1.1 Tropical curves

Let G be a graph, by which we mean a finite, connected multigraph with vertex and edge sets $V(G)$ and $E(G)$ , respectively. The valence of a vertex v, denoted $\operatorname {\mathrm {val}}(v)$ , is the number of half-edges incident to v. A leaf is an edge incident to a vertex of valence one. The genus of G is the quantity $\# E(G) - \# V(G) + 1$ .

Fix an arbitrary orientation on the edges of G; then each edge e has a head vertex $e^+$ and tail vertex $e^-$ . Given the additional data of a length function , we construct a topological space

where $[0,\ell (e)] \subset \mathbb {R}$ is a closed interval of length $\ell (e)$ and the gluing relations on endpoints are given by the incidence relations in G, that is, for all $e, f \in E(G)$ not necessarily distinct,

$$ \begin{align*} \begin{cases} (0,e) \sim (0,f) & \text{if } e^- = f^- \\ (0,e) \sim (\ell(f),f) & \text{if } e^- = f^+ \\ (\ell(e),e) \sim (\ell(f),f) & \text{if } e^+ = f^+, \end{cases} \end{align*} $$

where $(t,e)$ denotes the point t in the closed interval $[0,\ell (e)]$ associated to e in the disjoint union. Observe that C becomes a metric space with the ‘shortest distance’ metric induced by edge lengths. We say that a metric space C obtained via this construction is a tropical curve. We further say that the edge-weighted graph $(G,\ell )$ is a model for C, that G underlies C, and that C overlies G.

Many graphs can underlie the same tropical curve. More precisely, consider the relation on pairs $(G,\ell )$ generated by a single edge refinement: pick an edge e in G and subdivide it into two edges, $e_{1}$ and $e_{2}$ , to obtain a new graph $G^{\prime }$ ; fix the length function $\ell ^{\prime }$ on $G^{\prime }$ that equals $\ell $ away from e and satisfies $\ell ^{\prime }(e_{1}) + \ell ^{\prime }(e_{2}) = \ell (e)$ . The inverse operation is performed at vertices of valence two. If $(G,\ell )$ is a model for C, then every other model is obtained from $(G,\ell )$ by a sequence of refinements and inverse refinements. The genus of C is its first Betti number or, equivalently, the genus of any underlying graph. A tropical curve of genus at least two has a unique minimal model with no vertices of valence two.

We remark that our notion of tropical curve is what in the language of 1-dimensional rational polyhedral spaces should better be called a ‘smooth’ tropical curve; see, for instance, [Reference Gross and Shokrieh21, Section 2.3]. Also, our definition departs slightly from that of [Reference Mikhalkin and Zharkov26, Section 3.1], wherein leaves of a graph G are prescribed to have infinite length. This difference is of no critical importance: as we shall see in §4.2, our main object of interest, the Ceresa period, is not affected by contracting all of the leaves.

1.2 Real tori with integral structures

Given a ring R, an R-algebra S and an R-module M, we write as a shorthand and abbreviate the primitive elements $s \otimes m$ of $M_S$ by $sm$ .

Fix a free abelian group N of rank g. Then we may naturally regard N as a (full-rank) lattice inside of the g-dimensional $\mathbb {R}$ -vector space $N_{\mathbb {R}}$ . Fix another lattice $\Lambda \subset N_{\mathbb {R}}$ . The quotient group with the quotient topology and smooth structure induced from $N_{\mathbb {R}}$ is a real torus of dimension g. The tangent space at every point of X is canonically isomorphic to the universal cover $N_{\mathbb {R}}$ , so the lattice N defines what is known as an integral structure on the real torus X. We say that a tangent vector v on X is integral if $v \in N$ .

1.3 Smooth tropical homology

Recall that singular q-chains on a topological space X are integral formal sums of continuous maps $\Delta ^q \to X$ , where $\Delta ^q$ is the standard q-simplex

If X is a smooth manifold, we may restrict our attention to smooth simplices $\Delta ^q \to X$ ; the boundary map on singular chains takes smooth q-chains to smooth $(q-1)$ -chains, so we obtain the smooth singular chain complex $S_{\bullet }(X)$ . We write $H_{q}(X)$ for the q-th homology of $S_{\bullet }(X)$ . It is well-known that the inclusion of $S_{\bullet }(X)$ into the usual singular chain complex is a chain homotopy equivalence, and therefore that $H_{q}(X)$ is isomorphic to the singular homology of X.

We now restrict to the case where $X = N_{\mathbb {R}}/\Lambda $ is a real torus with integral structure N. Say that a smooth q-simplex $\sigma $ in X is affine if it is obtained from an affine map $\mathbb {R}^{q+1} \to N_{\mathbb {R}}$ by restricting to $\Delta ^q \subset \mathbb {R}^{q+1}$ and pushing forward by the quotient $N_{\mathbb {R}} \to X$ . Given points $u_i \in N_{\mathbb {R}}$ , we write $(u_0,\ldots ,u_q)$ for the unique affine q-simplex defined by mapping the i-th vertex of $\Delta ^q$ to $u_i$ . For any $\lambda \in \Lambda $ , $(u_0+\lambda ,\ldots ,u_q+\lambda )$ and $(u_0,\ldots ,u_q)$ represent the same affine simplex of the torus X.

