2.1 Stable graphs, tropical curves and tropical Jacobians

Given a graph G, denote by $V(G)$ and $E(G)$ its set of vertices and edges, respectively. An edge of G is a loop if it is adjacent to a single vertex, and a pair of nonloop edges $(f,f')$ are parallel if they join the same pair of vertices. A (integral, unweighted) tropical curve $\Gamma $ consists of a finite connected graph G, possibly with loops and parallel edges, together with a positive integer-valued function $c: E(G) \to \mathbb {Z}$ on the edge set $E(G)$ . We view G as the underlying graph of $\Gamma $ and $c(e)$ as the length of the edge e.Footnote ¹ The genus of $\Gamma $ , written as $g(\Gamma )$ , is the first Betti number of G, that is,

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

A tropical curve $\Gamma $ is said to be 2-connected if G has no cut-vertices; in particular, such a graph cannot have a loop or a bridge.

Let $\Gamma = (G,c)$ be a genus $g\geq 2$ tropical curve, and fix an orientation on G. The Jacobian of $\Gamma $ is the real g-dimensional torus

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

together with the positive definite symmetric quadratic form $Q_{\Gamma }$ on $H_1(G,\mathbb {R})$ given by

$$ \begin{align*} Q_{\Gamma}\left( \sum_{e\in E(G)} a_{e} \cdot e , \sum_{e\in E(G)} b_{e} \cdot e \right) = \sum_{e\in E(G)} a_e b_e \cdot c(e). \end{align*} $$

The valence of a vertex v is the number of half edges adjacent to it; in particular a loop edge contributes 2 to the valence. A connected graph is stable if every vertex has valence at least $3$ , and a tropical curve is stable if its underlying graph is stable. Two tropical curves are tropically equivalent if one can be obtained from the other by a sequence of the following moves:

• - adding or removing a 1-valent vertex and its incident edge, or

• - adding or removing a 2-valent vertex, preserving the underlying metric space.

Every tropical curve of genus $g\geq 2$ is tropically equivalent to a unique stable tropical curve.

A tropical curve is hyperelliptic if there is an involution $\sigma $ of $\Gamma $ such that the quotient $\Gamma /\sigma $ (in the sense of [Reference Chan8, § 2.2]) is a tree. A tropical curve is said to be of hyperelliptic type if its Jacobian is isomorphic to the Jacobian of a hyperelliptic tropical curve as a principally polarized tropical abelian variety. This notion is defined and studied in [Reference Corey9], where it is shown that $\Gamma $ is of hyperelliptic type if and only if its underlying graph has no $K_4$ or $L_3$ minor ([Reference Corey9, Theorem 1.1]); see Figure 1. Moreover, if $\Gamma $ is a 2-connected tropical curve of hyperelliptic type, then there is a hyperelliptic tropical curve $\Gamma '$ such that the underlying graph of $\Gamma $ is obtained from that of $\Gamma '$ by edge contractions.

2.2 Dual graphs and multitwists from tropical curves

We assume familiarity with surface topology and refer the reader to [Reference Farb and Margalit12] for a comprehensive treatment. Given an orientable topological real surface S, possibly with boundary or punctures, denote by $\operatorname {\mathrm {Mod}}(S)$ its mapping class group. We write $\Sigma _g$ for a closed genus-g surface and $\Sigma _g^1$ a genus g surface with one boundary component. We always identify $\Sigma _g^1$ as the subsurface of $\Sigma _g$ obtained by removing a small open disc. Given a curve $\gamma $ on $\Sigma _g$ or $\Sigma _g^1$ , write $T_{\gamma }$ for the (left-handed) Dehn twist about $\gamma $ .

Let $\Lambda $ be a finite collection of pairwise disjoint isotopy classes of simple closed curves on $\Sigma _g$ . Its (unweighted) dual graph is the graph with

• - a vertex $v_S$ for every connected component S of $\Sigma _g\setminus \bigcup _{\ell \in \Lambda } \ell $ , and

• - an edge $e_{\ell }$ between $v_S$ and $v_{S'}$ for each $\ell $ in the boundary of S and $S'$ . The curve $\ell $ is said to be dual to $e_\ell $ .

Any connected genus-g graph is the dual graph of such a configuration. We are primarily interested in the case where $\Lambda $ is Lagrangian in the sense of [Reference Corey, Ellenberg and Li10], that is, each component S of $\Sigma _g\setminus \bigcup _{\ell \in \Lambda } \ell $ has genus $0$ .

Let $\Lambda _1$ and $\Lambda _2$ be two Lagrangian arrangements of curves on $\Sigma _g$ as above such that their dual graphs, $G_1$ and $G_2$ , respectively, are stable. Then $G_1$ is isomorphic to $G_2$ if and only if there is a mapping class of $\Sigma _g$ that takes $\Lambda _1$ to $\Lambda _2$ ; this follows from the well-known identification of the quotient of the curve complex of $\Sigma _g$ by $\operatorname {\mathrm {Mod}}(\Sigma _g)$ with the tropical moduli space of genus g tropical curves of total edge length 1; see [Reference Chan, Galatius and Payne7, §1].

Given a genus-g tropical curve $\Gamma = (G,c)$ and $\Lambda $ an arrangement of curves on $\Sigma _g$ whose dual graph is G, define the multitwist

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

(Recall that we only consider tropical curves with integer edge lengths.) By the previous paragraph and the fact that $\sigma T_{\ell }\sigma ^{-1} = T_{\sigma (\ell )}$ for any $\sigma \in \operatorname {\mathrm {Mod}}(\Sigma _g)$ , the mapping class $T_{\Gamma }$ is well defined up to conjugation in $\operatorname {\mathrm {Mod}}(\Sigma _g)$ . Furthermore, $T_{\Gamma } = T_{\Gamma '}$ where $\Gamma '$ is the unique stable tropical curve tropically equivalent to $\Gamma $ .

2.3 The Johnson homomorphism

Let $\mathcal {I}_g^1 \leq \operatorname {\mathrm {Mod}}(\Sigma _g^1)$ , resp. $\mathcal {I}_g \leq \operatorname {\mathrm {Mod}}(\Sigma _g)$ , denote the Torelli group. Set

$$ \begin{align*} H = H_1(\Sigma_g^1,\mathbb{Z}) \cong H_1(\Sigma_g,\mathbb{Z}), \hspace{10pt} \text{ and } \hspace{10pt} L=\wedge^3H. \end{align*} $$

The intersection product on H induces a $2$ -form $\omega \in \wedge ^2H$ , and taking the exterior product with $\omega $ yields an injection $H\hookrightarrow L$ . The Johnson homomorphism $J:\mathcal {I}_g^1 \to L$ , resp. $J:\mathcal {I}_g \to L/H$ , may be characterized in the following way. By [Reference Powell16, Theorem 2], the Torelli group is generated by separating twists (Dehn twists about separating curves) and bounding pair maps (a product $T_{\gamma }T_{\gamma '}^{-1}$ where $\gamma ,\gamma '$ form a separating pair). If $\gamma $ is a separating curve on $\Sigma _g^{1}$ , then $J(T_{\gamma }) = 0$ . Now, suppose that $\gamma ,\gamma '$ are a separating pair. The removal of $\gamma \cup \gamma '$ from $\Sigma _g^1$ separates $\Sigma _g^1$ into two surfaces S and $S'$ ; let S be the subsurface not containing the boundary component of $\Sigma _g^1$ . The form $\omega $ restricts to the intersection 2-form on S; denote this by $\omega _S$ . Then

$$ \begin{align*} J(T_{\gamma}T_{\gamma'}^{-1}) = \omega_{S} \wedge [\gamma], \end{align*} $$

where $[\gamma ]$ is oriented so that S appears on its right. The Johnson homomorphism on $\mathcal {I}_g$ is defined in a similar way, except that one can choose either surface S or $S'$ in the above formula; the two possible expressions are equivalent modulo H.

