g, L_{\mathbb{Z}_\ell}/H_{\mathbb{Z}_\ell}) \end{align*} $$

which is the $\ell $ -adic analog of $\nu $ defined in §2. The class $\nu $ with $\mathbb {Z}_\ell $ coefficient corresponds to $(\hat {\rho }_{X}^{(\ell )})^{*}(\hat {n}^{(\ell )})$ under the comparison map

$$ \begin{align*} H^1(\Gamma_g,L/H)\otimes \mathbb{Z}_{\ell} \xrightarrow{\sim} H^1_{\text{\'et}}(\mathcal{M}_g\otimes \overline{K},L_{\mathbb{Z}_\ell}/H_{\mathbb{Z}_\ell}), \end{align*} $$

where

$$ \begin{align*} \hat{\rho}_{X}^{(\ell)}:\pi_1(\mathcal{M}_g,X_{\overline{K}})\to \operatorname{\mathrm{Out}} \pi_1^{(\ell)}(X_{\overline{K}}) \end{align*} $$

is the universal monodromy representation. Let $n^{(\ell )}(X)\in H^1(G_K,L_{\mathbb {Z}_\ell }/H_{\mathbb {Z}_\ell })$ denote the pullback of $\hat {n}^{(\ell )}$ under the natural action $G_K \to \operatorname {\mathrm {Out}} \pi _1^{(\ell )}(X_{\overline {K}})$ . The $\ell $ -adic Harris–Pulte theorem [Reference Hain and Matsumoto22, Theorem 10.5] asserts

(3.2) $$ \begin{align} n^{(\ell)}(X) = \nu^{(\ell)}(X). \end{align} $$

We now show that the $\ell $ -adic Ceresa class of a curve over $\mathbb {C}(\!(t)\!)$ is torsion. We obtain this by showing that the $\ell $ -adic Ceresa class is, in a natural sense, the $\ell $ -adic completion of the Ceresa class of product of Dehn twists attached to the curve in the previous section. This fact justifies calling that topologically defined class ‘the Ceresa class’, and allows us to apply Theorem 6.8 to the $\ell $ -adic Ceresa class.

Proof. We begin by reducing to the semistable reduction case. By the semistable reduction theorem, there is a positive integer n such that the pullback $X'$ of X by the map

$$ \begin{align*} \phi:\operatorname{\mathrm{Spec}} K \to \operatorname{\mathrm{Spec}} K \hspace{20pt} \phi^{\#}(t) = t^n \end{align*} $$

has semistable reduction. The map $\phi $ induces an endomorphism of $H^1(G_K,L_{\mathbb {Z}_\ell }/H_{\mathbb {Z}_\ell })$ which is multiplication by n. This means that $\nu ^{(\ell )}(X)$ is torsion if and only if $\nu ^{(\ell )}(X')$ is torsion.

Therefore, we may assume that X has a semistable model $\mathfrak {X}$ defined over with special fiber $X_0$ . The étale local equation in $\mathfrak {X}$ of each node of $X_0$ is $xy=t^{c_i}$ for some $c_i\in \mathbb {Z}_{>0}$ . Let

$$ \begin{align*} S = \operatorname{\mathrm{Spec}} \mathbb{C}[\![x_1,\ldots,x_{3g-3}]\!], \hspace{20pt} S' = \operatorname{\mathrm{Spec}} \mathbb{C}[\![x_1^{\pm},\ldots, x_{N}^{\pm}, x_{N+1}, \ldots x_{3g-3}]\!], \end{align*} $$

where N is the number of nodes of $X_0$ . The map is the pullback of the (algebraic) local universal family $\mathcal {X} \to S$ by a morphism of the form

$$ \begin{align*} i^{\#}(x_{k}) = a_kt^{c_k} + \text{higher order terms}, \hspace{20pt} \end{align*} $$

where $a_k\in \mathbb {C}^*$ . Define an analytic family $\mathfrak {Y} \to D$ by the pullback of $\mathcal {Y} \to \mathcal {D}$ under the map

$$ \begin{align*} j:D\to \mathcal{D}\hspace{20pt} t\mapsto (a_1t^{c_1},\ldots, a_N t^{c_N},0,\ldots,0). \end{align*} $$

Consider the following diagram.

(3.3)

The left and middle vertical arrows are profinite completions, the right arrow is the $\ell $ -adic completion and $\rho _{\mathcal {X}}$ is the monodromy map associated to $\mathcal {X} \to S$ . The left square commutes because $G_K\to \pi _1^{\operatorname {\mathrm {alg}}}(S')$ is the profinite completion of $\pi _1(D^*) \to \pi _1(\mathcal {D}^*)$ , and the composition of the bottom two arrows may be identified with the natural action $G_K \to \operatorname {\mathrm {Out}}(\Pi _g^{(\ell )})$ . Commutativity of the right square follows from commutativity of the following diagram

where $\mathcal {X}^*$ and $\mathcal {Y}^*$ are the smooth loci of $\mathcal {X}\to S$ and $\mathcal {Y}\to \mathcal {D}$ , respectively. Here, the vertical arrows are profinite completions and the rows are exact. Let $\gamma \in \operatorname {\mathrm {Out}}(\Pi _g)$ denote the image of the counterclockwise generator of $\pi _1(D^*)$ in $\operatorname {\mathrm {Out}}(\Pi _g)$ . Commutativity of the diagram in Equation (3.3) yields the following commutative square

Because the left arrow takes $\nu $ to $\hat {n}^{(\ell )}$ , the right arrow takes $\nu (\gamma )$ to $\nu ^{(\ell )}(X)$ . By Proposition 3.1, $\gamma $ acts as the multitwist $\prod _{i}T_{\ell _i}^{-c_i}$ on $\Pi _g$ . The theorem now follows from Theorem 6.8.

4.1 Tropical curves

A tropical curve $\Gamma $ consists of a vertex weighted graph $\mathbf {G}$ , together with a positive real-value $c_e$ associated to each edge e, recording its length.Footnote ¹ The genus of $\Gamma $ is

$$ \begin{align*} g(\Gamma) = |E(G)| - |V(G)| + 1 +|w|, \end{align*} $$

where G is the underlying graph of $\Gamma $ , and $|w|$ is the sum of the vertex weights. This is consistent with the interpretation of a weight on a vertex as an ‘infinitesimal loop’. The valence of a vertex v, denoted by $\operatorname {\mathrm {val}}(v)$ , is the number of half-edges adjacent to v; in particular, a loop edge contributes $2$ to the valence.

Given an arrangement of simple, closed, nonintersecting curves $\Lambda = \{\ell _1,\ldots , \ell _N\}$ in $\Sigma _g$ , its dual graph is the vertex weighted graph with:

1. - a vertex $v_S$ for each connected component S of $\Sigma _g\setminus \bigcup _{i=1}^N \ell _i$ whose weight is the genus of S, and

2. - an edge $e_i$ between $v_S$ and $v_{S'}$ for each loop $\ell _i$ between S and $S'$ .

Any vertex-weighted graph $\mathbf {G}$ of genus g may be realized as the dual graph to an arrangement of pairwise nonintersecting curves on $\Sigma _g$ in the following way. For each vertex v, let $\Sigma _v$ be a genus- $w_v$ surface with $\operatorname {\mathrm {val}}(v)$ boundary components. For each edge e of $\mathbf {G}$ between the (not necessarily distinct) vertices u and v, glue a boundary component of $\Sigma _u$ to a boundary component of $\Sigma _v$ ; denote the glued locus in the resulting surface by $\ell _e$ . This process yields a genus-g surface, together with an arrangement of pairwise nonintersecting curves $\Lambda = \{\ell _e \, | \, e\in E(\Gamma )\}$ whose dual graph is $\mathbf {G}$ . For an illustration, see Figure 4.1. If $\Gamma $ is a tropical curve with integral edge lengths $c_e$ , then we have a canonical multitwist

$$ \begin{align*} T_{\Gamma} = \prod_{e\in E(\Gamma)} T_{\ell_e}^{c_e} \end{align*} $$

and we let $\delta _{\Gamma }\in \operatorname {\mathrm {Sp}}(H)$ denote the action of $T_{\Gamma }$ on H. At this point, one may define the Ceresa class of $\Gamma $ to be $\nu (T_{\Gamma })\in H^1(\langle T_{\Gamma } \rangle , L/H) = H^1(\langle \delta _{\Gamma } \rangle , L/H)$ . However, when the edge lengths of $\Gamma $ are not integral, we cannot define $\delta _{\Gamma }$ and $\nu (\Gamma )$ in terms of a multitwist. We will define the Ceresa class for a tropical curve with real edge lengths and explain what it means for it to be trivial in §4.3.

Figure 4.1

Passing from a tropical curve to a multitwist.

4.2 The tropical Jacobian

Now, suppose $\Gamma $ has genus $g \geq 2$ and fix an orientation on the underlying graph G. Its Jacobian is the real g-dimensional torus

$$ \begin{align*} \operatorname{\mathrm{Jac}}(\Gamma) = (H_1(G,\mathbb{R}) \oplus \mathbb{R}^{|w|})/(H_1(G,\mathbb{Z}) \oplus \mathbb{Z}^{|w|}) \end{align*} $$

together with the semipositive quadratic form $Q_{\Gamma }$ which vanishes on $\mathbb {R}^{|w|}$ and on $H_1(G,\mathbb {R})$ is equal to

$$ \begin{align*} Q_{\Gamma} \left( \sum_{e\in E(G)} \alpha_{e}\cdot e \right) = \sum_{e\in E(G)} \alpha_e^2\cdot c_e. \end{align*} $$

The form $Q_{\Gamma }$ is positive definite when all vertex weights are $0$ , and $\det (Q_{\Gamma })$ is the first Symanzik polynomial of of G [Reference Amini, Bloch, Burgos Gil and Fresán2, Proposition 2.9]. That is,

(4.1) $$ \begin{align} \det(Q_{\Gamma}) = \sum_{T} c_{T}, \hspace{15pt} \text{where} \hspace{15pt} c_T = \prod_{e\notin E(T)} c_e \end{align} $$

and the sum is taken over all spanning trees T of G.