The tropical homology of rational polyhedral spaces was introduced by [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov22] and has been further studied in, for example, [Reference Jell, Rau and Shaw23, Reference Jell, Shaw and Smacka24, Reference Gross and Shokrieh20]. On a real torus X with integral structure as above, a tropical $(p,q)$ -simplex is a singular q-simplex $\sigma $ with the additional data of a p-fold wedge product of integral tangent vectors at some point in the image of $\sigma $ ; since we have canonically identified the tangent space at each point of X with the universal cover, this extra data corresponds to an element of $N^{\wedge p}$ , the p-th exterior power of N. However, we would like to be able to integrate over tropical chains, so we modify the definition by replacing singular chains with smooth ones:

The boundary maps on $S_{p,\bullet }(X)$ are induced by those on $S_{\bullet }(X)$ ; we write $H_{p,q}(X)$ for the corresponding q-th homology group. By the universal coefficient theorem and the equivalence of smooth and singular homology noted above, $H_{p,q}(X)$ is isomorphic to the tropical homology in the sense of [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov22], and in fact,

$$ \begin{align*} H_{p,q}(X) \cong N^{\wedge p} \otimes_{\mathbb{Z}} H_{q}(X). \end{align*} $$

By the Künneth theorem, $H_{q}(X) \cong \Lambda ^{\wedge q}$ . Fixing bases $\epsilon _1,\ldots ,\epsilon _g$ for N and $\lambda _1,\ldots ,\lambda _g$ for $\Lambda $ , the Eilenberg–Zilber map for singular homology determines explicit generators

(1) $$ \begin{align} \begin{aligned} H_{1,1}(X) &= \mathbb{Z}\langle{\epsilon_i \otimes (0,\lambda_j) \; | \; i,j \in [g]}\rangle \\ H_{1,2}(X) &= \mathbb{Z}\langle{\epsilon_i \otimes \big((0,\lambda_j,\lambda_j+\lambda_k) - (0,\lambda_k,\lambda_j+\lambda_k)\big)\; | \; i,j,k \in [g],\, j < k}\rangle, \end{aligned} \end{align} $$

where $[g] = \{1,\ldots ,g\}$ .

1.4 Tropical algebraic cycles

A (tropical) algebraic k-cycle in a rational polyhedral space X is a balanced, weighted, rational polyhedral complex of pure dimension k, defined up to refinement. For details, we refer to [Reference Allermann and Rau2, Section 2] and [Reference Allermann, Hampe and Rau1, Section 2]. In the case where $X = N_{\mathbb {R}}/\Lambda $ is a real torus with integral structure N, a polyhedral complex in X is a stratified closed subset that locally lifts to a polyhedral complex in $N_{\mathbb {R}}$ . We call it weighted if every facet is assigned an integer weight, and rational if the affine hull of every face has a basis that is integral in the sense of §1.2. Algebraic k-cycles form an abelian group $\mathscr {Z}_k(X)$ , and a morphism of rational polyhedral spaces induces a pushforward homomorphism . In analogy with the classical situation, there is a notion of rational equivalence of cycles, which is explored in [Reference Allermann, Hampe and Rau1, Section 3]. Likewise, there is a notion of algebraic equivalence defined by [Reference Zharkov28] and explored further in [Reference Gross and Shokrieh21, Section 5.1].

There exists a group homomorphism , called the cycle class map, from algebraic k-cycles to tropical $(k,k)$ -homology classes. We note that $\operatorname {\mathrm {cyc}}$ commutes with pushforward maps. When X is a real torus with integral structure as above and $Z \in \mathscr {Z}_1(X)$ , we may describe a representative $(1,1)$ -cycle $\operatorname {\mathrm {ch}}(Z)$ of the homology class $\operatorname {\mathrm {cyc}}(Z)$ as follows. (For the general case, see [Reference Jell, Rau and Shaw23, Section 4.C] or [Reference Gross and Shokrieh20, Section 5].) Fix a polyhedral complex structure P for Z. Each facet $\sigma $ of P is a line segment with rational slope, so we may lift it to a line segment in the universal cover $N_{\mathbb {R}}$ with endpoints $u_{\sigma }$ and $u^{\prime }_{\sigma }$ and primitive tangent vector $n_{\sigma } \in N$ ; that is, $n_{\sigma }$ is a positive scalar multiple of $u^{\prime }_{\sigma } - u_{\sigma }$ such that $a n_{\sigma } \in N$ for precisely $a \in \mathbb {Z}$ . Then

(2)

where the summation is taken over all facets $\sigma $ of P. The balancing condition at the vertices of P translates to $\operatorname {\mathrm {ch}}(Z)$ having trivial boundary. We caution that the ‘chain map’ so defined is highly non-canonical; in particular, it depends on a choice of polyhedral complex structure underlying Z for each algebraic 1-cycle Z. We shall have to contend with this fact in §1.8 when we define the Ceresa period $\alpha (C)$ .

1.5 Tropical Jacobians

The tropical Jacobian is a real torus with integral structure that we can canonically associate to any tropical curve. The original definition given by [Reference Mikhalkin and Zharkov26, Section 6.1] mimics that of the Jacobian of a complex algebraic curve; here, we present an equivalent definition that avoids mention of $1$ -forms.

Let C be a tropical curve of genus g. Given a model $(G,\ell )$ of C, let $C_1(G,\mathbb {Z})$ denote the oriented simplicial 1-chains on G. The corresponding simplicial homology $H_1(G,\mathbb {Z}) \subset C_1(G,\mathbb {Z})$ of G is isomorphic to the singular homology $H_1(C,\mathbb {Z})$ of C; in what follows, we conflate $H_1(G,\mathbb {Z})$ and $H_1(C,\mathbb {Z})$ without further remark. Furthermore, by the universal coefficient theorem, we may identify $H_1(C,\mathbb {R})^{\vee }$ with $H^1(C,\mathbb {R})$ . Fixing an orientation on the edges of G, we define a symmetric, bilinear map called the length pairing by

(3) $$ \begin{align} [e, e^{\prime}] = \begin{cases} \ell(e) &\text{ if } e = e^{\prime} \\ 0 &\text{ otherwise} \end{cases}. \end{align} $$

For any refinement $G^{\prime }$ of G, this pairing descends to $C_1(G^{\prime },\mathbb {Z}) \times C_1(G^{\prime },\mathbb {Z})$ in a way that is compatible with the restriction of either entry to $H_1(C,\mathbb {Z})$ . Likewise, $[\cdot ,\cdot ]$ formally extends to allow coefficients in $\mathbb {R}$ . By the $\otimes $ – $\operatorname {\mathrm {Hom}}$ adjunction, $[\cdot ,\cdot ]$ induces a homomorphism via $e \mapsto [e,\cdot ]$ . Consider the lattice

its dual lattice is defined by

The tropical Jacobian of C is the real torus

with integral structure given by N.