2.4 The tropical Ceresa class

Next, we briefly recall the definition of the Ceresa class of $\Gamma $ as was described in the Introduction; see [Reference Corey, Ellenberg and Li10] for more details. Fix a hyperelliptic quasi-involution $\tau $ , that is, a mapping class in $\operatorname {\mathrm {Mod}}(\Sigma _g)$ that acts as $-I$ on H. Consider the map

$$ \begin{align*} \nu_{\tau}: \operatorname{\mathrm{Mod}}(\Sigma_g) \to L/H \hspace{20pt} \gamma \mapsto J([\gamma,\tau]), \end{align*} $$

where $[\gamma ,\tau ] = \gamma \tau \gamma ^{-1}\tau ^{-1}$ is the commutator. This is a 1-cocycle, and its class in $H^1(\operatorname {\mathrm {Mod}}(\Sigma _g),L/H)$ is independent of the choice of $\tau $ [Ibid., Proposition 2.1]. The $\tau $ -Ceresa cocycle of $\Gamma $ , denoted by $\nu _{\tau }(\Gamma )$ , is the restriction of $\nu _{\tau }$ to $\langle T_{\Gamma }\rangle \cong \mathbb {Z}$ . Similarly, the Ceresa class of $\Gamma $ , denoted by $\nu (\Gamma )$ , is the class of $\nu _{\tau }(\Gamma )$ in $H^1(\mathbb {Z}, L/H)$ . Next, we will discuss several groups related to a filtration on L in which the Ceresa class lives and our computation will take place.

A subgroup Y of H is Lagrangian if $Y_{\mathbb {Q}}$ is a maximal isotropic subspace of the symplectic vector space $H_{\mathbb {Q}}$ (the symplectic form is induced by the intersection pairing on H) and $H/Y$ is torsion-free. Let $Y \leq H$ be the subgroup generated by the homology classes $\{[\ell ] \, : \, \ell \in \Lambda \}$ . This is a Lagrangian subgroup; see [Ibid., §6.1]. For values $q=0,1,2,3$ , define the following filtrations on L and $L/H$ :

$$ \begin{align*} F_q L = (\wedge^q Y) \wedge (\wedge^{3-q} H), \hspace{15pt} F_q (L/H) = \frac{F_{q}L + H}{H}, \hspace{15pt} \operatorname{\mathrm{gr}}_q^{F}(L/H) = \frac{F_q(L/H)}{F_{q-1}(L/H)}. \end{align*} $$

These filtrations and groups associated with them are discussed in detail in [Ibid., §5].

Since $H\subset F_1L$ and $F_3L \cap H = \{0\}$ , we have

(2.1) $$ \begin{align} F_0(L/H) = \frac{L}{H}, \hspace{10pt} F_1(L/H) = \frac{F_1L}{H}, \hspace{10pt} F_2(L/H) = \frac{F_2L+H}{H}, \hspace{10pt} F_3(L/H) \cong \frac{F_3L}{F_3L \cap H} \cong F_3L. \hspace{10pt} \end{align} $$

Denote by $\delta _{\Gamma }:H\to H$ the pushforward map on H induced by $T_{\Gamma }$ . Recall from [Ibid., Lemma 5.1] that $\delta _\Gamma (F_qL)\subset F_{q+1}L $ and $\delta _\Gamma $ preserves $\omega $ . Then the map $\delta _{\Gamma }-I$ restricts to the graded components

$$ \begin{align*} \delta_{\Gamma}-I : \operatorname{\mathrm{gr}}_{q}^F(L/H) \to \operatorname{\mathrm{gr}}_{q+1}^F(L/H). \end{align*} $$

Define groups

$$ \begin{align*} A(\delta_{\Gamma}) &= \operatorname{\mathrm{im}}( H^1(\langle \delta_{\Gamma} \rangle, F_2(L/H)) \to H^1(\langle \delta_{\Gamma} \rangle, L/H)) \cong \frac{F_2L + H}{(\delta_{\Gamma}-I)(F_1L) + H} \\ B(\delta_{\Gamma}) &= \operatorname{\mathrm{coker}}(\delta_{\Gamma} - I: \operatorname{\mathrm{gr}}_1^{F}(L/H) \to \operatorname{\mathrm{gr}}_2^{F}(L/H)) \cong \frac{F_2L + H}{(\delta_{\Gamma}-I)(F_1L) + F_3L + H} \\ C(\delta_{\Gamma}) &= \operatorname{\mathrm{coker}}\left( (\delta_{\Gamma} - I)^2: \operatorname{\mathrm{gr}}_1^{F}(L/H) \to \operatorname{\mathrm{gr}}_3^{F}(L/H)\right) \cong \frac{F_3L}{(\delta_{\Gamma}-I)^2(F_1L)}. \end{align*} $$

The isomorphisms for $A(\delta _{\Gamma })$ and $B(\delta _{\Gamma })$ follow from [Ibid., Proposition 5.4]. These three groups are finite since $(\delta _{\Gamma }-I): \operatorname {\mathrm {gr}}_{q}^F(L/H) \to \operatorname {\mathrm {gr}}_{q+1}^F(L/H)$ is rationally surjective for $q=1,2$ [Ibid.]. The identity map on $F_2L + H$ induces a homomorphism $A(\delta _{\Gamma }) \to B(\delta _{\Gamma })$ and we have an induced homomorphism $(\delta _{\Gamma } - I) : B(\delta _{\Gamma }) \to C(\delta _{\Gamma })$ . By [Ibid., Theorem 6.6], there is a hyperelliptic quasi-involution $\tau $ such that $\nu _{\tau }(\Gamma )$ lies in $F_2(L/H)$ . In practice, whenever we compute the tropical Ceresa class, we always use such a $\tau $ . In particular, the Ceresa class $\nu (\Gamma )$ lies in $A(\delta _{\Gamma }) \subset H^1(\langle \delta _\Gamma \rangle ,L/H)$ . This implies the main result of [Ibid.], that $\nu (\Gamma )$ is torsion. Now, we fix the following notation

$$ \begin{align*} \mathbf{v}(\Gamma) = \text{image of } \nu(\Gamma) \text{ in } B(\delta_{\Gamma}) \hspace{20pt} \mathbf{w}(\Gamma) = (\delta_{\Gamma}-I)(\mathbf{v}(\Gamma)) \in C(\delta_\Gamma). \end{align*} $$