When $\Gamma $ has integral edge lengths, $\delta _{\Gamma }$ and $Q_{\Gamma }$ are related in the following way. First, embed G into $\Sigma _g$ so that each vertex v maps to a point in $\Sigma _v$ , and each edge e maps to a simple arc intersecting the loop $\ell _{e}$ exactly one time, and no other $\ell _{e'}$ . This embedding, which we denote by $\iota :G\hookrightarrow \Sigma _g$ , induces an injective map on integral homology groups. Then

(4.2) $$ \begin{align} Q_{\Gamma}(\gamma) = \hat{i}( [\iota(\gamma)], (\delta_{\Gamma}-I)[\iota(\gamma)]). \end{align} $$

Here is a more explicit description of the relationship between $Q_{\Gamma }$ and $\delta _{\Gamma }$ . Enumerate the edge set $E(G) = \{e_1,\ldots ,e_N\}$ so that $E(G)\setminus \{e_1,\ldots ,e_h\}$ are the edges of a spanning tree T. The graph $T\cup \{e_i\}$ has a unique cycle; denote by $\gamma _i$ the image of this cycle under $\iota $ . The cycles $[\gamma _1],\ldots ,[\gamma _h]$ form a basis for an isotropic subspace of H. Orient $\gamma _i$ and $\ell _{e_i}$ so that $\hat {i}([\gamma _i], [\ell _{e_i}]) = 1$ , for $1\leq i\leq h$ . Setting $\alpha _i = [\gamma _{i}]$ and $\beta _i = [\ell _{e_i}]$ yields a symplectic basis of a symplectic subspace of H. This extends to a symplectic basis $\alpha _1,\ldots ,\alpha _g,\beta _1,\ldots ,\beta _g$ on all of H, allowing us to identify $Q_{\Gamma }$ with a symmetric $g\times g$ matrix. Then

(4.3) $$ \begin{align} \delta_{\Gamma} = \begin{pmatrix} I & 0 \\ Q_{\Gamma} & I \end{pmatrix}. \end{align} $$

In particular, we may identify $Q_{\Gamma }$ with the restriction $\delta _{\Gamma }-I:H/Y \to Y$ , where Y is the $\mathbb {Z}$ -submodule of H spanned by the $\beta _i$ for $i=1,\ldots ,h$ .

4.3 The tropical Ceresa class

We saw in §4.1 how one may define the Ceresa class of a tropical curve with integral edge lengths in terms of a multitwist. When the edge lengths of $\Gamma $ are not integral, then we do not have access to such a multitwist. Instead, we will define what it means for a tropical curve to be Ceresa trivial.

The kernel of the Johnson homomorphism, denoted by $\mathcal {K}_g$ , is a normal subgroup of $\Gamma _g$ , which allows us to form the quotient $\mathcal {G}_g = \Gamma _g/\mathcal {K}_g$ . This follows from the fact that $\mathcal {K}_g$ is the kernel of the map $\Gamma _g \to \operatorname {\mathrm {Out}}(N_3)$ . Let $\mathbf {G}$ be a vertex-weighted graph, and denote by the subgroup of $\mathcal {G}_g$ generated by the twists $T_{\ell _e}$ for $e\in E(\mathbf {G})$ . This is a free $\mathbb {Z}$ -module because the $\ell _e$ ’s are nonintersecting, and it has rank $N-s$ , where s is the number of separating edges in $\mathbf {G}$ . Given a hyperelliptic quasi-involution $\tau \in \mathcal {G}_g$ , define

where $C_{\mathcal {G}_g}(\tau )$ denotes the centralizer of $\tau $ in $\mathcal {G}_g$ . Let

We say that $\Gamma $ is Ceresa trivial if there exists a hyperelliptic quasi-involution $\tau $ such that . Proposition 4.2 below shows that this notion agrees with the Ceresa class associated to the multitwist $T_\Gamma $ being trivial in the case where $\Gamma $ has integral edge length, but first we will need the following Lemma.

Proof. The ‘if’ direction is clear. Suppose $\nu (T_{\Gamma }) = 0$ . The class $\nu (T_{\Gamma })$ is represented by the cocycle $\gamma \mapsto J([\gamma ,\tau ])$ for some hyperelliptic quasi-involution $\tau $ , and hence

$$ \begin{align*} J([T_{\Gamma},\tau]) = T_{\Gamma}\cdot h-h \end{align*} $$

for some $h\in L/H$ . Because the Johnson homomorphism is surjective, there is a $t\in \mathcal {I}_g$ such that $J(t)=h$ . By rearranging the above equality, we see that $J([\gamma ,t^{-1}\tau ]) = 0$ . The lemma now follows from the fact that $t^{-1}\tau $ is also a hyperelliptic quasi-involution.

We end this section by showing that the Ceresa class vanishes for hyperelliptic tropical curves. First, we recall some terminology. Let $\mathbf {G}$ be a vertex-weighted graph. A vertex v of a vertex-weighted graph $\mathbf {G}$ is stable if

$$ \begin{align*} 2w_v-2+\operatorname{\mathrm{val}}(v)>0, \end{align*} $$

and $\mathbf {G}$ is stable if all of its vertices are stable. A tropical curve is stable if its underlying weighted graph is stable. Two tropical curves are tropically equivalent if one can be obtained from the other via a sequence of the following moves:

1. - adding or removing a 1-valent vertex v with $w_v = 0$ and the edge incident to v, or

2. - adding or removing a 2-valent vertex v with $w_v = 0$ , preserving the underlying metric space.

Every tropical curve $\Gamma $ of genus $g\geq 2$ is tropically equivalent to a unique tropical curve whose underlying weighted graph is stable [Reference Caporaso13, Section 2].

Suppose $\Gamma $ is a stable hyperelliptic tropical curve with underlying vertex-weighted graph $\mathbf {G}$ , and $\sigma $ the hyperelliptic involution of $\Gamma $ . That is, $\sigma :\Gamma \to \Gamma $ is an isometry that induces an involution of $\mathbf {G}$ such that all vertices of positive weight are fixed and $\Gamma /\sigma $ is a metric tree. By [Reference Corey18, Proposition 2.5], the edge set of $\Gamma $ partitions into the subsets

1. - $\{e\} $ for separating edges e and $\sigma $ restricted to e is the identity,

2. - $\{e,f\}$ where $e\neq f$ form a separating pair of edges and $\sigma (e) = f$ , and

3. - $\{e\}$ where e is any other edge, and $\sigma $ takes e to itself, interchanging its endpoints.

If $\{e,f\}$ is a separating pair, then $c_{e} = c_f$ because $\sigma $ is an isometry.

Proof. Let $\Gamma _1,\ldots ,\Gamma _k$ be a block decomposition of $\Gamma $ in the sense of [Reference Corey18, §2]. The hyperelliptic involution $\sigma $ restricts to a hyperelliptic involution on each $\Gamma _i$ because $\sigma $ fixes vertices of positive weight and acts as $-I$ on $\operatorname {\mathrm {Jac}}(\Gamma )$ [Reference Baker and Norine5, Theorem 5.19]. If $\Gamma _i$ is a single vertex of weight 1, then let $\Sigma _i$ be a genus-1 surface with one boundary component and $\tau _i:\Sigma _i\to \Sigma _i$ be a orientation-preserving homeomorphism that acts as $-I$ on $H_1(\Sigma _i,\mathbb {Z})$ and restricts to the identity on $\partial \Sigma _i$ . If $\Gamma _i$ is a single vertex with a loop edge e, then let $\Sigma _i$ be a genus-1 surface with one boundary component and $\tau _i:\Sigma _i\to \Sigma _i$ be a orientation-preserving homeomorphism that acts as $-I$ on $H_1(\Sigma _i,\mathbb {Z})$ , $\tau (\ell _e) = \ell _e$ , and restricts to the identity on $\partial \Sigma _i$ .

Otherwise, $\Gamma _i$ is 2-vertex connected and genus $g_i\geq 2$ . Form $\Sigma _{g_i}$ as in §4.1. For each $u\in V(\Gamma _i)$ fixed by $\sigma $ , remove a small open disc $S_u$ from $\Sigma _u^{\circ }$ ; denote the resulting boundary curve by $\ell _u$ and the resulting surface by $\Sigma _i$ . For each $u\in V(\Gamma _i)$ (resp. $e\in E(\Gamma )$ ) choose a point $p_u\in \Sigma _u^{\circ }$ (resp. $p_e\in \ell _e$ ). For each half-edge h of $\Gamma _i$ , define a simple path $\eta _h$ in $\Sigma _i$ satisfying the following.

1. - If $\sigma (u)\neq u$ , and h is a half-edge of e adjacent to u, then $\eta _h$ is a simple path in $\Sigma _u$ from $p_u$ to $p_e$ meeting $\partial \Sigma _u$ only at $p_e$ .