Fix a point $v \in C$ . Following [Reference Baker and Faber8, Section 4], we define the Abel–Jacobi map based at v by

where $\delta $ is a path in $C_1(G,\mathbb {Z})$ from v to w for some model $(G,\ell )$ of C containing both v and w as vertices. We claim that $\Phi _v(w)$ is well defined. Indeed, the definition does not depend on $\delta $ , since any other path $\delta ^{\prime }$ from v to w satisfies $\delta ^{\prime } - \delta \in H_1(C,\mathbb {Z})$ , and hence, $\pi (\delta ^{\prime }) \equiv \pi (\delta )\ \pmod \Lambda $ . Moreover, it is independent of the choice of model $(G,\ell )$ because the length pairing is preserved under refinement.

1.6 Tropical Ceresa cycle

We identify C with its fundamental algebraic cycle in $\mathscr {Z}_1(C)$ – that is, the unique 1-cycle with support equal to all of C and with weight one on every edge. We also write for the inversion map. Following [Reference Zharkov28], we define the tropical Ceresa cycle based at $v \in C$ by

where $\Phi _{v,*}$ and $[-1]_{*}$ are the induced pushforwards on algebraic cycles mentioned in §1.4. Given two basepoints v and w, it is not hard to see that $\Phi _{v,*}C$ and $\Phi _{w,*}C$ are related by translation. Then it follows from [Reference Gross and Shokrieh21, Proposition 5.5] that, modulo algebraic equivalence, ${\mathbf {Cer}}_{v}(C)$ does not depend on the choice of basepoint v.

Fix a model $(G,\ell )$ of C for which $v \in V(G)$ . This determines a polyhedral complex structure on $\Phi _{v,*}C$ ; with respect to this choice, we define

where $\operatorname {\mathrm {ch}}$ is the chain map defined in §1.4. By construction, this means that $\operatorname {\mathrm {Cer}}_{v,G}(C)$ is a $(1,1)$ -cycle whose class in $H_{1,1}(\operatorname {\mathrm {Jac}}(C))$ is $\operatorname {\mathrm {cyc}}({\mathbf {Cer}}_{v}(C))$ . In fact, using the generators of $H_{1,1}(\operatorname {\mathrm {Jac}}(C))$ given by (1), one sees that $[-1]_*$ acts as the identity on $H_{1,1}(\operatorname {\mathrm {Jac}}(C))$ . Then the Ceresa cycle is homologically trivial:

$$ \begin{align*} \operatorname{\mathrm{cyc}}\left({\mathbf{Cer}}_{v}(C)\right) &= \operatorname{\mathrm{cyc}}(\Phi_{v,*}C) - [-1]_{*}\operatorname{\mathrm{cyc}}(\Phi_{v,*}C)\\ &= \operatorname{\mathrm{cyc}}(\Phi_{v,*}C) - \operatorname{\mathrm{cyc}}(\Phi_{v,*}C)\\ &= 0. \end{align*} $$

A straightforward computation using the expression for $\operatorname {\mathrm {ch}}$ in (2) shows that

(4) $$ \begin{align} \operatorname{\mathrm{Cer}}_{v,G}(C) = \sum_{e \in E(G)} \ell(e)^{-1}\pi(e) \otimes \big( (\pi(\delta_e),\pi(\delta_e+e)) + (-\pi(\delta_e),-\pi(\delta_e+e)) \big), \end{align} $$

where $\delta _e$ is a path in G from v to $e^-$ . This formula will be key in defining a universal version of the Ceresa cycle in §2.3.

1.7 Integration map

Let $\epsilon _1,\ldots ,\epsilon _g$ denote a basis for N and $z_1,\ldots ,z_g$ the associated coordinate functions on $N_{\mathbb {R}}$ ; then the differential 1-forms $dz_i$ on $N_{\mathbb {R}}$ descend to $\operatorname {\mathrm {Jac}}(C)$ . Recall from §1.3 that $S_{1,2}(\operatorname {\mathrm {Jac}}(C))$ is generated by elements of the form $n \otimes \sigma $ , where $n = \sum _{i=1} n_i\epsilon _i \in N$ and is a smooth $2$ -simplex. Given a differential form $\omega $ on $\operatorname {\mathrm {Jac}}(C)$ , we define the integral of $\omega $ over $\sigma $ by

Let $\binom {[g]}{3}$ denote the collection of all subsets of $[g]$ of cardinality $3$ . Given $I \in \binom {[g]}{3}$ , we write $I = \{i_1,i_2,i_3\}$ for $i_1 < i_2 < i_3$ and adopt the multi-index notation . Mimicking the ‘determinantal 2-form’ for $K_4$ introduced by [Reference Zharkov28], we define the integration map via

(5)

In the case that $\sigma = (u_0,u_1,u_2)$ is affine (see §1.3), (5) reduces to

$$ \begin{align*} \Theta(n \otimes (u_0,u_1,u_2)) = \sum_{I \in \binom{[g]}{3}} \det(M_{I,[3]}) \epsilon_I, \end{align*} $$

where M is the $g \times 3$ matrix whose columns are the entries of n, $u_1-u_0$ , and $u_2-u_0$ with respect to the basis $\epsilon _1,\ldots ,\epsilon _g$ . We see that $\Theta $ is coordinate-independent at least on affine $(1,2)$ -chains by rewriting this expression in terms of exterior products:

(6) $$ \begin{align} \Theta(n \otimes (u_0,u_1,u_2)) = n \wedge (u_1-u_0) \wedge (u_2-u_0). \end{align} $$

The boundary map sends $n \otimes \sigma \mapsto n \otimes \partial \sigma $ . Then by Stokes’ theorem, $\Theta (n \otimes \partial \sigma ) = 0$ , so $\Theta $ descends to a map on homology $H_{1,2}(\operatorname {\mathrm {Jac}}(C)) \to N_{\mathbb {R}}^{\wedge 3}$ . Let