If $\tau $ satisfies $\nu _\tau (\Gamma ) \in F_2(L/H)$ , by [Ibid., Proposition 6.7], the class $\mathbf {v}(\Gamma )$ has a particularly nice representative

(2.2) $$ \begin{align} \mathbf{v}_{\tau}(\Gamma) = \sum_{e\in E(G)} c(e)\, J([T_{\ell_e}, \tau]). \end{align} $$

The tropical curve $\Gamma $ is Ceresa trivial if $\nu (\Gamma )=0$ . If $\Gamma $ is Ceresa trivial, then $\mathbf {v}(\Gamma ) = 0$ . Given the above formula, $\mathbf {v}(\Gamma )$ is easier to compute than $\nu (\Gamma )$ , and therefore a good strategy to show that $\Gamma $ is Ceresa nontrivial is to show that $\mathbf {v}(\Gamma ) \neq 0$ in $B(\delta _{\Gamma })$ .

3.1 Multitwist on the homology of a surface

Throughout, given $\mathbb {Z}$ -modules A and R, we set $A_R = A\otimes _{\mathbb {Z}} R$ . Let $\Lambda $ be a collection of pairwise disjoint isotopy classes of simple closed curves on $\Sigma _{g}^{1}$ or $\Sigma _{g}$ . Define a polynomial ring

$$ \begin{align*} R[\Lambda] = \mathbb{Z}[x_{\ell} \, : \, \ell \in \Lambda] \end{align*} $$

which we denote by R when $\Lambda $ is clear from the context. Denote by $\langle a,b \rangle $ the intersection product of homology classes $a,b\in H = H_1(\Sigma _g,\mathbb {Z})$ . For any loop $\ell \in \Lambda $ , let $\delta _{\ell }: H_{R} \to H_{R}$ be the homomorphism defined on simple tensors by

(3.1) $$ \begin{align} \delta_{\ell}(h\otimes a) = h\otimes a + \langle h, [\ell] \rangle [\ell] \otimes x_{\ell} a. \end{align} $$

The map $\delta _{\ell }$ is an isomorphism with inverse

(3.2) $$ \begin{align} \delta_{\ell}^{-1}(h\otimes a) = h\otimes a - \langle h, [\ell] \rangle [\ell] \otimes x_{\ell} a. \end{align} $$

Because the loops in $\Lambda $ are pairwise disjoint, the $\delta _{\ell }$ ’s pairwise commute. The homomorphism $\delta _\ell $ is related to the pushforward $T_{\ell })_{*}:H\to H$ in the following way. Given a function $c:\Lambda \to \mathbb {Z}$ , let $R\to \mathbb {Z}$ be the evaluation ring homomorphism $x_{\ell }\mapsto c(\ell )$ . Then, under the identification $H \cong H\otimes _{\mathbb {Z}} R \otimes _{R} \mathbb {Z}$ , the map $\delta _{\ell } \otimes _{R} \operatorname {\mathrm {id}}_{\mathbb {Z}}$ is the pushforward of $T_{\ell }^{c(\ell )}$ on H by [Reference Farb and Margalit12, Proposition 6.3].

Proposition 3.1. Given integers $\{a_{\ell } \, : \, \ell \in \Lambda \}$ , we have

$$ \begin{align*} \left(\prod_{\ell \in \Lambda} \delta_{\ell}^{a_{\ell}} \right) - I = \sum_{\ell\in \Lambda} a_{\ell} (\delta_{\ell}-I) \hspace{10pt} \text{ as maps } \hspace{10pt} H_R \to H_R. \end{align*} $$

Proof. It suffices to show

$$ \begin{align*} &\delta_{\ell}\delta_{\ell'} - I = (\delta_{\ell} - I) + (\delta_{\ell'} - I), \text{ and } \\ &\delta_{\ell}^{-1} - I = -(\delta_{\ell}-I). \end{align*} $$

The second formula follows readily from formulas (3.1) and (3.2). For the first formula, we have

$$ \begin{align*} (\delta_{\ell}\delta_{\ell'}-I)(h\otimes a) &= \delta_{\ell}(h \otimes a + \langle h, [\ell'] \rangle [\ell'] \otimes x_{\ell'} a)-h\otimes a \\ &= (\delta_{\ell}-I)(h\otimes a) + \langle h, [\ell'] \rangle \delta_{\ell}( [\ell'] \otimes x_{\ell'} a) \\ &= (\delta_{\ell}-I)(h\otimes a) + \langle h, [\ell'] \rangle ([\ell'] \otimes x_{\ell'} a + \langle [\ell'], [\ell] \rangle [\ell] \otimes x_{\ell}x_{\ell'} a) \\ &= (\delta_{\ell}-I)(h\otimes a) + \langle h, [\ell'] \rangle[\ell']\otimes x_{\ell'} a \\ &=(\delta_{\ell} - I)(h\otimes a) + (\delta_{\ell'} - I)(h\otimes a).\\[-34pt] \end{align*} $$

Next, define $\mathcal {B}=\langle \delta _{\ell } \, :\, \ell \in \Lambda \rangle $ , which is a free abelian subgroup of $\operatorname {\mathrm {Aut}}(H_R)$ whose rank is equal to the number of nonseparating loops in $\Lambda $ . Recall that $Y=\operatorname {\mathrm {span}}_{\mathbb {Z}}\{[\ell ] \, : \, \ell \in \Lambda \}$ , which is a Lagrangian subspace of H. The following is a straight-forward consequence of Equation (3.1) and Proposition 3.1.

Proposition 3.2. For any $f\in \mathcal {B}$ , we have

$$ \begin{align*} \mathrm{(1)} \;\; (f-I)(H_R) \subset Y_R, \hspace{20pt} \mathrm{(2)} \;\; (f-I)(Y_R) = 0, \hspace{20pt} \mathrm{(3)} \;\; (f-I)^2(H_R) = 0. \end{align*} $$

Denote by $\delta _{\Lambda }:H_R \to H_R$ the map

$$ \begin{align*} \delta_{\Lambda} = \prod_{\ell \in \Lambda} \delta_{\ell}. \end{align*} $$

By Proposition 3.1, as a map $H_R\to H_R$ , we have

$$ \begin{align*} \delta_{\Lambda}-I = \sum_{\ell \in \Lambda} (\delta_{\ell}-I). \end{align*} $$

3.2 The action of $\delta _{\Lambda }$ on $L_R$

The third exterior power of $\delta _{\Lambda }$ is a R-module homomorphism $\wedge ^3(H_R) \to \wedge ^3(H_R)$ . However, the natural R-module homomorphism

$$ \begin{align*} \wedge^3(H_R) \to L_R \hspace{20pt} (h_1\otimes f_1) \wedge (h_2\otimes f_2) \wedge (h_3\otimes f_3) \mapsto (h_1\wedge h_2 \wedge h_3) \otimes (f_1f_2f_3). \end{align*} $$

is an isomorphism; thus we may view the third exterior power of $\delta _{\Lambda }$ as an R-module endomorphism of $L_R$ which we still denote by $\delta _{\Lambda }$ . Next, we show that $\delta _{\Lambda }$ is an endomorphism of $L_R/H_R$ , but we need the following more general setup.