2. - If $\sigma (u)= u$ , and h is a half-edge of e adjacent to u, then $\eta _h$ is a simple path in $\Sigma _u\setminus S_u$ from $p_u$ to $p_e$ meeting $\partial (\Sigma _u\setminus S_u)$ only at $p_e$ .

3. - If $h,h'$ are adjacent to u, then $\eta _{h} \cap \eta _{h'} = p_u$ .

We claim that there are orientation-preserving homeomorphisms $\tau _u:\Sigma _u \to \Sigma _{\sigma (u)}$ so that

1. - $\tau _u(\eta _h) = \eta _{\sigma (h)}$ ,

2. - $\tau _u|\ell _e = \tau _v|\ell _e$ , and

3. - the restriction of $\tau _u$ to $\ell _u$ is the identity, for each $u\in V(\Gamma _i)$ fixed by $\sigma $ .

Suppose $\sigma (u) = v\neq u$ . Order the half edges of u (resp. v) by $h_1,\ldots ,h_a$ (resp. $k_1,\ldots ,k_a$ ) such that $\sigma (h_j) = k_j$ , and denote by $e_j$ the edge containing $h_j$ (resp. $f_j$ the edge containing $k_j$ ). Let D be an oriented $3a$ -gon, and label the edges of D (counterclockwise) by $\eta _{h_1},\ell _{e_1},\eta _{h_1}^{-1},\ldots , \eta _{h_a},\ell _{e_a},\eta _{h_a}^{-1}$ . Gluing $\eta _{h_j}$ along $\eta _{h_j}^{-1}$ (for $j=1,\ldots ,a$ ) yields a quotient map $\pi _u:D \to \Sigma _u$ ; see Figure 4.2 for an illustration. Now. relabel the edge $\eta _{h_j}$ by $\eta _{k_j}$ , $\ell _{e_j}$ by $\ell _{f_j}$ and $\eta _{h_j}^{-1}$ by $\eta _{k_j}^{-1}$ . Gluing $\eta _{k_j}$ along $\eta _{k_j}^{-1}$ (for $j=1,\ldots ,a$ ) yields a quotient map $\pi _v:D \to \Sigma _v$ . This map induces a homeomorphism on the quotient $\tau _u:\Sigma _u\to \Sigma _v$ such that $\tau _u(\eta _{h_j}) = \eta _{k_j}$ and $\tau _u(\ell _{e_j}) = \tau _u(\ell _{f_j})$ . In particular, the composition $\tau _{v}\tau _{u}$ is the identity on $\Sigma _u$ . If $\Gamma $ is 2-vertex connected and stable, it has no vertices fixed by $\sigma $ , and we may glue these $\tau _u$ to give a hyperelliptic involution $\tau $ .

Figure 4.2

The gluing map $\pi _u:D\to \Sigma _u$ when $a=3$ .

Now, suppose $\sigma (u) = u$ . Label the half-edges of u by $h_1,\ldots ,h_a,k_1,\ldots ,k_a$ such that $\sigma (h_j) = k_j$ , and denote by $e_j$ the edge containing $h_j$ (resp. $f_j$ the edge containing $k_j$ ). Let $\eta _u$ be a simple path in $\Sigma _u\setminus S_u$ from $p_u$ to a point p on $\ell _u$ that meets the other $\eta _{h_i}$ ’s and $\eta _{k_i}$ ’s only at $p_u$ . Let D be an oriented $6a+3$ -gon. Label the edges of D (counterclockwise) by

$$ \begin{align*} \eta_{h_1},\ell_{e_1},\eta_{h_1}^{-1},\ldots, \eta_{h_a},\ell_{e_a},\eta_{h_a}^{-1}, \eta_{u},\ell_u,\eta_{u}^{-1},\eta_{k_a},\ell_{f_a},\eta_{k_a}^{-1},\ldots, \eta_{k_1},\ell_{f_1},\eta_{k_1}^{-1}. \end{align*} $$

Gluing $\eta _u$ along $\eta _{u}^{-1}$ , $\eta _{h_j}$ along $\eta _{h_j}^{-1}$ , and $\eta _{k_j}$ along $\eta _{k_j}^{-1}$ (for $j=1,\ldots ,a$ ) yields a quotient map $\pi _u:D \to \Sigma _u$ . Now, relabel $\eta _{h_j}$ by $\eta _{k_j}$ , $\eta _{h_j}^{-1}$ by $\eta _{k_j}^{-1}$ , and $\ell _{e_j}$ by $\ell _{f_j}$ ( $j=1,\ldots ,a$ ). Gluing $\eta _u$ along $\eta _{u}^{-1}$ , $\eta _{h_j}$ along $\eta _{h_j}^{-1}$ , and $\eta _{k_j}$ along $\eta _{k_j}^{-1}$ (for $j=1,\ldots ,a$ ) yields another quotient map $\pi _u':D \to \Sigma _u$ . This map induces a homeomorphism on the quotient $\tau _u:\Sigma _u\to \Sigma _u$ such that $\tau _u(\eta _{h_j}) = \eta _{k_j}$ , $\tau _u(\ell _{e_j}) = \ell _{f_j}$ , (for $j=1,\ldots ,a$ ) and is the identity on $\ell _u$ .

Finally, we may modify the $\tau _u$ ’s in a collar neighborhood of $\partial \Sigma _u$ so that $\tau _u|\ell _e = \tau _v|\ell _e$ when e is an edge between u and v. Having done so, the resulting $\tau _u$ ’s glue to give an orientation-preserving homeomorphism $\tau _i:\Sigma _i\to \Sigma _i$ that restricts to the identity on $\partial \Sigma _i$ , and which sends $\ell _e$ to $\ell _{\sigma (e)}$ for all edges e.

Define a homology basis of $\Sigma _{g_i}$ in the following way. Let $e_1,\ldots , e_{g_i}\in E(\Gamma _i)$ be a collection of edges whose removal from $\Gamma _i$ is a spanning tree T. Denote by $h_j$ the unique cycle in $T\cup \{e_j\}$ . Let $\gamma _i$ be the simple closed curve in $\Gamma _{g_i}$ formed by the paths $\eta _h$ for all half-edges in the path $h_j$ . Orient $\ell _i$ and $\gamma _i$ so that $\hat {i}([\gamma _i],[\ell _{e_i}]) = 1$ ; the cycles

$$ \begin{align*} [\ell_{e_1}],\ldots, [\ell_{e_{g_i}}], [\gamma_1],\ldots, [\gamma_{g_i}] \end{align*} $$

form a symplectic basis of $H_1(\Sigma _{g_i},\mathbb {Z})$ .

Next, we claim, for $j=1,\ldots ,g_i$ , that

(4.4) $$ \begin{align} & \tau_*([\ell_{e_j}]) = -[\ell_{e_j}], \end{align} $$

(4.5) $$ \begin{align} & \tau_*([\gamma_j]) = -[\gamma_j]. \end{align} $$

Consider Equation (4.4). Denote the vertices of $e_{j}$ by $u_j$ and $v_j$ . Without loss of generality, suppose $\Sigma _{u_j}$ is on the left of $\ell _{e_j}$ . Because $\tau _{u_j}$ is orientation preserving, $\Sigma _{v_j}$ appears on the left of $\tau (\ell _{e_j})$ . If $e_{j}$ is flipped, then $\tau (\ell _{e_j}) = \ell _{e_j}$ and because $\Sigma _u$ appears on the left of $\ell _e$ but on the right of $\tau (\ell _e)$ , we have $\tau _*([\ell _{e_j}]) = -[\ell _{e_j}]$ . Now, suppose $\{e_j,f_j\}$ is a separating pair. Orient $\ell _{f_j}$ so that $[\ell _{e_j}] = [\ell _{f_j}]$ . Their removal splits $\Sigma _{g_i}$ into two surfaces $S,S'$ with boundary curves $\ell _{e_j},\ell _{f_j}$ . The subsurfaces $\Sigma _{u_j}$ and $\Sigma _{v_j}$ belong to the same surface; suppose it is S. Because S lies on the left of both $\ell _{e_j}$ and $\tau (\ell _{e_j})$ , we must have that $\tau _*([\ell _{e_j}]) = -[\ell _{f_j}] = -[\ell _{e_j}]$ , and therefore Equation (4.4).

Now, consider Equation (4.5). Because $\tau _i$ is orientation-preserving, we have

$$ \begin{align*} \hat{i}(\tau_*([\gamma_j]), \tau_*([\ell_{e_j}])) = \hat{i}(\tau_*([\gamma_j]), -[\ell_{e_j}]) = 1. \end{align*} $$

Together with the fact that $\tau _i(\gamma _j) = \gamma _j$ , we have $\tau _{*}([\gamma _j]) = -[\gamma _j]$ .

Finally, we will piece together the $\tau _i$ ’s to get the requisite hyperelliptic quasi-involution. For each cut-vertex u of $\Gamma $ , let $\Sigma _{u}$ be a genus-0 surface with $n_u$ boundary components, where $n_u$ is the number of blocks attached to u. For each block $\Gamma _i$ attached at u, glue $\Sigma _i$ to $\Sigma _u$ along the corresponding boundary components. The orientation-preserving homeomorphism $\tau :\Sigma _g\to \Sigma _g$ given by

$$ \begin{align*} \tau|\Sigma_{i} &= \tau_i \hspace{5pt} \text{ for } \hspace{5pt} i=1,\ldots,k \\ \tau|\Sigma_u &= \operatorname{\mathrm{id}}_{\Sigma_u} \hspace{5pt} \text{ if }u\text{ is a cut-vertex of } \Gamma \end{align*} $$

acts as $-I$ on $H_{1}(\Sigma _g,\mathbb {Z})$ and satisfies $\tau (\ell _e) = \ell _{\sigma (e)}$ for all $e\in E(\Gamma )$ , as required.