Fixing a basis $\lambda _1,\ldots ,\lambda _g$ for $\Lambda $ , we may apply (6) to the affine generating set of $H_{1,2}(\operatorname {\mathrm {Jac}}(C))$ given by (1) to find that

(7) $$ \begin{align} \mathscr{P} = \mathbb{Z}\langle{2\epsilon_i \wedge \lambda_j \wedge \lambda_k \; | \; i,j,k \in [g],\, j < k}\rangle. \end{align} $$

1.8 Ceresa period

We saw in §1.6 that $\operatorname {\mathrm {Cer}}_{v,G}(C)$ is trivial in $H_{1,1}(\operatorname {\mathrm {Jac}}(C))$ , so we may choose some $\Sigma \in S_{1,2}(\operatorname {\mathrm {Jac}}(C))$ for which $\partial \Sigma = \operatorname {\mathrm {Cer}}_{v,G}(C)$ . Define the Ceresa period $\alpha (C) \in N_{\mathbb {R}}^{\wedge 3}/\mathscr {P}$ via

As the notation suggests, $\alpha (C)$ is independent of the choice of v, $\Sigma $ , and G. We prove independence from the basepoint v in Corollary 3.4. Given $\Sigma ^{\prime } \in S_{1,2}(\operatorname {\mathrm {Jac}}(C))$ with the same boundary as $\Sigma $ , we have that $\Sigma ^{\prime } - \Sigma \in H_{1,2}(\operatorname {\mathrm {Jac}}(C))$ , and hence, $\Theta (\Sigma ^{\prime }) \equiv \Theta (\Sigma )\ \pmod {\mathscr {P}}$ . Finally, any model $(G^{\prime },\ell ^{\prime })$ of C is related to $(G,\ell )$ by a series of edge refinements, so it suffices to prove independence in the case that $G^{\prime }$ is obtained from G by subdividing a single edge e into $e_{1}$ and $e_{2}$ . Without loss of generality, we orient both edges in the direction of e, with $e^{-} = e_{1}^{-}$ . Then we have both that $\ell ^{\prime }(e_{1})^{-1}\pi (e_{1}) = \ell ^{\prime }(e_{2})^{-1}\pi (e_{2}) = \ell (e)^{-1}\pi (e)$ , and that $\delta _{e_{1}} = \delta _{e}$ and $\delta _{e_{2}} = \delta _{e} + e_{1}$ . Given $\Sigma $ as above, let

Comparing the e term in $\operatorname {\mathrm {Cer}}_{v,G}(C)$ to the $e_{1}$ and $e_{2}$ terms in $\operatorname {\mathrm {Cer}}_{v,G^{\prime }}(C)$ , one sees that $\partial \Sigma ^{\prime } = \operatorname {\mathrm {Cer}}_{v,G^{\prime }}(C)$ . Moreover, since $\pi (\delta _{e})$ , $\pi (\delta _{e}+e_{1})$ , and $\pi (\delta _{e}+e)$ are colinear in $N_{\mathbb {R}}$ , they vanish under the integration map $\Theta $ . This forces $\Theta (\Sigma ) = \Theta (\Sigma ^{\prime })$ , as desired.

Given the data of an algebraic equivalence between $Z_{1}$ and $Z_{2}$ in $\mathscr {Z}_{1}(\operatorname {\mathrm {Jac}}(C))$ , Zharkov constructs in [Reference Zharkov28, §3.1] an affine $(1,2)$ -chain that satisfies $\partial \Sigma = \operatorname {\mathrm {ch}}(Z_{1}) - \operatorname {\mathrm {ch}}(Z_{2})$ [Reference Zharkov28, Lemma 4]. As noted in [Reference Zharkov28, Lemma 5], by construction, each $n_{i}$ lies in the affine hull of the corresponding simplex $\sigma _{i}(\Delta ^{2})$ . The following result is a straightforward generalization of [Reference Zharkov28, Lemma 5].

Proof. Suppose that $\Phi _{v,*}C$ and $[-1]_{*}\Phi _{v,*}C$ are algebraically equivalent. Then we construct $\Sigma $ as above so that

$$ \begin{align*} \partial\Sigma = \operatorname{\mathrm{ch}}\left(\Phi_{v,*}C\right) - \operatorname{\mathrm{ch}}\left([-1]_{*}\Phi_{v,*}C\right) = \operatorname{\mathrm{Cer}}_{v,G}(C). \end{align*} $$

The fact that $n_i$ lies in the affine hull of $\sigma _i(\Delta ^2)$ forces $\Theta (n_i \otimes \sigma _i) = 0$ by (6). This implies that $\Theta (\Sigma ) = 0$ , as desired.

2.1 Universal Jacobian

Given a graph G of genus g with arbitrary edge orientations, we define a polynomial ring . Let , and consider the free R-module $N_R \cong H^1(G,R) \cong \operatorname {\mathrm {Hom}}(H_1(G,\mathbb {Z}),R)$ . We define a pairing via

(8) $$ \begin{align} [e, e^{\prime}] = \begin{cases} x_e &\text{ if } e = e^{\prime} \\ 0 &\text{ otherwise} \end{cases}, \end{align} $$

which induces a homomorphism by $e \mapsto [e,\cdot ]$ . Then is a free $\mathbb {Z}$ -submodule of $N_R$ of rank g. Formally, we define the universal Jacobian $\operatorname {\mathrm {Jac}}(G)$ of G to be the triple $(N_{ {R}}/\Lambda , N_{{R}}, N)$ , although we shall conflate $\operatorname {\mathrm {Jac}}(G)$ with the quotient group $N_{ {R}}/\Lambda $ whenever the other data are clear from context.