Let $\iota : S\hookrightarrow \Sigma _g$ be a subsurface (possibly with boundary) of $\Sigma _g$ . The homology of S splits as $H_{1}(S,\mathbb {Z}) \cong V \oplus W$ , where V is the subgroup of H generated by the boundary curves of S and W is a symplectic subspace of H whose symplectic form, which we denote by $\omega _S \in \wedge ^2W$ , is obtained by restricting the intersection of $\Sigma _g$ to S. Let $\mathcal {B}_S$ be the subgroup of $\mathcal {B}$ generated by the $\delta _{\ell }$ such that $\ell $ is isotopic to a curve in S.

Proof of Proposition 3.3.

For (1), it suffices to show that

(3.3) $$ \begin{align} \delta_{\ell}(\omega_{S}) \equiv \omega_S\quad \mod (V\wedge W)_{R}. \end{align} $$

Choose a symplectic basis $\alpha _1,\ldots ,\alpha _g$ , $\beta _1,\ldots ,\beta _g$ of H such that $\alpha _1,\ldots ,\alpha _h,\beta _1,\ldots ,\beta _h$ is a symplectic basis of W. This means that

$$ \begin{align*} \omega_S = \sum_{i=1}^{h} \alpha_i \wedge \beta_i. \end{align*} $$

Write $[\ell ] = \mu + \lambda $ , where $\mu \in V$ and $\lambda \in W$ . We have

$$ \begin{align*} \lambda = \sum_{i=1}^h \left( - \langle \beta_i, [\ell] \rangle \alpha_i + \langle \alpha_i, [\ell] \rangle \beta_i \right). \end{align*} $$

We compute

$$ \begin{align*} \delta_{\ell}(\omega_S) &= \sum_{i=1}^{h} (\alpha_i + \langle \alpha_i, [\ell] \rangle [\ell] \otimes x_{\ell} ) \wedge (\beta_i + \langle \beta_i, [\ell] \rangle [\ell] \otimes x_{\ell} ) \\ &= \omega_S + \sum_{i=1}^{h}\left( \langle \beta_i, [\ell] \rangle \alpha_i\wedge [\ell] + \langle \alpha_i, [\ell] \rangle [\ell] \wedge \beta_i \right) \otimes x_{\ell} \\ &= \omega_S + [\ell] \wedge \sum_{i=1}^{h} \left( -\langle \beta_i, [\ell] \rangle \alpha_i + \langle \alpha_i, [\ell] \rangle \beta_i \right) \otimes x_{\ell} \\ & = \omega_S + \mu \wedge \lambda \otimes x_{\ell}, \end{align*} $$

from which formula (3.3) follows. For (2), we have

$$ \begin{align*} (f-I)(h\wedge \omega_{S}) = f(h) \wedge f(\omega_{S}) - h\wedge \omega_{S} = (f-I)(h) \wedge \omega_{S} + f(h) \wedge \eta \end{align*} $$

for some $\eta \in (V\wedge W)_R$ .

Similar to the integral setup in Section 2.4, we define a filtration on $ L_R$ and $L_R/H_R$ :

$$ \begin{align*} F_q L_R := (F_qL)_R \cong (\wedge^qY_R)\wedge (\wedge^{3-q}H_R) \hspace{20pt} \text{ and } \hspace{20pt} F_q (L_R/H_R) := (F_q(L/H))_R \end{align*} $$

and denote by $\operatorname {\mathrm {gr}}_{q}^{F}(L_R/H_R)$ the graded piece

$$ \begin{align*} \operatorname{\mathrm{gr}}_{q}^{F}(L_R/H_R) := \frac{F_{q}(L_R/H_R)}{F_{q+1}(L_R/H_R)}. \end{align*} $$

Proposition 3.5. Given $f\in \mathcal {B}$ and $h_1,h_2,h_3\in H$ , we have that

$$ \begin{align*} (f-I)(h_1\wedge h_2 \wedge h_3) & \; = \; (f - I)h_1 \wedge h_2 \wedge h_3 + h_1 \wedge (f-I) h_2 \wedge h_3 + h_1 \wedge h_2 \wedge (f - I) h_3 \\ &\quad + (f - I)h_1 \wedge (f - I) h_2 \wedge h_3 + (f - I) h_1 \wedge h_2 \wedge (f - I) h_3\\ &\quad + h_1 \wedge (f - I) h_2 \wedge (f - I) h_3 \\ &\quad + (f - I)h_1 \wedge (f - I) h_2 \wedge (f - I) h_3. \end{align*} $$

In particular, we have that $(f-I)(F_{q}(L_R)) \subset F_{q+1}(L_R)$ .

As a consequence of this proposition, the map $\delta _{\Lambda } -I$ on $L_R/H_R$ takes $F_{q}(L_R/H_R)$ to $F_{q+1}(L_R/H_R)$ and hence induces a map on graded components:

$$ \begin{align*} \delta_{\Lambda}-I:\operatorname{\mathrm{gr}}_{q}^{F}(L_R/H_R) \to \operatorname{\mathrm{gr}}_{q+1}^{F}(L_R/H_R). \end{align*} $$

Proposition 3.6. As maps $\operatorname {\mathrm {gr}}_{q}^{F}(L_R/H_R) \to \operatorname {\mathrm {gr}}_{q+1}^{F}(L_R/H_R)$ , we have

$$ \begin{align*} (\delta_{\Lambda}-I)(\alpha) = \sum_{\ell\in \Lambda} (\delta_{\ell}-I)(\alpha). \end{align*} $$

Proof. By Proposition 3.5, we have that

$$ \begin{align*} (\delta_{\Lambda}-I)(h_1\wedge h_2 \wedge h_3) \equiv (\delta_{\Lambda}-I)h_1 \wedge h_2 \wedge h_3 + h_1 \wedge (\delta_{\Lambda}-I) h_2 \wedge h_3 + h_1 \wedge h_2 \wedge (\delta_{\Lambda}-I) h_3\quad \mod F_{q}(L_R/H_R). \end{align*} $$

Now, apply Proposition 3.1.

In summary, the maps $\delta _{\Lambda }-I$ and $\sum _{\ell \in \Lambda }(\delta _{\ell }-I)$ coincide when viewed as maps on $H_R$ or the graded components $\operatorname {\mathrm {gr}}_{q}^F(L_R/H_R)$ but are different when viewed on the whole $L_R/H_R$ . This difference is important in the proof of Proposition 4.2.