Proof. By Lemma 4.4, we may assume that $\Gamma $ is stable. Denote by $\mathbf {G}^2$ (resp. $\Gamma ^2$ ) the 2-edge connectivization of $\mathbf {G}$ (resp. $\Gamma $ ) (this is obtained by contracting all separating edges; see [Reference Caporaso and Viviani14, Definition 2.3.6]). Because and $\nu (\Gamma )=\nu (\Gamma ^2)$ , we may assume that $\Gamma $ is 2-edge connected. Let $\tau $ be the hyperelliptic involution from Lemma 4.5. If e is a separating edge, then $T_{\ell _e}$ is a separating twist, which is trivial in . If $\{e,f\}$ is a pair of separating edges, then . If e is any other edge, then . Decompose $\nu (\Gamma )$ as

$$ \begin{align*} \nu(\Gamma) = \sum (c_eT_{\ell_e} + c_fT_{\ell_f}) + \sum c_eT_{\ell_e}, \end{align*} $$

where the sum on the left is over all separating pairs, and the sum on the right is over all nonseparating edges not in a separating pair. Because $c_e=c_f$ whenever $\{e,f\}$ is a separating pair, we have that , and hence $\Gamma $ is Ceresa trivial.

5.1 Filtrations on $H^1(\mathbb {Z},\wedge ^kH)$

In this section, we set up a rather general framework for abelian groups with a unipotent automorphism, which we will apply in the case of the singular homology of a genus-g topological surface acted upon by a multitwist.

Let H be a finitely generated free $\mathbb {Z}$ -module and $\delta \in \operatorname {\mathrm {SL}}(H)$ be an element such that $(\delta -I)^2 = 0$ . We consider the action of the cyclic group $\langle \delta \rangle \cong \mathbb {Z}$ on H, which induces an action of $\langle \delta \rangle $ on $\wedge ^k H$ for any $k \geq 0$ . Let Y be the saturation of $\operatorname {\mathrm {im}}(\delta -I)$ in H. By the hypothesis on $\delta $ , we have $Y\leq \ker (\delta -I)$ , that is, $\delta $ acts trivially on Y. Consider the following descending filtration on $\wedge ^kH$ :

(5.1) $$ \begin{align} F_{q}\wedge^kH = (\wedge^qY) \wedge (\wedge^{k-q} H). \end{align} $$

Note that $F_{q}\wedge ^kH$ is saturated in $\wedge ^kH$ , so the graded piece $\operatorname {\mathrm {gr}}_q^F\wedge ^kH := F_{q}\wedge ^kH / F_{q+1}\wedge ^kH$ is torsion-free. The following lemma shows that $(\delta -I)(F_{q}\wedge ^kH) \leq F_{q+1}\wedge ^kH$ for any $k \ge q \ge 0$ .

Lemma 5.1. For $y\in \wedge ^{q}Y$ and $h=h_1\wedge \ldots \wedge h_{k-q} \in \wedge ^{k-q}H$ ,

$$ \begin{align*} (\delta-I)(y \wedge h) = \sum_{i=1}^{k-q} y\wedge h_1 \wedge \ldots \wedge (\delta-I) h_i \wedge \ldots \wedge h_{k-q}\quad \mod F_{q+2}\wedge^k H. \end{align*} $$

In particular, $(\delta -I)(F_q\wedge ^kH) \leq F_{q+1}\wedge ^kH$ .

Proof. Because $\delta y= y$ , we can write

$$ \begin{align*} (\delta - I)(y\wedge h) = y\wedge (\delta - I + I)h_1 \wedge \ldots \wedge (\delta - I + I)h_{k-q} - y\wedge h \end{align*} $$

and expand the latter as

$$ \begin{align*} &\sum_{i=1}^{k-q} y \wedge h_1 \wedge \ldots \wedge (\delta -I) h_i \wedge \ldots \wedge h_{k-q} \\ &+ \sum_{1\leq i<j\leq k-q} y \wedge h_1 \wedge \ldots \wedge (\delta -I) h_i \wedge \ldots \wedge (\delta-I) h_j \wedge \ldots \wedge h_{k-q} + \ldots \\[-45pt] \end{align*} $$

This means, in particular, that $\delta -I$ induces a map $\operatorname {\mathrm {gr}}_{q-1}^F\wedge ^kH \rightarrow \operatorname {\mathrm {gr}}_{q}^F\wedge ^kH$ for all q. While these maps are typically not surjective, what we will see in the lemma below is that at least half of them are surjective rationally; that is, their cokernels are finite. Compare this to the ‘Lefschetz property’ of the weight filtration associated to a nilpotent endomorphism as in [Reference Peters and Steenbrink33, Lemma-Definition 11.9].

Lemma 5.2. The map

$$ \begin{align*} (\delta-I)_{\mathbb{Q}}: \operatorname{\mathrm{gr}}_{q-1}^F\wedge^kH_{\mathbb{Q}} \rightarrow \operatorname{\mathrm{gr}}_{q}^F\wedge^kH_{\mathbb{Q}} \end{align*} $$

is surjective for $q> k/2$ .

Proof. Let $y = y_1 \wedge \ldots \wedge y_{q} \in \wedge ^qY_{\mathbb {Q}}$ and $h = h_1 \wedge \ldots \wedge h_{k-q} \in \wedge ^{k-q}H_{\mathbb {Q}}$ . It suffices to show that $y \wedge h$ lies in the image of $(\delta -I)_{\mathbb {Q}}\bmod F_{q+1}\wedge ^kH_{\mathbb {Q}}$ for all such $y,h$ . Choose $x_i\in H_{\mathbb {Q}}$ such that $(\delta -I) x_i = y_i$ . For $I \in {[q] \choose p}$ , let $\hat {y}_{I}$ be obtained from y by replacing $y_{i}$ with $x_{i}$ for each $i \in I$ . Similarly, for $J \in {[k-q]\choose p}$ let $\hat {h}_J$ be obtained from h by replacing $h_{j}$ with $(\delta -I) h_{j}$ for each $j\in J$ . We define

$$ \begin{align*} \mathfrak{y}(a,b) = \sum_{I\in {[q] \choose a},J \in {[k-q] \choose b}} \hat{y}_{I} \wedge \hat{h}_{J} \in F_{q-a+b}\wedge^kH_{\mathbb{Q}}. \end{align*} $$

Note that $\mathfrak {y}(a,b) \neq 0$ only if $0 \leq a\leq q$ and $0\leq b \leq k-q$ , and $\mathfrak {y}(0,0) = y \wedge h$ . By Lemma 5.1,

$$ \begin{align*} (\delta-I)_{\mathbb{Q}} (\hat{y}_{I} \wedge \hat{h}_{J}) \equiv \sum_{i\in I} \hat{y}_{I\setminus\{i\}} \wedge \hat{h}_{J} + \sum_{j\notin J} \hat{y}_{I} \wedge \hat{h}_{J\cup \{j\}} \bmod F_{q-a+b+2}\wedge^kH_{\mathbb{Q}}. \end{align*} $$

In particular,

(5.2) $$ \begin{align} (\delta-I)_{\mathbb{Q}} (\mathfrak{y}(a,b)) \equiv (q-a+1)\cdot \mathfrak{y}(a-1,b) + (b+1)\cdot \mathfrak{y}(a,b+1) \bmod F_{q-a+b+2}\wedge^kH_{\mathbb{Q}}. \end{align} $$

Claim. For $0\leq p\leq k-q$ , $\mathfrak {y}(p,p) $ is in $\operatorname {\mathrm {im}}(\delta -I)_{\mathbb {Q}} \bmod F_{q+1}\wedge ^kH_{\mathbb {Q}}$ .

The case $p=0$ is exactly the statement that $y\wedge h \in \operatorname {\mathrm {im}} (\delta -I)_{\mathbb {Q}} \bmod F_{q+1}\wedge ^kH_{\mathbb {Q}}$ . We will proceed by downward induction on p. Because $\mathfrak {y}(k-q+1, k-q+1) = 0$ , Equation (5.2) yields

$$ \begin{align*} (\delta-I)_{\mathbb{Q}}\left(\mathfrak{y}(k-q+1, k-q)\right) \equiv (2q-k) \cdot \mathfrak{y}(k-q, k-q) \bmod F_{q+1}\wedge^kH_{\mathbb{Q}}, \end{align*} $$

and therefore the claim holds when $p=q$ , noting that $2q> k$ . Assuming the claim is true for $p+1$ , we will show that it is true for p. Again by Equation 5.2,

$$ \begin{align*} (\delta-I)_{\mathbb{Q}}\left(\mathfrak{y}(p+1,p) \right) \equiv (q-p)\cdot\mathfrak{y}(p,p) + (p+1) \cdot\mathfrak{y}(p+1,p+1) \bmod F_{q+1}\wedge^kH_{\mathbb{Q}}. \end{align*} $$

By the inductive hypothesis, $\mathfrak {y}(p+1,p+1)$ is in the image of $(\delta -I)_{\mathbb {Q}}\bmod F_{q+1}\wedge ^kH_{\mathbb {Q}}$ and therefore so is $(q-p)\mathfrak {y}(p,p)$ ; by the hypothesis that $q> k/2$ , we have $q-p \neq 0$ , so $\mathfrak {y}(p,p)$ is in the image of $(\delta -I)_{\mathbb {Q}}\bmod F_{q+1}\wedge ^kH_{\mathbb {Q}}$ and we are done.