It should be noted that, although we use the symbol $\operatorname {\mathrm {Jac}}(G)$ , our universal Jacobian is distinct from the Jacobian of a finite graph, also known as the abelian sandpile group or critical group, which has been studied in various contexts in physics, arithmetic geometry, and graph theory. For more on this subject, see, for instance, [Reference Bak, Tang and Wiesenfeld7, Reference Lorenzini25, Reference Dhar18, Reference Gabrielov19, Reference Bacher, de La Harpe and Nagnibeda6, Reference Nagnibeda27, Reference Baker and Norine9].

2.2 Universal homology

We define homology theories on $\operatorname {\mathrm {Jac}}(G)$ as follows. Let $C_q(\operatorname {\mathrm {Jac}}(G))$ denote the free abelian group generated by $N_R^{q+1}/{\sim }$ , equivalence classes of ordered q-simplices, where we identify

$$ \begin{align*} (u_0+\lambda,\ldots,u_q+\lambda) \sim (u_0,\ldots,u_q) \end{align*} $$

for all $\lambda \in \Lambda $ . This becomes a chain complex $C_{\bullet }(\operatorname {\mathrm {Jac}}(G))$ via the usual boundary maps

$$ \begin{align*} \partial(u_0,\ldots,u_q) = \sum_{i=0}^q (-1)^i(u_0,\ldots,\hat u_i,\ldots,u_q), \end{align*} $$

where $\hat {\cdot }$ means that the corresponding entry is omitted. Let $H_q(\operatorname {\mathrm {Jac}}(G))$ denote the q-th homology of $C_{\bullet }(\operatorname {\mathrm {Jac}}(G))$ . We further define , with corresponding q-th homology $H_{p,q}(\operatorname {\mathrm {Jac}}(G))$ .We call a $(p,q)$ -chain degenerate if every q-simplex that it contains has repeated entries.

The universal coefficient theorem yields

(9) $$ \begin{align} H_{p,q}(\operatorname{\mathrm{Jac}}(G)) \cong N^{\wedge p} \otimes_{\mathbb{Z}} H_q(\operatorname{\mathrm{Jac}}(G)). \end{align} $$

Comparing the following technical result to (1), this says that $\operatorname {\mathrm {Jac}}(G)$ has the homology we expect of a g-dimensional real torus.

Proof. By restriction of scalars, the R-module $N_R$ inherits the structure of a free $\mathbb {Z}$ -module of infinite rank, which we denote by M. Then $M_{\mathbb {R}}$ is an infinite-dimensional $\mathbb {R}$ -vector space that naturally contains M as a $\mathbb {Z}$ -submodule. We endow the g-dimensional subspace $\Lambda _{\mathbb {R}}$ of $M_{\mathbb {R}}$ with the Euclidean topology. Choose a subspace W complementary to $\Lambda _{\mathbb {R}}$ and give it the Euclidean norm with respect to some basis $w_1,w_2,\ldots $ . Give $M_{\mathbb {R}} = \Lambda _{\mathbb {R}} \times W$ the product topology. The action of $\Lambda $ on $\Lambda _{\mathbb {R}}$ by translation extends to an action on $M_{\mathbb {R}}$ ; define with the quotient topology. Equivalently, we may write $J = \Lambda _{\mathbb {R}}/\Lambda \times W$ . It is a straightforward exercise to show that $W \cong \varinjlim _n \mathbb {R}^n$ is contractible via a straight-line homotopy to $0$ ; therefore, the singular homology $H_q(J)$ of J is isomorphic to that of the g-dimensional real torus $\Lambda _{\mathbb {R}}/\Lambda $ . In particular, just as in §1.3, one can show using the Eilenberg–Zilber map that $H_1(J)$ is freely generated by the affine simplices $(0,\lambda _i)$ in $M_{\mathbb {R}}$ for each i, while $H_2(J)$ is freely generated by $(0,\lambda _i,\lambda _i+\lambda _j) - (0,\lambda _j,\lambda _i+\lambda _j)$ for $i < j$ .

Let $S_{\bullet }$ denote the singular chain complex functor. There is a natural surjection $S_{\bullet }(M_{\mathbb {R}}) \to S_{\bullet }(J)$ that identifies simplices in $M_{\mathbb {R}}$ that are $\Lambda $ -translates of each other, so we may refer to simplices of J by their representatives in $M_{\mathbb {R}}$ . There is an injective chain map sending the ordered q-simplex $(u_0,\ldots ,u_q)$ to the affine singular simplex $(u_0,\ldots ,u_q)$ ; we naturally identify $C_{\bullet }(\operatorname {\mathrm {Jac}}(G)) \cong \operatorname {\mathrm {im}}\Phi $ . It is clear that the given generators for $H_q(J)$ for $q \in \{1,2\}$ are in $\operatorname {\mathrm {im}}\Phi $ , so the induced map $H_q(\operatorname {\mathrm {Jac}}(G)) \to H_q(J)$ is surjective. We claim that it is in fact an isomorphism.

It suffices to show that the injection $\Phi $ splits, since then it must induce an injection on homology. Indeed, let $A_{\bullet }(J) \subset S_{\bullet }(J)$ denote the subcomplex generated by affine simplices; by construction, $\operatorname {\mathrm {im}}\Phi \subset A_{\bullet }(J)$ . We define chain maps sending a singular simplex to the affine simplex with the same vertices and sending $(u_0,\ldots ,u_q) \mapsto ( \left \lfloor {u_0} \right \rfloor ,\ldots , \left \lfloor {u_q} \right \rfloor )$ , where $ \left \lfloor {\cdot } \right \rfloor $ applies the usual floor function to each coordinate of the argument with respect to the basis $\lambda _1,\ldots ,\lambda _g,w_1,w_2,\ldots $ of $M_{\mathbb {R}}$ .

The restriction of $GF$ to $\operatorname {\mathrm {im}}\Phi $ is the identity, as desired.

Since $H_q(\operatorname {\mathrm {Jac}}(G)) \to H_q(J)$ is an isomorphism for $q \in \{1,2\}$ , the chosen generators of $H_q(J)$ pull back to corresponding generators for $H_q(\operatorname {\mathrm {Jac}}(G))$ . The final result then follows from (9).