3.3 Configuration of loops from graphs

Let G be a connected genus-g stable graph and $\Lambda = \{\ell _e \, : \, e\in E(G)\}$ a collection of pairwise disjoint isotopy classes of simple closed curves on $\Sigma _g$ whose dual graph is G, as in §2.2. The polynomial ring

$$ \begin{align*} R[G] = \mathbb{Z}[x_e \, : \, e\in E(G)] \end{align*} $$

is identified with $R[\Lambda ]$ by the relabeling $x_{e} = x_{\ell _e}$ . We simply write R when the graph G is clear from context. Define the endomorphism $\delta _{G}$ on $H_{R}$ (respectively, $L_R$ and $L_{R}/H_{R}$ ) by $\delta _{G} = \delta _{\Lambda }$ . The map $\delta _G$ is closely related to the polarization of the Jacobian of a tropical curve, as we see in the following construction.

Define the symmetric quadratic form $Q_{G}$ on $H_1(G,\mathbb {Z})\otimes R$

$$ \begin{align*} Q_{G}\left( \sum_{e\in E(G)} a_{e} \cdot e , \sum_{e\in E(G)} b_{e} \cdot e \right) = \sum_{e\in E(G)} a_e b_e \otimes x_{e}. \end{align*} $$

If $\Gamma = (G,c)$ is a tropical curve, then $Q_{\Gamma }$ is obtained from $Q_{G}$ by the substitution $x_e \mapsto c(e)$ . Let us give an explicit description of $Q_{G}$ . Enumerate the edges of G by $e_1,\ldots ,e_n$ so that $e_{g+1},\ldots ,e_{n}$ are the edges of a spanning tree T. Fix an arbitrary orientation on G. For $j=1,\ldots , g$ , the graph $T\cup \{e_j\}$ contains a unique cycle $\gamma _j$ ; orient $\gamma _j$ in the direction of $e_j$ . The cycles $\gamma _1,\ldots ,\gamma _g$ form a basis of $H_1(G,\mathbb {Z})$ . Let $x_i = x_{e_i}$ . The $(i,j)$ –entry of the matrix of $Q_{G}$ with respect to this basis is a linear form in the $x_k$ ’s: $x_k$ appears with coefficient $1$ if $\gamma _i$ and $\gamma _j$ traverse $e_k$ in the same direction, $-1$ if they traverse in the opposite direction and $0$ if they do not meet at $e_k$ . For details and examples, see [Reference Bolognese, Brandt and Chua5, §4].

Now, we give the promised comparison to $\delta _{G}$ . Let $\ell _{k} = \ell _{e_k}$ . Orient $\ell _k$ so that the subsurface corresponding to the target of $e_k$ lies to the left of $\ell _k$ . Viewing G as a one-dimensional CW-complex, there is an embedding $\iota : G\hookrightarrow \Sigma _g$ so that $\iota (e_j) \cap \ell _j$ is a point. For $j=1,\ldots ,g$ , denote by $\alpha _j$ and $\beta _j$ the homology classes $-[\iota (\gamma _j)]$ and $[\ell _j]$ , respectively; the orientations are chosen so that the signed intersection number of $\alpha _i$ with $\beta _i$ is 1. Then $\alpha _1,\ldots ,\alpha _g,\beta _1,\ldots ,\beta _g$ is a symplectic basis for H.

Since the $\ell \in \Lambda $ are pairwise nonintersecting, we have that $\delta _{G}(\beta _j) = \beta _j$ . For the $\alpha _j$ ’s, we have

$$ \begin{align*} \delta_{G}(\alpha_j) = \alpha_{j} + \sum_{k=1}^{n} \langle \alpha_j, [\ell_{k}] \rangle [\ell_{k}] \otimes x_{k}. \end{align*} $$

The $\beta _{i}$ –coefficient of $\delta _{G}(\alpha _j)$ is

$$ \begin{align*} \langle \alpha_i, \delta_{G}(\alpha_j) \rangle = \sum_{k=1}^{n} \langle \alpha_{i}, [\ell_k] \rangle \langle \alpha_j, [\ell_k] \rangle \otimes x_{k}. \end{align*} $$

This is a linear form in $x_{k}$ : $x_{k}$ appears with coefficient 1 if $\alpha _{i}$ and $\alpha _{j}$ have the same intersection pairing with $\ell _{k}$ , $-1$ if they have opposite intersection pairing and 0 otherwise. Thus, the matrix of $\delta _G:H_R\to H_R$ with respect to the ordered basis $\alpha _1,\ldots ,\alpha _g,\beta _1,\ldots ,\beta _g$ is given by

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

This allows us to view $Q_G$ as a map $Y^{\perp }_{R} \to Y_R$ , where

$$ \begin{align*} Y=\operatorname{\mathrm{span}}_{\mathbb{Z}}\{[\ell] \, : \, \ell \in \Lambda\} = \operatorname{\mathrm{span}}_{\mathbb{Z}}\{\beta_1,\ldots,\beta_g\}, \hspace{10pt} \text{ and } \hspace{10pt} Y^{\perp}=\operatorname{\mathrm{span}}_{\mathbb{Z}}\{\alpha_1,\ldots,\alpha_g\}. \end{align*} $$

Example 3.7. Consider $G=K_4$ , the complete graph on $4$ vertices, with the orientation as in Figure 2. To its right is an arrangement of curves on $\Sigma _3$ whose dual graph is $K_4$ . Observe that each $\ell _i$ is oriented so that the subsurface corresponding to the target of $e_i$ lies to the left of $\ell _i$ . Also, observe that $\alpha _i$ is oriented so that the signed intersection product of $\alpha _i$ and $\beta _i:=[\ell _i]$ is 1. The matrix $Q_G$ is

(3.5) $$ \begin{align} Q_{G} = \begin{pmatrix} x_1 + x_5 + x_6 & -x_6 & -x_5 \\ -x_6 & x_2+x_4+x_6 & -x_4 \\ -x_5 & -x_4 & x_3+x_4+x_5 \end{pmatrix}. \end{align} $$

Figure 2

An arrangement of curves on $\Sigma _3$ with dual graph $K_4$ .