Remark 5.3. In the case where $\dim H = 2\dim Y$ , the largest possible dimension of Y, the bound $q>k/2$ in Lemma 5.2 is sharp. Set $d=\dim Y$ , $e = \dim H$ , and $u(q) = \dim (\operatorname {\mathrm {gr}}_q^F\wedge ^kH)$ . Then

$$ \begin{align*} u(q) = {d \choose q} {e-d\choose k-q}. \end{align*} $$

When $e=2d$ and $0\leq q\leq k/2$

$$ \begin{align*} \frac{u(q)}{u(q-1)} = \frac{(d-q+1)(k-q+1)}{q(d-k+q)} \geq \frac{(d-q+1)(q+1)}{q(d-q)}> 1, \end{align*} $$

and therefore u is a strictly increasing function on this interval. This means that $(\delta -I)_{\mathbb {Q}}$ as in Lemma 5.2 cannot be surjective.

We denote by $A_q(\delta )$ and $B_q(\delta )$ the groups:

$$ \begin{align*} A_q(\delta) &= \operatorname{\mathrm{im}}( H^1(\langle \delta \rangle, F_{q}\wedge^{2q-1}H) \to H^1(\langle \delta \rangle, \wedge^{2q-1}H)), \\ B_q(\delta) &= \operatorname{\mathrm{coker}}(\delta-I:\operatorname{\mathrm{gr}}_{q-1}^F\wedge^{2q-1}H \to \operatorname{\mathrm{gr}}_{q}^F\wedge^{2q-1}H). \end{align*} $$

Proposition 5.4. We have isomorphisms of groups

$$ \begin{align*} A_q(\delta) \cong \frac{F_q\wedge^{2q-1}H}{((\delta-I)\wedge^{2q-1}H) \cap (F_q\wedge^{2q-1}H)}, \hspace{10pt} B_q(\delta) \cong \frac{F_q\wedge^{2q-1}H}{((\delta-I)F_{q-1}\wedge^{2q-1}H) + (F_{q+1}\wedge^{2q-1}H)}. \end{align*} $$

In particular, $A_q(\delta )$ and $B_q(\delta )$ are finite.

Proof. It is a standard fact that

$$ \begin{align*} H^1(\langle \delta \rangle, \wedge^kH) \xrightarrow{} \wedge^kH/(\delta-I)\wedge^kH \hspace{20pt} [\zeta] \mapsto \zeta(\delta) \end{align*} $$

is an isomorphism. This yields the isomorphism involving $A_q(\delta )$ , and the one involving $B_q(\delta )$ is clear. Because each $(\delta -I)_{\mathbb {Q}}:\operatorname {\mathrm {gr}}_{i-1}^F\wedge ^{2q-1}H_{\mathbb {Q}} \to \operatorname {\mathrm {gr}}_{i}^F\wedge ^{2q-1}H_{\mathbb {Q}}$ is surjective for $i\geq q$ by Lemma 5.2, we see that $F_q\wedge ^{2q-1}H_{\mathbb {Q}}$ is contained in $(\delta -I)(F_{q-1}\wedge ^{2q-1}H_{\mathbb {Q}})$ . Therefore, $A_q(\delta )$ and $B_q(\delta )$ are finite.

Proposition 5.5. If $\delta -I:\operatorname {\mathrm {gr}}_{q-2}^F\wedge ^kH \to \operatorname {\mathrm {gr}}_{q-1}^F\wedge ^kH$ is injective, then

$$ \begin{align*} (\delta-I)(F_{q-2}\wedge^kH) \cap F_{q} \wedge^kH = (\delta-I)F_{q-1}\wedge^kH. \end{align*} $$

In particular, this means that if $\delta -I:\operatorname {\mathrm {gr}}_{i-2}^F\wedge ^{2q-1}H \to \operatorname {\mathrm {gr}}_{i-1}^F\wedge ^{2q-1}H$ is injective for all $i\leq q$ , then $A_q(\delta ) \cong F_q\wedge ^{2q-1}H/(\delta -I)F_{q-1}\wedge ^{2q-1}H$ , and hence there is a natural surjection $A_q(\delta )\to B_q(\delta )$ .

In §7, we will need to show that certain classes in $H^1(\langle \delta \rangle , \wedge ^k H)$ arising from topology are trivial, for which we will need the following explicit computation.

Proof. Choose $x \in H$ such that $(\delta -I)x = y$ . Then

$$ \begin{align*} (\delta-I)(x\wedge z_1\wedge\ldots \wedge z_{k-1}) =\ & \delta x \wedge \delta z_1 \wedge \ldots \wedge \delta z_{k-1} - x\wedge z_1\wedge \ldots \wedge z_{k-1} \\ =\ & \delta x \wedge z_1 \wedge \ldots \wedge z_{k-1} - x\wedge z_1\wedge \ldots \wedge z_{k-1} \\ =\ & y\wedge z_1 \wedge \ldots \wedge z_{k-1}.\\[-36pt] \end{align*} $$

The main application of Proposition 5.4 will be in the case $k=3$ and we will denote $L=\wedge ^3 H$ as before. For this reason, we simply write $A(\delta )$ and $B(\delta )$ for the subgroups $A_2(\delta )$ and $B_2(\delta )$ , respectively. These groups are finite by Proposition 5.4, and in particular any element of $H^1(\langle \delta \rangle , L)$ which lies in $A(\delta )$ is torsion.

5.2 The maximal rank case

An important subcase is that where $\dim Y$ is as large as possible, namely $\dim Y= g=\tfrac {1}{2}\dim H$ . Because $Y\subset H$ is saturated, there is a subgroup $X\subset H$ such that $H=X\oplus Y$ . Let $Q:X \to Y$ denote the restriction of $\delta -I$ to X and $q_1\leq \cdots \leq q_g$ its invariant factors, with $q_i | q_{i+1}$ . Because Q is rationally surjective, each $q_i$ is positive and

$$ \begin{align*} \operatorname{\mathrm{coker}}(Q) \cong \prod_{i=1}^g \mathbb{Z}/(q_i). \end{align*} $$

is a finite group. Choose bases $\{x_1,\ldots ,x_g\}$ of X and $\{y_1,\ldots ,y_g\}$ of Y such that $Q(x_i) = q_iy_i$ . To compute $A(\delta )$ , we decompose $L = \wedge ^3H$ as the direct sum of

$$ \begin{align*} &V_1 = \operatorname{\mathrm{span}}\{x_i\wedge x_j \wedge x_k \, | \, i<j<k\}, \hspace{28pt} V_2 = \operatorname{\mathrm{span}}\{x_i\wedge x_j \wedge y_j \, | \, i\neq j\}, \\ &V_3 = \operatorname{\mathrm{span}}\{x_i\wedge x_j \wedge y_k \, | \, i<j, k\neq i,j\}, \hspace{10pt} V_4 = \operatorname{\mathrm{span}}\{x_i\wedge y_i \wedge y_j \, | \, i\neq j\},\\ &V_5 = \operatorname{\mathrm{span}}\{x_i\wedge y_j \wedge y_k \, | \, j<k, i\neq j,k\}, \hspace{10pt} V_6 = \operatorname{\mathrm{span}}\{y_i\wedge y_j \wedge y_k \, | \, i<j<k\}. \end{align*} $$

Let A denote the matrix of $\delta -I$ with respect to the basis above, and $A_{ij}$ the block whose rows correspond to $V_i$ and columns correspond to $V_j$ .

Proof. The matrix $A_{13}$ is nonsingular because its rows each have exactly one nonzero entry. Indeed, the row corresponding to $x_i\wedge x_j\wedge x_k$ has $q_k$ in the column corresponding to $x_i\wedge x_j\wedge y_k$ and 0 for the remaining entries. Next, $A_{24}$ is a square $g(g-1)$ matrix and $(\delta -1)(x_i \wedge x_j \wedge y_j) = q_i \cdot y_i \wedge x_j \wedge y_j \mod F_3L$ . Therefore, $A_{24}$ is nonsingular and its cokernel is of the desired form. Now, consider $A_{35}$ . Given $i<j<k$ , set

$$ \begin{align*} V_{ijk} &= \operatorname{\mathrm{span}}\{x_i\wedge x_j \wedge y_k,\, x_j\wedge x_k \wedge y_i,\, x_k\wedge x_i \wedge y_j\} \\ W_{ijk} &= \operatorname{\mathrm{span}}\{x_k\wedge y_i \wedge y_j,\, x_i\wedge y_j \wedge y_k,\, x_j\wedge y_k \wedge y_i \}. \end{align*} $$

The matrix $A_{35}$ may be arranged into block-diagonal form, where each block has rows indexed by $W_{ijk}$ and columns by $V_{ijk}$ . With respect to the bases above, this block is

$$ \begin{align*} \begin{pmatrix} 0 & q_j & q_i \\ q_j & 0 & q_k \\ q_i & q_k & 0 \end{pmatrix} \end{align*} $$

whose invariant factors are $q_i,q_i,2q_jq_k/q_i$ . Therefore, $\operatorname {\mathrm {coker}} A_{35}$ is isomorphic to the product of the $(\mathbb {Z}/(q_i))^2\times \mathbb {Z}/(q_jq_k/q_i)$ . Because each block matrix is nonsingular, $A_{35}$ is also nonsingular. This completes the proof of (1), (2) and (3).