Suppose that G and $G^{\prime }$ are graphs with universal Jacobians $\operatorname {\mathrm {Jac}}(G) = N_R/\Lambda $ and $\operatorname {\mathrm {Jac}}(G^{\prime }) = N^{\prime }_{R^{\prime }}/\Lambda ^{\prime }$ , respectively. Then any $\mathbb {Z}$ -linear map for which $f(\Lambda ) \subset \Lambda ^{\prime }$ and $f(N) \subset N^{\prime }$ induces a chain map via

(10)

2.3 Universal Ceresa cycle

Fix a basepoint $v \in V(G)$ and define the Abel–Jacobi map by sending $w \mapsto \pi (\delta )$ , where $\delta $ is a path from v to w. Just as for the Abel–Jacobi map associated to a tropical curve, $\Phi _v$ is well defined modulo $\Lambda = \pi (H_1(G,\mathbb {Z}))$ . By an abuse of notation, we also define by

(11)

where $\delta _e$ is a path from v to $e^-$ . Notice that $\pi (\delta _e)$ and $\pi (\delta _e + e)$ are representatives in $N_R$ of $\Phi _v(e^-)$ and $\Phi _v(e^+)$ , respectively.

Taking (4) as inspiration, we define the (universal) Ceresa cycle of G based at v by

(12)

where the inversion map induces a pushforward as in (10).

Proof. Let $\bar e$ be the edge e with the opposite orientation. We observe that $\bar e = -e$ in $C_1(G,\mathbb {Z})$ and $\bar {e}^- = e^+$ , so we may take $\delta _{\bar {e}} = \delta _e + e$ . We also have $x_{\bar {e}} = x_e$ . Then

$$ \begin{align*} \Phi_v(\bar e) &= x_{\bar{e}}^{-1}\pi(\bar{e}) \otimes (\pi(\delta_{\bar{e}}),\pi(\delta_{\bar{e}}+\bar{e}))\\ &= -x_e^{-1}\pi(e) \otimes (\pi(\delta_e + e),\pi(\delta_e + e + \bar{e}))\\ &= -x_e^{-1}\pi(e) \otimes (\pi(\delta_e + e),\pi(\delta_e))\\ &= \Phi_v(e) - \partial\left(x_e^{-1}\pi(e) \otimes \big((\pi(\delta_e),\pi(\delta_e+e),\pi(\delta_e)) + (\pi(\delta_e),\pi(\delta_e),\pi(\delta_e))\big)\right).\\[-37pt]\end{align*} $$

Proof. We compute

$$ \begin{align*} \partial \Phi_v(e) = x_e^{-1}\pi(e) \otimes ((\pi(\delta_e+e)) - (\pi(\delta_e))). \end{align*} $$

Then $\partial \left (\sum _{e\in E(G)}\Phi _v(e)\right )$ is supported on the subset $\Phi _v(V(G)) \subset \operatorname {\mathrm {Jac}}(G)$ , so it suffices to fix $w \in V(G)$ and show that the part of the boundary supported on $\Phi _v(w)$ vanishes. The only edges that have a boundary component at $\Phi _v(w)$ are those that are incident to w. We further restrict our attention to the non-loop edges, since if e is a loop edge, $\pi (e) \in \Lambda $ , and hence, $\partial \Phi _v(e) = 0$ .

Let $\delta $ be a path from v to w, and label the non-loop edges adjacent to w by $e_1,\ldots ,e_n$ . By Lemma 2.2, up to adding a boundary, we may assume that each $e_i$ is oriented away from w. We write $\Phi _v(e_i) = x_{e_i}^{-1}\pi (e_i) \otimes (\pi (\delta ),\pi (\delta +e_i))$ . This contributes a boundary component of $-x_{e_i}^{-1}\pi (e_i) \otimes (\pi (\delta ))$ at $\Phi _v(w)$ , so we need only show that $\sum _{i=1}^n x_{e_i}^{-1}\pi (e_i) = 0$ or, equivalently, that $\sum _{i=1}^n x_{e_i}^{-1}[e_i,\gamma ] = 0$ for all $\gamma \in H_1(G,\mathbb {Z})$ . Indeed, we may write $\gamma = \sum _{i=1}^n c_i e_i + \gamma ^{\prime }$ , where $\gamma ^{\prime }$ is supported away from the edges $e_1,\ldots ,e_n$ . The fact that $\partial \gamma = 0$ forces $\sum _{i=1}^n c_i = 0$ , so

$$ \begin{align*} \sum_{i=1}^n x_{e_i}^{-1}[e_i,\gamma] = \sum_{i=1}^n c_i = 0.\\[-48pt] \end{align*} $$

Immediately, we get that $\operatorname {\mathrm {Cer}}_{v}(G)$ defines a $(1,1)$ -cycle, and so descends to a homology class in $H_{1,1}(\operatorname {\mathrm {Jac}}(G))$ . In fact, more is true:

Proof. By the construction of $\operatorname {\mathrm {Cer}}_v(G)$ and Lemma 2.3, it suffices to show that acts as the identity. We need only check this on the generators described in Lemma 2.1:

$$ \begin{align*} [-1]_{*}(\epsilon_i \otimes (0,\lambda_j)) &= -\epsilon_i \otimes (0,-\lambda_j)\\ &= -\epsilon_i \otimes (\lambda_j,0) \\ &= \epsilon_i \otimes (0,\lambda_j) - \partial \left( \epsilon_i \otimes (0,\lambda_j,0) + \epsilon_i \otimes (0,0,0) \right), \end{align*} $$

where the second equality follows by equivalence of simplices under translation by $\Lambda $ .