Combining the above connection between $\delta _{G}$ and $Q_{G}$ with Proposition 3.5, we deduce the following identities:

(3.6) $$ \begin{align} (\delta_G-I)(\alpha_{i} \wedge \alpha_{j} \wedge \beta_{k} ) &= Q_G(\alpha_i) \wedge \alpha_j \wedge \beta_k + \alpha_i \wedge Q_G(\alpha_j) \wedge \beta_k + Q_G(\alpha_i) \wedge Q_G(\alpha_j) \wedge \beta_k,\notag \\ (\delta_G-I)(\alpha_{i} \wedge \beta_{j} \wedge \beta_{k} ) &= Q_G(\alpha_i) \wedge \beta_j \wedge \beta_k, \\ (\delta_G-I)^2(\alpha_{i} \wedge \alpha_{j} \wedge \beta_{k} ) &= 2\, Q_G(\alpha_i) \wedge Q_G(\alpha_j) \wedge \beta_k. \notag \end{align} $$

4.1 Ceresa–Zharkov triviality

Recall that we view $\Sigma _g^1$ as a subsurface of $\Sigma _g$ obtained by removing a small open disc D from $\Sigma _g$ . Recall that a hyperelliptic quasi-involution is an element of $\operatorname {\mathrm {Mod}}(\Sigma _g^1)$ or $\operatorname {\mathrm {Mod}}(\Sigma _g)$ whose action on $H = H_1(\Sigma _g^1,\mathbb {Z}) \cong H_{1}(\Sigma _g,\mathbb {Z})$ is $-I$ . Let $\tau '$ be a hyperelliptic quasi-involution on $\Sigma _g^{1}$ and $\tau $ its image under the natural map $\operatorname {\mathrm {Mod}}(\Sigma _g^1) \to \operatorname {\mathrm {Mod}}(\Sigma _g)$ . Let $\Lambda '$ be a finite collection of pairwise disjoint isotopy classes of simple closed curves on $\Sigma _g^1$ and $\Lambda $ the image of this collection under the inclusion $\Sigma _g^1\hookrightarrow \Sigma _g$ ; note that any finite collection of simple closed curves can be isotoped away from the disc D, so $\Lambda $ may represent any finite collection of isotopy classes of simple closed curves on $\Sigma _g$ .

The $\tau '$ -Ceresa cocycle of $\Lambda '$ , respectively the $\tau $ -Ceresa cocycle of $\Lambda $ , is defined as

$$ \begin{align*} \boldsymbol \mu_{\tau'}(\Lambda') = \sum_{\ell \in \Lambda'} J([T_{\ell},\tau']) \otimes x_{\ell} \in L_R, \hspace{5pt} \text{respectively,} \hspace{5pt} \mathbf{v}_{\tau}(\Lambda) = \sum_{\ell \in \Lambda} J([T_{\ell},\tau]) \otimes x_{\ell} \in L_R/H_R, \end{align*} $$

where J is the Johnson homomorphism; see §2.3. Define the $\tau $ -Ceresa–Zharkov cocycle as

$$ \begin{align*} \mathbf{w}_{\tau}(\Lambda) = (\delta_{\Lambda}-I)(\mathbf{v}_{\tau}(\Lambda)). \end{align*} $$

When G is a genus-g graph, we write $\mathbf {v}_{\tau }(G) = \mathbf {v}_{\tau }(\Lambda )$ and $\mathbf {w}_{\tau }(G) = \mathbf {w}_{\tau }(\Lambda )$ , where $\Lambda $ is an arrangement of curves whose dual graph is G. We say that G is Ceresa–Zharkov trivial if there is a $\tau $ such that $\mathbf {w}_{\tau }(G) = 0$ . In Lemma 4.3, we show that this notion is a well-defined property of the graph G. While our primary interest is in $\mathbf {w}_{\tau }(G)$ , we occasionally need the class $\boldsymbol \mu _{\tau '}(\Lambda ')$ in §5. Compare these to the definitions of the Ceresa–Zharkov class and Ceresa–Zharkov triviality of a tropical curve defined in §2.4. In practice, when we compute the tropical Ceresa class for a tropical curve $\Gamma $ , we use a hyperelliptic involution $\tau $ such that $\nu _{\tau }(\Gamma )$ lies in $F_2(L/H)$ as in §2.4. We do something similar in the graph-theoretic case. As before, $R_d\subset R$ denotes the linear subspace of degree d forms.

Proposition 4.1. There exists a hyperelliptic quasi-involution $\tau '$ of $\Sigma _g^1$ such that

$$ \begin{align*} \boldsymbol \mu_{\tau'}(\Lambda') \in F_2L \otimes R_1. \end{align*} $$

In particular, $\mathbf {v}_{\tau }(G)$ lies in $F_2(L/H) \otimes R_1$ and $\mathbf {w}_{\tau }(G)$ lies in $F_3L \otimes R_2$ , where $\tau $ is the image of $\tau '$ under $\operatorname {\mathrm {Mod}}(\Sigma _g^1) \to \operatorname {\mathrm {Mod}}(\Sigma _g)$ .

Here is a characterization for Ceresa–Zharkov triviality when we use such a $\tau $ .

Proposition 4.2. If G is Ceresa–Zharkov trivial and $\mathbf {v}_{\tau }(G) \in F_2(L/H) \otimes R_1$ , then

(4.1) $$ \begin{align} \mathbf{w}_{\tau}(G) \in (\delta_{G}-I)^2(F_1(L/H)). \end{align} $$

Conversely, if there exists a $\tau $ such that Equation (4.1) holds, then G is Ceresa–Zharkov trivial.

To prove this proposition, we first derive a characterization of Ceresa–Zharkov triviality that works for any $\tau $ but is more cumbersome to state due to the fact that the endomorphism $\delta _G-I$ of $L_R/H_R$ does not split as the sum $\sum _{e\in E(G)}(\delta _e-I)$ as discussed in §3.2.

Denote by $\psi _G:L_R/H_R \to L_R/H_R$ the composition

$$ \begin{align*} \psi_G = (\delta_G-I) \circ \left(\sum_{e\in E(G) }(\delta_e-1) \right). \end{align*} $$

Proof. Given hyperelliptic quasi-involutions $\tau , \tilde {\tau }$ , we have

$$ \begin{align*} (J([T_{\ell_e},\tilde{\tau}]) - J([T_{\ell_e},\tau])) \otimes x_e = (T_{\ell_e}-I)_*(J(\tilde{\tau}^{-1} \tau)) \otimes x_e = (\delta_e-I)(J(\tilde{\tau}^{-1} \tau)) \end{align*} $$

and therefore

(4.2) $$ \begin{align} \mathbf{v}_{\tilde{\tau}}(G) - \mathbf{v}_{\tau}(G) = \sum_{e\in E(G)} (\delta_{e}-I)(J(\tilde{\tau}^{-1} \tau)). \end{align} $$

Suppose G is Ceresa–Zharkov trivial, say $\mathbf {w}_{\tau }(G)= 0$ for some $\tau $ . Applying $\delta _G-I$ to formula (4.2), we see that (1) $ \Rightarrow $ (2). The implication (2) $\Rightarrow $ (3) is trivial. Now, suppose (3) is true, say $\mathbf {w}_{\tau }(G) = \psi _G(\mathbf {u})$ for some $\mathbf {u} \in L/H$ . By surjectivity of the Johnson homomorphism, there is a $t\in \mathcal {I}_g$ such that $J(t)=\mathbf {u}$ . The function $\tilde {\tau } = \tau t$ is a hyperelliptic quasi-involution, and by the above equation, we have

$$ \begin{align*} \mathbf{v}_{\tilde{\tau}}(G) = \mathbf{v}_{\tau}(G) - \sum_{e\in E(G)} (\delta_{e}-I)(\mathbf{u}). \end{align*} $$

Applying $\delta _G-I$ , we get

$$ \begin{align*} \mathbf{w}_{\tilde{\tau}}(G) = \mathbf{w}_{\tau}(G) - \psi_G(\mathbf{u}) = 0, \end{align*} $$

as required.