Now, consider (4). Because $(\delta -I)(x_i\wedge y_j\wedge y_k) = q_i \cdot y_i\wedge y_j\wedge y_k$ , each column of $A_{56}$ has exactly one nonzero entry. By only performing column operations, we may form a diagonal matrix $B_{56}$ from $A_{56}$ so that each diagonal entry is the $\gcd $ of the integers in its corresponding row. Because the nonzero entries of the row in $A_{56}$ corresponding to $y_{i}\wedge y_{j} \wedge y_{k}$ are $q_i,q_j,q_k$ , the corresponding diagonal entry in $B_{56}$ is $\gcd (q_i,q_j,q_k) = q_i$ . From this, we see that $\operatorname {\mathrm {coker}}(A_{56})$ has the desired form.

Finally, consider (5). The matrices $A_{ij}$ for $i\geq j$ are all 0 because the matrix of $\delta -I$ , with respect to the above decomposition, is strictly lower-triangular. The remaining listed matrices are 0 because $\delta -I:X\to Y$ is diagonal with respect to the given bases.

Proposition 5.8. The maps $\delta -I:\operatorname {\mathrm {gr}}_{q-1}^FL_{\mathbb {Q}} \to \operatorname {\mathrm {gr}}_{q}^FL_{\mathbb {Q}}$ are surjective when $q=2,3$ and injective when $q=1,2$ . In particular,

$$ \begin{align*} A(\delta) \cong \frac{F_2L}{(\delta-I)F_1L}. \end{align*} $$

Proposition 5.9. We have an isomorphism

$$ \begin{align*} B(\delta) \cong \operatorname{\mathrm{coker}}(Q)^{g-1} \times \prod_{i<j<k}(\mathbb{Z}/(q_i))^{2} \times \mathbb{Z}/(2q_jq_k/q_i). \end{align*} $$

Proof. In terms of the decomposition above, $\delta -I: \operatorname {\mathrm {gr}}_F^1L \to \operatorname {\mathrm {gr}}_F^2L$ is given by the block matrix

$$ \begin{align*} A = \begin{pmatrix} A_{24} & A_{25} \\ A_{34} & A_{35} \end{pmatrix}. \end{align*} $$

The proposition now follows from Lemma 5.7.

Proposition 5.10. We have an isomorphism

$$ \begin{align*} A(\delta) \cong B(\delta) \times \prod_{i=1}^{g-2}(\mathbb{Z}/(q_i))^{{g-i\choose 2}}. \end{align*} $$

Proof. Under the identifications $A(\delta ) \cong F_2L/(\delta -I)F_1L$ and $B(\delta ) \cong F_2L/((\delta -I)F_1L + F_3L)$ from Propositions 5.8 and 5.4, the map $A(\delta ) \to B(\delta )$ given by the projection

$$ \begin{align*} \frac{F_2L}{(\delta-I)F_1L} \xrightarrow{} \frac{F_2L}{(\delta-I)F_1L + F_3L} \end{align*} $$

is surjective. Its kernel is isomorphic to $F_3L/(F_3L\cap (\delta -I)F_1L)$ . The map $\delta -I:\operatorname {\mathrm {gr}}_1^FL\to \operatorname {\mathrm {gr}}_2^FL$ is injective by Proposition 5.8, so $F_3L\cap (\delta -I)F_1L = (\delta -I)F_2L$ by Proposition 5.5. Therefore, we have an exact sequence

$$ \begin{align*} 0 \to \frac{F_3L}{(\delta-I)F_2L} \to A(\delta) \to B(\delta) \to 0. \end{align*} $$

We claim that this exact sequence splits. The decomposition $F_2L\cong \operatorname {\mathrm {gr}}_2^FL\times F_3L$ yields a projection $\pi :F_2L\to F_3L/(\delta -I)F_2L$ . Given $a\in (\delta -I)F_1$ , express it as $a = a_1+a_2$ with $a_1\in \operatorname {\mathrm {gr}}_2^FL$ and $a_2\in F_3$ . In fact, $a_2 \in (\delta -I)F_2$ since $\delta -I:\operatorname {\mathrm {gr}}_1^FL \to \operatorname {\mathrm {gr}}_2^FL$ is injective. Therefore, $a\in \ker \pi $ , and hence $\pi $ induces a splitting of $F_3L/(\delta -I)F_2L \to A(\delta )$ . Finally, $F_3L/(\delta -I)F_2L \cong \prod _{i=1}^{g-2}(\mathbb {Z}/(q_i))^{{g-i\choose 2}}$ by Lemma 5.7(4) and (5).

Corollary 5.11. When Y has maximal rank,

$$ \begin{align*} |A(\delta)| = 2^{{g\choose 3}}\det(Q)^{{g\choose 2}}\prod_{i=1}^{g-2}q_i^{{g-i\choose 2}} \hspace{10pt} \text{and} \hspace{10pt} |B(\delta)| = 2^{{g\choose 3}}\det(Q)^{{g\choose 2}}. \end{align*} $$

5.3 The symplectic case

Finally, we consider the case where H is equipped with a symplectic form $\omega \in \wedge ^2H$ , and $\delta $ is an element of $\operatorname {\mathrm {Sp}}(H)$ such that $(\delta -I)^2=0$ . We embed H into L via $h\mapsto \omega \wedge h$ . Because $\delta $ preserves the form $\omega $ , it acts on $L/H$ . We define

$$ \begin{align*} F_q(L/H) = (F_qL + H)/H. \end{align*} $$

Recall that Y is the saturation of $\operatorname {\mathrm {im}}(\delta -I)$ , which is isotropic since $\delta $ is symplectic, and X a subgroup such that $H=X\oplus Y$ . Because $H\subset F_1L$ , $F_2L + H = F_2L \oplus X$ , and $F_3L\cap H = 0$ , each $F_q(L/H)$ is saturated in $L/H$ . In particular, the graded pieces $\operatorname {\mathrm {gr}}_q^F(L/H)$ are free. By Lemma 5.1, $(\delta -I)$ takes $F_q(L/H)$ to $F_{q+1}(L/H)$ , hence induces a map $\operatorname {\mathrm {gr}}_{q}^F(L/H) \to \operatorname {\mathrm {gr}}_{q+1}^F(L/H)$ . We denote by $\overline {A(\delta )}$ and $\overline {B(\delta )}$ the groups

$$ \begin{align*} \overline{A(\delta)} &= \operatorname{\mathrm{im}}( H^1(\langle \delta \rangle, F_{2}(L/H)) \to H^1(\langle \delta \rangle, L/H)), \\ \overline{B(\delta)} &= \operatorname{\mathrm{coker}}(\delta-I:\operatorname{\mathrm{gr}}_{1}^F(L/H) \to \operatorname{\mathrm{gr}}_{2}^F(L/H)). \end{align*} $$

As we will show in the next section, the Ceresa class $\nu (f)$ of a positive multitwist f on a closed surface, with symplectic representation $\delta _f$ , lives in $\overline {A(\delta _f)}$ , provided Y has maximal rank. In this subsection, we will show that $\overline {A(\delta )}$ is finite, from which we conclude that the Ceresa class is torsion. When Y has maximal rank, $\overline {A(\delta )}$ naturally surjects onto $\overline {B(\delta )}$ . It is much easier to compute the image of $\nu (f)$ in $\overline {B(\delta _{f})}$ ; see Equation (6.4). We use this in §7 to determine nontriviality of $\nu (f)$ in some examples.

The following three propositions, and their proofs, are similar to Propositions 5.4, 5.8 and 5.10.

Proposition 5.12. We have isomorphisms

$$ \begin{align*} \overline{A(\delta)} \cong \frac{F_2L + H}{((\delta-I)L + H) \cap (F_2L + H)}, \hspace{10pt} \overline{B(\delta)} \cong \frac{F_2L+H}{(\delta-I)(F_{1}L) + F_{3}L + H}. \end{align*} $$

In particular, $\overline {A(\delta )}$ and $\overline {B(\delta )}$ are finite.

Proposition 5.13. The map $(\delta -I)_{\mathbb {Q}}:\operatorname {\mathrm {gr}}_{q-1}^F(L/H)_{\mathbb {Q}} \to \operatorname {\mathrm {gr}}_{q}^F(L/H)_{\mathbb {Q}}$ is surjective when $q=2,3$ . When Y has maximal rank, this map is injective for $q=1,2$ and

$$ \begin{align*} \overline{A(\delta)} \cong \frac{F_2L + H}{(\delta-I)F_1L + H}. \end{align*} $$

Proposition 5.14. If Y has maximal rank, then

$$ \begin{align*} \overline{A(\delta)} \cong \overline{B(\delta)} \times \prod_{i=1}^{g-2}(\mathbb{Z}/(q_i))^{{g-i\choose 2}}. \end{align*} $$

The next two propositions compare $A(\delta )$ and $B(\delta )$ from the previous subsection to their counterparts $\overline {A(\delta )}$ and $\overline {B(\delta )}$ .

Proposition 5.15. If Y has maximal rank, then we have an exact sequence

$$ \begin{align*} 0 \to \operatorname{\mathrm{coker}}(Q) \to B(\delta) \to \overline{B(\delta)} \to 0. \end{align*} $$

Proof. Consider the following commutative diagram

whose rows are exact. The map $\delta -I:F_1(L/H) \to F_2(L/H)$ is injective because it becomes an isomorphism after tensoring with $\mathbb {Q}$ by Proposition 5.13. We now get the desired exact sequence by the snake lemma.