2.4 Universal Ceresa period

We take exterior powers of $N_R \cong R^g$ with respect to its R-module structure. We no longer have a topology, let alone a smooth structure for integration, but we can still mimic the effect of $\Theta $ on affine simplices given by (6). Therefore, we define by

A direct computation shows that $\Theta \circ \partial = 0$ , so is well defined. Let $\Sigma \in C_{1,2}(\operatorname {\mathrm {Jac}}(G))$ be such that $\partial \Sigma = \operatorname {\mathrm {Cer}}_v(G)$ . Then the (universal) Ceresa period of G is the image of $\Theta (\Sigma )$ in $N_R^{\wedge 3}/\mathscr {P}$ . Just as for $\alpha (C)$ in §1.8, $\alpha (G)$ depends neither on the choice of $\Sigma $ nor on the basepoint $v \in V(G)$ , although the latter fact will not be proved until Corollary 3.3. Although we will need the tools of §3 to compute $\alpha (G)$ efficiently, see §4.1 for the minimal examples of nontrivial Ceresa periods.

2.5 Evaluation

Fix a tropical curve C and a model $(G,\ell )$ . We let $[\cdot ,\cdot ]$ continue to denote the length pairing with values in R and adopt the notation $[\cdot ,\cdot ]_C$ for the pairing defined by (3). Likewise, we adopt the notations $\pi _C$ , $\Lambda _C$ , $\Theta _C$ , and $\mathscr {P}_C$ for the ‘real edge length’ analogues of the ‘universal’ $\pi $ , $\Lambda $ , $\Theta $ , and $\mathscr {P}$ , respectively. The length function $\ell $ induces a ring map via $x_e \mapsto \ell (e)$ . Then we may write $\operatorname {\mathrm {Jac}}(G) = N_R/\Lambda $ and $\operatorname {\mathrm {Jac}}(C) = N_{\mathbb {R}}/\Lambda _C$ , where $N = H^1(G,\mathbb {Z})$ . The evaluation map $\mathrm {ev}$ induces a map $N_R \to N_{\mathbb {R}}$ that we also denote by $\mathrm {ev}$ ; this in turn induces a map $N_R^{\wedge 3} \to N_{\mathbb {R}}^{\wedge 3}$ .

We naturally associate to a graph G a moduli space of isomorphism classes of tropical curves overlying G as follows (but for more general constructions, see, for example, [Reference Chan14, Section 4.1]). The group $\operatorname {\mathrm {Aut}}(G)$ acts on $E(G)$ ; this induces an action on $\mathbb {R}_{>0}^{E(G)}$ by permuting the coordinates. Then let

Each point of $\mathscr {M}_G^{\mathrm {tr}}$ determines a length function $\ell $ on G and hence the tropical curve with $(G,\ell )$ as a model. In analogy to the classical setting, we say that a property is true of a very general tropical curve overlying G if it is true of every tropical curve corresponding to a point on $\mathscr {M}_G^{\mathrm {tr}} \setminus H$ , where H is some countable union of hypersurfaces. Then Corollary 2.6 has the following partial converse.

Proof. Let C be a tropical curve with $(G,\ell )$ as a model, and adopt the notation used in the proof of Proposition 2.5. If we choose generators $p_1,\ldots ,p_r$ of $\mathscr {P}$ , then $\mathrm {ev}(p_1),\ldots ,\mathrm {ev}(p_r)$ generate $\mathscr {P}_C$ . Suppose that $\alpha (C) = 0$ . Then by definition, $\Theta _C(\Sigma _C) \in \mathscr {P}_C$ , so we may write

$$ \begin{align*} \mathrm{ev}(\Theta(\Sigma)) = \Theta_C(\Sigma_C) = \sum_{i=1}^r a_i \mathrm{ev}(p_i) = \mathrm{ev}\left(\sum_{i=1}^r a_ip_i\right) \end{align*} $$

for some $a_i \in \mathbb {Z}$ , and hence,

(13)

Observe that $S \in N_R^{\wedge 3}$ . Since $N_R^{\wedge 3} \cong R \otimes _{\mathbb {Z}} N^{\wedge 3}$ is a free R-module of finite rank, we may fix a basis and write S as an R-linear combination of the basis elements. Then $S \in \ker \mathrm {ev}$ precisely when all of its finitely many R-coefficients vanish under $\mathrm {ev}$ . The fact that $\alpha (G) \neq 0$ implies that $S \neq 0$ , so these coefficients cut out a subvariety of $\mathbb {R}_{> 0}^{E(G)}$ of positive codimension. The desired statement follows by considering all possible choices of the $a_i$ , of which there are countably many.

3.1 Explicit $(1,2)$ -chain $\Upsilon $ whose boundary is $\operatorname {\mathrm {Cer}}_v(G)$

Given a graph G with basepoint $v \in V(G)$ , we shall define a $(1,2)$ -chain $\Upsilon $ for which $\partial \Upsilon = \operatorname {\mathrm {Cer}}_v(G)$ . To that end, we fix orientations on the edges of G and a spanning tree T and label the edges of $G \setminus T$ by $e_1,\ldots ,e_n$ . Each $e_i$ determines a simple cycle $\gamma _i$ that uses only edges from $E(T) \cup \{e_i\}$ and that has positive $e_i$ -coefficient. Order the edges of $\gamma _i$ cyclically. Then $H_1(G,\mathbb {Z})$ has $\gamma _1,\ldots ,\gamma _g$ as a basis; let $\epsilon _1,\ldots ,\epsilon _g$ be the dual basis for N, that is,

$$ \begin{align*} \epsilon_i(\gamma_j) = \begin{cases} 1 & \text{ if } i = j \\ 0 & \text{ otherwise} \end{cases}. \end{align*} $$

For $e \in E(G)$ , the integer counts the multiplicity of e in $\gamma _i$ ; by our choice of homology basis, $f_i(e) \in \{0,\pm 1\}$ . Observe also that

$$ \begin{align*} \pi(e) = \sum_{i=1}^g [e,\gamma_i]\epsilon_i = x_e\sum_{i=1}^gf_i(e)\epsilon_i. \end{align*} $$