Proof of Proposition 4.2.

Let $\alpha _1,\ldots ,\alpha _g,\beta _1,\ldots ,\beta _g$ be the symplectic basis of H from §3.3. We first record the image of the simple wedges under the map $\psi _G$ using Proposition 3.5:

$$ \begin{align*} \psi_G(\alpha_{i} \wedge \alpha_{j} \wedge \alpha_{k} ) &= 2(Q_{G} \alpha_i \wedge Q_{G} \alpha_j \wedge \alpha_k + Q_{G} \alpha_i \wedge \alpha_j \wedge Q_{G} \alpha_k + \alpha_i \wedge Q_{G} \alpha_j \wedge Q_{G}\alpha_k) \\ &\quad +3 \, Q_G \alpha_i \wedge Q_{G} \alpha_j \wedge Q_{G} \alpha_k \\ \psi_G(\alpha_{i} \wedge \alpha_{j} \wedge \beta_{k} ) &= 2\, Q_{G} \alpha_i \wedge Q_{G} \alpha_j \wedge \beta_k \\ \psi_G(\alpha_{i} \wedge \beta_{j} \wedge \beta_{k} ) &= \psi_G(\beta_{i} \wedge \beta_{j} \wedge \beta_{k} ) = 0. \end{align*} $$

By this computation, formula (3.6) and the fact that $(\delta _G-I)^2|_{F_2(L/H)} = 0$ , we deduce that $ (\delta _{G}-I)^2|_{F_1(L/H)} = \psi _{G} |_{F_1(L/H)}$ . Therefore,

(4.3) $$ \begin{align} (\delta_{G}-I)^2(F_1(L/H)) = \psi_G(F_1(L/H)) = \psi_G(L/H) \cap (F_3(L/H) \otimes R_2). \end{align} $$

Suppose G is Ceresa–Zharkov trivial and that $\mathbf {v}_{\tau }(G) \in F_2(L/H) \otimes R_1$ . Then the class $\mathbf {w}_{\tau }(G)$ lies in $\psi _G(L/H) \cap (F_3(L/H) \otimes R_2)$ by Lemma 4.3. So formula (4.1) follows from Equation (4.3), as required. Conversely, if there exists a $\tau $ such that formula (4.1) holds, then G is Ceresa–Zharkov trivial by Lemma 4.3 and formula (4.3).

4.2 Edge-contraction

In this section, we prove that Ceresa–Zharkov triviality is preserved under edge contraction. We prove in §5 that Ceresa–Zharkov triviality is a minor-closed property.

Suppose G is a connected graph, $f\in E(G)$ is a nonloop edge and $G/f$ the graph obtained from G by contracting f. Denote by $\zeta _f:R[G] \to R[G/f]$ the map that evaluates $x_f$ to 0; this is a retract of the natural inclusion $R[G/f] \subset R[G]$ .

Proposition 4.4. With G and f as above, we have an equality of maps $(L/H)_{R[G]} \to (L/H)_{R[G/f]} $ :

(4.4) $$ \begin{align} (1\otimes \zeta_f)\circ(\delta_{G}-I) = (\delta_{G/f}-I)\circ(1\otimes \zeta_f). \end{align} $$

Also,

$$ \begin{align*} (1\otimes \zeta_f) (\mathbf{v}_{\tau}(G)) = \mathbf{v}_{\tau}(G/f). \end{align*} $$

In particular, if G is Ceresa–Zharkov trivial, then so is $G/f$ .

Note that G and $G/f$ have the same genus g and both can be embedded into $\Sigma _g$ as described in §3.3. So the same $\tau \in \operatorname {\mathrm {Mod}}(\Sigma _g)$ gives Ceresa cocycles of G and $G/f$ .

Proof. First, consider the maps $1\otimes \zeta _f$ and $\delta _e-I$ on $H_R$ . We have

$$ \begin{align*} (1\otimes \zeta_f) (\delta_e-I) = (\delta_e-I) (1\otimes \zeta_f) \hspace{10pt} \text { when } e\neq f, \hspace{10pt} \text{ and } (1\otimes \zeta_f) (\delta_f-I) = 0. \end{align*} $$

As a map $H_R \to H_R$ , by Proposition 3.1, we have

$$ \begin{align*} (1\otimes \zeta_f) (\delta_G-I) = (1\otimes \zeta_f) \sum_{e\in E(G)}(\delta_e-I) = \sum_{e\in E(G/f)} (\delta_e-I) (1\otimes \zeta_f) = (\delta_{G/f}-I)(1\otimes \zeta_f). \end{align*} $$

Thus, the identity in formula (4.4) holds as maps $H_{R[G]} \to H_{R[G/f]}$ , and one readily extends this as an identity of morphisms $(L/H)_{R[G]} \to (L/H)_{R[G\setminus f]}$ using Proposition 3.5. Next, consider what happens for the Ceresa class. We have

$$ \begin{align*} (1\otimes \zeta_f)(\mathbf{v}_{\tau}(G)) = (1\otimes \zeta_f) \sum_{e\in E(G)} J([T_{\ell_e}, \tau]) \otimes x_e = \sum_{e\in E(G/f)} J([T_{\ell_e}, \tau]) \otimes x_e = \mathbf{v}_{\tau}(G/f). \end{align*} $$

Finally, suppose G is Ceresa–Zharkov trivial, say $\mathbf {w}_{\tau }(G) = 0$ for some $\tau $ . Then,

$$ \begin{align*} (\delta_{G/f}-I)(\mathbf{v}_{\tau}(G/f)) = (\delta_{G/f}-I)\circ (1\otimes \zeta_f)(\mathbf{v}_{\tau}(G)) = (1\otimes \zeta_f) \circ (\delta_{G}-I) (\mathbf{v}_{\tau}(G)) = 0, \end{align*} $$

and therefore $G/f$ is Ceresa–Zharkov trivial.

4.3 Tropical equivalence

Similar to the setting of tropical curves, two graphs are tropically equivalent if one can be obtained from the other via the following moves:

• - adding or removing a 1-valent vertex together with its adjacent edge,

• - subdividing a edge,

• - contracting exactly one edge adjacent to a 2-valent vertex.

In this section, we show Ceresa–Zharkov triviality is preserved under all these moves and thus is a property of tropical equivalence classes. This fact is analogous to [Reference Corey, Ellenberg and Li10, Lemma 4.4] which asserts that tropically equivalent tropical curves have the same Ceresa class.