Proposition 5.16. If Y has maximal rank, then we have an exact sequence

$$ \begin{align*} 0 \to \operatorname{\mathrm{coker}}(Q) \to A(\delta) \to \overline{A(\delta)} \to 0. \end{align*} $$

Proof. First, observe that $A(\delta )\to \overline {A(\delta )}$ induced by

$$ \begin{align*} \frac{F_2L}{(\delta-I)F_1L} \to \frac{F_2L + H}{(\delta-I)F_1L + H} \end{align*} $$

is surjective. Let K denote its kernel. Then we have the following commutative diagram

whose rows are exact. The vertical map on the left is an isomorphism by Propositions 5.10 and 5.14 and the snake lemma.

Corollary 5.17. When Y has maximal rank,

$$ \begin{align*} |\overline{A(\delta)}| = 2^{{g\choose 3}}\det(Q)^{{g\choose 2}-1}\prod_{i=1}^{g-2}q_i^{{g-i\choose 2}} \hspace{10pt} \text{and} \hspace{10pt} |\overline{B(\delta)}| = 2^{{g\choose 3}}\det(Q)^{{g\choose 2}-1}. \end{align*} $$

6.1 Dehn twists and multitwists

In this subsection, we recall some basic facts about Dehn twists. We refer the reader to [Reference Farb and Margalit19] for a more detailed treatment.

As before, set $H = H_1(\Sigma _g,\mathbb {Z}) = H_1(\Sigma _g^1,\mathbb {Z})$ . We write $[a]$ for the homology class of an isotopy class a of a simple closed curve, and $\hat {i}$ for the algebraic intersection number on H. The induced map $(T_{a})_* \in \operatorname {\mathrm {Sp}}(H)$ only depends on the homology class $[a]$ , and

(6.1) $$ \begin{align} (T_a)_* ([b]) = [b] + \hat{i}([a],[b])[a]. \end{align} $$

See [Reference Farb and Margalit19, Proposition 6.3].

Let $\Lambda $ be a collection of isotopy classes of pairwise nonintersecting essential simple closed curves in $\Sigma _g^1$ (resp. $\Sigma _g$ ). Define

The group is free and abelian by Lemma 6.1(2), and Y is saturated in H since any $d=\operatorname {\mathrm {rank}} Y$ collection of simple closed curves $\lambda _1,\ldots ,\lambda _d$ whose homology classes are linearly independent extends to an integral basis of H. Given , we write $\delta _{f} \in \operatorname {\mathrm {Sp}}(H)$ for the image of f under the symplectic representation (we could also call this $f_*$ , but use $\delta _f$ to better match the notation of §5). An element of the form

with $c_{\ell }>0$ for all $\ell $ is called a positive multitwist supported on $\Lambda $ .

Proof. We temporarily denote by $Y'$ the saturation of $\operatorname {\mathrm {im}} (\delta _{f}-I)$ in H. Applying Equation (6.1) to any $f = \prod T_{\ell }^{c_{\ell }}$ yields

(6.2) $$ \begin{align} (\delta_{f}-I)(h) = \sum_{\ell \in \Lambda} c_{\ell} \, \hat{i}([\ell],h)[\ell], \end{align} $$

and therefore $(\delta _{f}-I)(H) \leq Y$ . Applying this to a positive multitwist f, we have $Y'\leq Y$ . Because of this and the fact that Y and $Y'$ are saturated subgroups of H, the equality $Y_{\mathbb {Q}} = Y_{\mathbb {Q}}'$ will imply $Y=Y'$ . Denote by $(Y_{\mathbb {Q}})^{\omega } \leq H_{\mathbb {Q}}$ the symplectic complement to $Y_{\mathbb {Q}}$ , that is,

$$ \begin{align*} (Y_{\mathbb{Q}})^{\omega} = \{h\in H_{\mathbb{Q}} \, | \, \hat{i}( y,h )= 0 \text{ for all } y\in Y_{\mathbb{Q}}\} \end{align*} $$

and set $W_{\mathbb {Q}} = H_{\mathbb {Q}}/(Y_{\mathbb {Q}})^{\omega }$ . Because $Y_{\mathbb {Q}}$ is an isotropic subspace of $H_{\mathbb {Q}}$ , we have $\dim W_{\mathbb {Q}} = \dim Y_{\mathbb {Q}}$ . Since $(Y_{\mathbb {Q}})^{\omega }$ lies in the kernel of $(\delta _{f}-I)_{\mathbb {Q}}$ by Equation (6.2), the following

$$ \begin{align*} (\alpha,\alpha') = \hat{i}( \alpha, (\delta_{f}-I)\alpha') \end{align*} $$

defines a bilinear form on $W_{\mathbb {Q}}$ . We claim that this is actually an inner product. Indeed, we have

$$ \begin{align*} \hat{i}( \alpha,(\delta_{f}-I)\alpha') = \sum_{\ell\in \Lambda} c_{\ell}\, \hat{i}([\ell], \alpha) \hat{i}([\ell], \alpha'), \end{align*} $$

and hence this bilinear form is symmetric and positive semidefinite. If $\alpha $ is a nonzero element of $W_{\mathbb {Q}}$ , then there is some $\ell \in \Lambda $ such that $\hat {i}( [\ell ],\alpha ) \neq 0$ . By the above equation, we see that $(\alpha ,\alpha )> 0$ , which establishes the positive definiteness. We conclude that $(\delta _{f}-I)_{\mathbb {Q}}: W_{\mathbb {Q}} \to Y_{\mathbb {Q}}'$ is an injective linear map of vector spaces that have the same dimension and hence is also surjective. This proves $ Y_{\mathbb {Q}} = Y^{\prime }_{\mathbb {Q}}$ , and the proposition follows.

The following proposition will be used for ‘computing’ the Ceresa class of a Lagrangian collection of curves, in the sense of §6.3.

Proposition 6.3. Take elements , let f denote their product and let $\tau $ be a hyperelliptic quasi-involution. Then

$$ \begin{align*} J([f,\tau]) = \sum_{i=1}^M (g_i)_* \cdot J([f_i,\tau]) \end{align*} $$

for some .

Proof. We proceed by induction on M. When $M=1$ the Lemma is clear. Suppose that the Lemma is true for $M-1$ . Then by Proposition 2.1, we have

$$ \begin{align*} J([f,\tau]) =J([f_1,\tau])+(f_1)_{*} \cdot J([f_2\cdots f_M,\tau]) =J([f_1,\tau]) + (f_1)_{*} \sum_{i=2}^M (g_{i})_{*}\cdot J([f_i,\tau]). \end{align*} $$

Since each $f_i$ lies in , we have placed $J([f,\tau ])$ in the desired form.

6.2 Handlebodies and the Luft–Torelli group

Let V be a handlebody with boundary $\Sigma _g$ . Let $D\subset \Sigma _g$ be a small open disc so that $\Sigma _g^1 = \Sigma _{g} \setminus D$ . The handlebody group is the subgroup of $\Gamma _g^1$ consisting of mapping classes that are restrictions of homeomorphisms of V. Denote by the kernel of the homomorphism

The Luft–Torelli group of $\Sigma _g^{1}$ is

A meridian is a nontrivial isotopy class of a simple closed curve in $\Sigma _g^1$ that bounds a properly embedded disc in V. Note that $T_{\ell }$ lies in if $\ell $ is a meridian. A contractible bounding pair is a bounding pair $(\gamma ,\gamma ')$ such that $\gamma $ and $\gamma '$ are meridians, and a contractible bounding pair map is the product of Dehn twists $T_{\gamma }T_{\gamma '}^{-1}$ , where $(\gamma ,\gamma ')$ is a contractible bounding pair. The following is [Reference Pitsch34, Theorem 9].

Theorem 6.4. For $g\geq 3$ , the Luft–Torelli group $\mathbf {LT}_g^1(V)$ is generated by contractible bounding pair maps.

Figure 6.1

A basis of $\Sigma _g^1$ to compute $J(T_{\gamma }T_{\gamma '}^{-1})$ ; here, the $\beta _1,\ldots ,\beta _g$ are meridians of a handlebody.

Given a handlebody V, the kernel $Y_{V}$ of the map

$$ \begin{align*} H_{1}(\Sigma_g^1, \mathbb{Z}) \to H_1(V, \mathbb{Z}) \end{align*} $$

induced by the inclusion $\Sigma _g^1 \hookrightarrow V$ is a Lagrangian subspace of H. Define a filtration of L similar to the one in Equation (5.1):

$$ \begin{align*} F_q^{V}L = (\wedge^{q} Y_V) \wedge (\wedge^{3-q} H). \end{align*} $$

Proof. By Theorem 6.4, it suffices to prove the first statement. Let $(\gamma ,\gamma ')$ be a contractible bounding pair; we may assume that $\gamma $ and $\gamma '$ are as in Figure 6.1. Using the basis in this figure, we compute

$$ \begin{align*} J(T_{\gamma}T_{\gamma'}^{-1}) = \sum_{i=1}^{d-1} \alpha_i \wedge \beta_i \wedge [\gamma]. \end{align*} $$

Since $\beta _1,\ldots ,\beta _{d-1},[\gamma ]$ lie in Y, we see that $J(T_{\gamma }T_{\gamma '}^{-1})$ lies in $F_2^{V}L$ , as required.