Let G be a graph, and suppose $f \in E(G)$ is subdivided into two edges $e_1,e_2$ , producing the graph $G'$ . Consider the ring map $\phi _f:R[G] \to R[G']$ given by

$$ \begin{align*} \phi_f(x_e) = \left\{ \begin{array}{ll} x_e & \text{ if } e\neq f\\ x_{e_1} + x_{e_2} & \text{ if } e = f. \end{array} \right. \end{align*} $$

Lemma 4.6. With the above notation, we have

(4.5) $$ \begin{align} (1\otimes \phi_f) \circ (\delta_{G}-I) = (\delta_{G'}-I) \circ (1\otimes \phi_f) \end{align} $$

as morphisms $(L/H)_{R[G]} \to (L/H)_{R[G']}$ , and

$$ \begin{align*} (1 \otimes \phi_f) (\mathbf{v}_{\tau}(G)) = \mathbf{v}_{\tau}(G'). \end{align*} $$

Proof. First, observe that the loops $\ell _{f}$ , $\ell _{e_1}$ and $\ell _{e_2}$ are isotopic to each other, thus

$$ \begin{align*} T_{\ell_f} = T_{\ell_{e_1}} = T_{\ell_{e_2}} \end{align*} $$

as elements of $\operatorname {\mathrm {Mod}}(\Sigma _g)$ , and as a consequence,

$$ \begin{align*} \delta_{\ell_f} = \delta_{\ell_{e_1}} = \delta_{\ell_{e_2}}. \end{align*} $$

As morphisms $H_{R[G]} \to H_{R[G']}$ , we have

$$ \begin{align*} (1\otimes \phi_f) \circ (\delta_{e}-I) = \left\{ \begin{array}{ll} (\delta_{\ell_e}-I) \circ (1\otimes \phi_f) & \text{if } e \neq f \\ ((\delta_{\ell_{e_1}}-I) + (\delta_{\ell_{e_2}}-I) ) \circ (1\otimes \phi_f) & \text{if } e = f. \end{array} \right. \end{align*} $$

Therefore,

$$ \begin{align*} (1\otimes \phi_f) \circ (\delta_{G}-I) &= (1\otimes \phi_f) \circ \sum_{e\in E(G)} (\delta_{\ell_{e}}-I) \\ &= \left((\delta_{\ell_{e_1}}-I) + (\delta_{\ell_{e_2}}-I) + \sum_{e\in E(G)\setminus f} (\delta_{\ell_{e}}-I) \right) \circ (1\otimes \phi_f) \\ &= (\delta_{G'}-I) \circ (1\otimes \phi_f). \end{align*} $$

Thus, the identity in Formula 4.5 holds as morphisms $H_{R[G]} \to H_{R[G']}$ . One readily extends this as an identity of morphisms $(L/H)_{R[G]} \to (L/H)_{R[G']}$ using Proposition 3.5.

For the statement regarding the Ceresa class, we compute

$$ \begin{align*} (1\otimes \phi_f)(\mathbf{v}_{\tau}(G)) &= J([T_{\ell_f},\tau]) \otimes (x_{e_1} + x_{e_2}) + \sum_{e\in E(G) \setminus f} J([T_{\ell_e},\tau]) \otimes x_e \\ &= \sum_{e\in E(G')} J([T_{\ell_e},\tau]) \otimes x_e \\ &= \mathbf{v}_{\tau}(G').\\[-36pt] \end{align*} $$

Proof. If $G'$ is obtained from G by adding or contracting an edge adjacent to a 1-valent vertex, then the claim follows from Lemma 4.5. Now, suppose $G'$ is obtained from G by subdividing an edge $f \in E(G)$ into $e_1$ and $e_2$ . If $G'$ is Ceresa–Zharkov trivial, then G is Ceresa–Zharkov trivial by Proposition 4.4. Conversely, if G is Ceresa–Zharkov trivial, say $\mathbf {w}_{\tau }(G) = 0$ for some $\tau $ , then by Lemma 4.6 we have

$$ \begin{align*} \mathbf{w}_{\tau}(G') = (\delta_{G}-I) \circ (1\otimes \phi_f) (\mathbf{v}_{\tau}(G)) = (1\otimes \phi_f)(\mathbf{w}_{\tau}(G)) = 0 \end{align*} $$

and hence $G'$ is Ceresa–Zharkov trivial.

4.4 Relation to the tropical Ceresa class

Let $\Gamma = (G,c)$ be a tropical curve. Recall from Section 2.4 that $\mathbf {v}(\Gamma ) \in B(\delta _{\Gamma })$ is the image of the Ceresa class $\nu (\Gamma )$ under the map $A(\delta ) \to B(\delta )$ . Given $c:E(G) \to \mathbb {Z}_{>0}$ , define homomorphisms

$$ \begin{align*} \epsilon_{c}^{1}: F_2(L/H) \otimes R_1 \to B(\delta_{\Gamma}) \hspace{20pt} \epsilon_{c}^{2}: F_3(L/H) \otimes R_2 \to C(\delta_{\Gamma}) \end{align*} $$

by the compositions

$$ \begin{align*} &F_2(L/H)\otimes R_1 \to F_2(L/H) \to \operatorname{\mathrm{gr}}_2^F(L/H) \to B(\delta_{\Gamma}), \\ &F_3(L/H)\otimes R_2 \to F_3(L/H) \to \operatorname{\mathrm{gr}}_3^F(L/H) \to C(\delta_{\Gamma}), \end{align*} $$

respectively, where the first maps $ F_2(L/H)\otimes R_1 \to F_2(L/H)$ and $F_3(L/H)\otimes R_2 \to F_3(L/H)$ are given by the evaluation map $h\otimes f \mapsto h\otimes f(c)$ . We say that $\Gamma $ is Ceresa–Zharkov trivial if $\mathbf {w}(\Gamma ) = 0$ in $C(\delta _{\Gamma })$ .

Proposition 4.8. Let $\Gamma = (G,c)$ be a tropical curve, and let $\tau $ be a hyperelliptic quasi-involution such that $\mathbf {v}_{\tau }(G)$ lies in $F_2(L/H)\otimes R_1$ . Then

$$ \begin{align*} \epsilon_{c}^1(\mathbf{v}_{\tau}(G)) = \mathbf{v}(\Gamma), \hspace{15pt} \text{and} \hspace{15pt} \epsilon_{c}^{2}(\mathbf{w}_{\tau}(G)) = \mathbf{w}(\Gamma). \end{align*} $$

In particular, if G is Ceresa–Zharkov trivial, then so is $\Gamma $ .

By [Reference Corey, Ellenberg and Li10, Proposition 4.6], if a tropical curve $\Gamma $ is hyperelliptic, then $\Gamma $ is Ceresa trivial. Assuming Theorem 5.11, we derive a similar statement for tropical curves of hyperelliptic type and Ceresa–Zharkov triviality.

However, the converse to Proposition 4.9 is not true by Remark 5.8.