6.3 The Lagrangian case

In this section and the next, we will prove our main result, that the Ceresa class is torsion. We begin by focusing on the case $\Sigma _g^1$ ; the closed surface case follows readily from this, as we explain in §6.5.

We say that a collection of nonintersecting simple closed curves $\Lambda $ is Lagrangian if the rank of Y is half the rank of H, the largest possible rank of Y. Given any Lagrangian $\Lambda $ on $\Sigma _g^1$ or $\Sigma _g$ , there is a handlebody V of $\Sigma _g$ such that each curve in $\Lambda $ is a meridian of V; see the proof of [Reference Hensel, Ji, Papadopoulos and Yau26, Lemma 5.7]. In this case, the subgroup $Y<H$ has three characterizations:

1. 1. Y is the integral span of the curves $[\ell ]$ for $\ell \in \Lambda $ (by definition),

2. 2. Y is the saturation of $\operatorname {\mathrm {im}}(\delta _{f}-I)$ for any positive multitwist f supported on $\Lambda $ (by Proposition 6.2),

3. 3. Y is the kernel $Y_V$ of the homomorphism $H_1(\Sigma _g,\mathbb {Z}) \to H_{1}(V,\mathbb {Z})$ .

Therefore, the filtrations $F_q^V L$ and $F_q L$ agree.

Proof. Suppose $g\geq 3$ . The commutator $[f,\tau ]$ lies in since each $T_{\ell }$ lies in and is a normal subgroup of . The commutator also lies in $\mathcal {I}_g^1$ since $\tau $ maps to the center of $\operatorname {\mathrm {Sp}}(H)$ . So $[f,\tau ] \in \mathbf {LT}_g^1(V)$ and therefore $J([f,\tau ]) \in F_2^V(L)$ by Proposition 6.5.

Next, suppose $g=2$ . There are $15$ arrangements $\Lambda $ on $\Sigma _2^1$ ; it suffices to consider the three maximal arrangements as illustrated in Figure 6.2. For each surface, regard the ‘inside’ as the handlebody V and $\tau $ is rotation by $180^{\circ }$ about the axis horizontal to the page. For the left or middle case, label the isotopy classes in $\Lambda $ in any order by $\ell _1, \ell _2, \ell _3, \ell _4$ and let $c_i=c_{\ell _i}$ . The eight isotopy classes $\{\ell _i,\tau (\ell _i) \, : \, i=1,2,3,4\}$ pairwise have geometric intersection number $0$ . Thus, the corresponding collection of Dehn twists commute, so

$$ \begin{align*} [f,\tau] = \prod_{i=1}^4 (T_{\ell_i} T_{\tau(\ell_i)}^{-1})^{c_i}. \end{align*} $$

Each $T_{\ell _i} T_{\tau (\ell _i)}^{-1}$ is either the identity or a contractible bounding pair, whence $J([f,\tau ]) \in F_2^V(L)$ by Proposition 6.5.

Figure 6.2

Maximal arrangements of nonintersecting curves on $\Sigma _2^1$ .

Finally, consider the arrangement on the right in Figure 6.2 and label the isotopy classes left to right by $\ell _1,\ell _2,\ell _3,\ell _4$ ; the curves $\ell _2$ and $\ell _3$ are separating. Since $T_{\ell _1}T_{\tau (\ell _1)}^{-1} = T_{\ell _2}T_{\tau (\ell _2)}^{-1} = \operatorname {\mathrm {id}}$ and $\ell _3, \tau (\ell _3)$ are separating curves, we have

$$ \begin{align*} J([f,\tau]) = c_{4} J(T_{\ell_4}T_{\tau(\ell_4)}^{-1}) \end{align*} $$

which lies in $F_2(L)$ by Proposition 6.5 since $(\ell _4, \tau (\ell _{4}))$ is a contractible bounding pair.

The last statement of the theorem follows from the fact that $F_2^V(L) = F_2(L)$ in the Lagrangian case and the finiteness of $A(\delta _{f})$ .

Suppose now that $\Lambda $ is Lagrangian and f is a positive multitwist supported on $\Lambda $ . Because $\mu (f) \in A(\delta _{f})$ , we can consider the image of this class in $B(\delta _{f})$ , which we denote by $w_{f}$ . The reason to consider $w_{f}$ is that it admits a relatively simple formula.

Proposition 6.7. If $\Lambda $ be a Lagrangian arrangement of curves on $\Sigma _g$ and $f = \prod T_{\ell }^{c_{\ell }}$ is a positive multitwist supported on $\Lambda $ , then

$$ \begin{align*} w_{f} = \sum_{\ell\in \Lambda} c_{\ell} \cdot J([T_{\ell},\tau]). \end{align*} $$

Proof. By Proposition 6.3,

$$ \begin{align*} J([f,\tau]) = \sum_{\ell \in \Lambda} c_{\ell} \cdot (g_{\ell})_* \cdot J([T_{\ell},\tau]) \hspace{20pt} \text{in } \hspace{5pt} B(\delta_{f}), \end{align*} $$

where each . The proposition now follows from the fact that acts trivially on $B(\delta _{f})$ .

6.4 The non-Lagrangian case

As we will see in Example 7.7, when the collection of homology classes of the curves in $\Lambda $ do not span a Lagrangian subspace of H, then the Ceresa class of a positive multitwist f need not live in $A(\delta _{f})$ . Nevertheless, this class is still torsion.

The proof will occupy this whole subsection. As usual, let $Y = \langle [\ell ] \, :\, \ell \in \Lambda \rangle $ , and $d = \dim Y$ . Choose a subset of loops $\ell _1, \ldots , \ell _d$ in $\Lambda $ which are linearly independent (and thus rationally span Y). Fix a collection of isotopy classes of simple closed curves $a_1,\ldots ,a_g, b_1,\ldots ,b_g$ on $\Sigma _g^1$ such that

1. 1. the classes $a_i$ and $b_j$ for $i,j=1,\ldots ,g$ are pairwise nonintersecting except $i(a_i,b_i)=1$ ;

2. 2. the classes $a_i$ and $b_j$ for $i,j=1,\ldots ,g-d$ do not intersect the classes in $\Lambda $ ;

3. 3. the homology classes $\alpha _i = [a_i]$ and $\beta _j = [b_j]$ form a symplectic basis for H;

4. 4. $b_{g+1-j} = \ell _j$ for $j=1,\ldots , d$ .

Let us explain why such a collection of curves exists. Let $\mathbf {G} = (G,w)$ be the weighted dual graph of $\Lambda $ . Recall that the sum of the weights $w(v)$ over all vertices v is $g-d$ . For each vertex v with $w(v)>0$ , choose a subsurface $S\cong \Sigma _{w(v)}^1$ of $\Sigma _v$ so that $\partial \Sigma _v$ lies in $\Sigma _v \setminus S$ . Now, define $w(v)$ of the $a_i$ and $b_i$ for $i\leq g-d$ by the basis in Figure 6.1. We have already specified $b_{g-d+1},\ldots ,b_g \in \Lambda $ . Finally, we may choose the remaining $a_i$ to complete the symplectic basis by [Reference Putman36, Lemma A.3].

Figure 6.3

A handlebody such that each $b_i$ is a meridian.

Choose closed regular neighborhoods $N_i$ of $a_i \cup b_i$ ; each $N_i$ is homeomorphic to a torus with one boundary component, and $\partial N_i$ is a separating curve. Choose $N_i$ small enough so that, for each $i\leq g-d$ , the boundary $\partial N_i$ does not intersect the curves in $\Lambda $ . The mapping class group $\Gamma _1^1$ is isomorphic to $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and the mapping class $\tau _i$ corresponding to $-I\in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ takes $a_i$ to $a_i$ and $b_i$ to $b_i$ , reversing their orientations. We claim that if M is a handlebody of $N_i$ , then . Since any two handlebody subgroups are conjugate to each other [Reference Hensel, Ji, Papadopoulos and Yau26, § 3] and $-I$ lies in the center of $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ , it suffices to show that $\tau _i$ lies in for just one M. Take $\Sigma _1^{1}$ to be the surface in Figure 6.1 (for $g=1$ ), and let M be the handlebody on the inside. A representative of the mapping class $\tau _i$ is given by a rotation of $180^{\circ }$ horizontal to the page and applying a suitable isotopy so that it is the identity on D. Extending by the identity defines $\tau _i$ on $\Sigma _{g}^{1}$ , and the product $\tau = \tau _1 \cdots \tau _g$ is a hyperelliptic quasi-involution on $\Sigma _{g}^{1}$ .

For $i=1,\ldots ,g-d$ , let $A_i$ and $B_i$ be handlebodies for $N_i$ so that $a_i$ is a meridian in $A_i$ and $b_i$ is a meridian in $B_i$ . The surface $S = \Sigma _g^1 \setminus \cup _{i=1}^{g-d} N_i^{\circ }$ is homeomorphic to $\Sigma _{d}^{g-d+1}$ (recall that $N_i^{\circ }$ denotes the relative interior of $N_i$ ). Let V be a handlebody for the surface obtained by capping off the boundary components of S so that $\ell $ is a meridian of V for each $\ell \in \Lambda $ . For any two-part partition $I,J$ of $\{1,\ldots ,g-d\}$ , let $V_{IJ}$ be the handlebody of $\Sigma _g^1$ obtained by attaching $A_i$ and $B_j$ to V for $i\in I$ and $j\in J$ . By the previous paragraph, we have that .