From (12), we write

(14) $$ \begin{align} \operatorname{\mathrm{Cer}}_v(G) &= \sum_{e\in E(G)} (\Phi_v(e) - [-1]_{*} \Phi_v(e))\nonumber\\ &= \sum_{e\in E(G)} \left( x_e^{-1}\pi(e) \otimes (\pi(\delta_e),\pi(\delta_e+e)) + x_e^{-1}\pi(e) \otimes (-\pi(\delta_e),-\pi(\delta_e+e)) \right)\nonumber\\ &= \sum_{e \in E(G)} \sum_{i=1}^gf_i(e)\epsilon_i \otimes \big( (\pi(\delta_e),\pi(\delta_e+e)) + (-\pi(\delta_e),-\pi(\delta_e+e)) \big)\nonumber\\ &= \sum_{i=1}^g \epsilon_i \otimes \sum_{e \in E(G)} f_i(e) \big((\pi(\delta_e),\pi(\delta_e+e)) + (-\pi(\delta_e),-\pi(\delta_e+e)) \big). \end{align} $$

Fix the unique path $\delta _i$ from v to a vertex $p_{i,0}$ in $\gamma _i$ so that $\delta _i$ is contained in T and does not share any edges with $\gamma _i$ . Following the cyclic orientation of $\gamma _i$ , label the subsequent vertices by $p_{i,1},p_{i,2},\ldots $ with $p_{i,n_i} = p_{i,0}$ . Write $e_{i,j}$ for the edge in $\gamma _i$ with vertices $p_{i,j}$ and $p_{i,j+1}$ (see Figure 1). For $0 \leq j \leq n_i$ , we define

Notice that $\delta _i + \sum _{k=0}^{j-1} f_i(e_{i,k})e_{i,k}$ is a path from v to $p_{i,j}$ that passes through $p_{i,0},p_{i,1},\ldots ,p_{i,j-1}$ ; in particular, $u_{i,j}$ is a lift of $\Phi _v(p_{i,j})$ to $N_R$ , and (14) reduces to

(15) $$ \begin{align} \operatorname{\mathrm{Cer}}_v(G) = \sum_{i=1}^g \epsilon_i \otimes \sum_{j=0}^{n_i-1} \begin{cases} (u_{i,j},u_{i,j+1}) + (-u_{i,j},-u_{i,j+1}) & \text{if } f_i(e_{i,j}) = 1 \\ -(u_{i,j+1},u_{i,j}) - (-u_{i,j+1},-u_{i,j}) & \text{if } f_i(e_{i,j}) = -1 \end{cases}, \end{align} $$

the $\epsilon _i$ -component of which is depicted in Figure 2(a). This cumbersome notation is necessary because, unlike for simplicial homology, we are not able to permute the vertices of a simplex at the cost of introducing a sign. The alternative that works in the current context is the following:

(16) $$ \begin{align} \begin{aligned} -(u_{i,j+1},u_{i,j}) - (-u_{i,j+1},-u_{i,j}) &= (u_{i,j},u_{i,j+1}) + (-u_{i,j},-u_{i,j+1})\\ & \quad- \partial(u_{i,j},u_{i,j+1},u_{i,j}) - \partial(u_{i,j},u_{i,j},u_{i,j})\\ & \quad- \partial(-u_{i,j},-u_{i,j+1},-u_{i,j}) - \partial(-u_{i,j},-u_{i,j},-u_{i,j}). \end{aligned} \end{align} $$

In other words, we must add the boundary of a degenerate $(1,2)$ -chain in order to permute the vertices of each $(1,1)$ -simplex.

Figure 1 Notation for the cycle $\gamma _i.$

Figure 2 Notation for and $\Upsilon$ .

Now that we have expressed $\operatorname {\mathrm {Cer}}_v(G)$ more explicitly, we define , where

(17)

Using the fact that $u_{i,n_i} = u_{i,0} + \lambda _i$ , a straightforward computation yields

which by (16) equals the $\epsilon _i$ -component of (15). This is depicted without orientations or degenerate simplices in Figure 2. We conclude that $\partial \Upsilon = \operatorname {\mathrm {Cer}}_v(G)$ .

3.2 Explicit representative $\Theta (\Upsilon )$ for $\alpha (G)$

Since $\Theta $ kills degenerate $(1,2)$ -simplices, we may ignore the second summation in (17) in the following computation:

(18) $$ \begin{align} \Theta(\Upsilon) &= \sum_{i=1}^g \Theta(\epsilon_i \otimes \Upsilon_i)\nonumber\\ &= \sum_{i=1}^g\epsilon_i \wedge \left( \sum_{j=1}^{n_i-1} \left( (u_{i,j}-u_{i,0}) \wedge (u_{i,j+1}-u_{i,0}) + (-u_{i,j}+u_{i,0}) \wedge (-u_{i,j+1}+u_{i,0})\right)\right. \nonumber\\ &\left.\hphantom{\sum_{i=1}^g\epsilon_i \wedge \Bigg(}\quad+(u_{i,n_i}-u_{i,0}) \wedge (-2u_{i,0}) + (-2u_{i,0}) \wedge (-u_{i,n_i}-u_{i,0}) \vphantom{\sum_{j=1}^{n_i-1} \left( (u_{i,j}-u_{i,0})\right)}\right) \nonumber\\ &= \sum_{i=1}^g\epsilon_i \wedge \left( 2\sum_{j=1}^{n_i-1} (u_{i,j}-u_{i,0}) \wedge (u_{i,j+1}-u_{i,j}) - 4 (u_{i,n_i}-u_{i,0}) \wedge u_{i,0} \right) \\ &= \sum_{i=1}^g\epsilon_i \wedge \left( 2\sum_{j=1}^{n_i-1} \left(\sum_{k=0}^{j-1}\pi(f_i(e_{i,k})e_{i,k})\right) \wedge \pi(f_i(e_{i,j})e_{i,j}) - 4 \pi(\gamma_i) \wedge \pi(\delta_i) \right)\nonumber\\ &= \sum_{i=1}^g\epsilon_i \wedge \left( 2\sum_{0 \leq k < j \leq n_i-1} \pi(f_i(e_{i,k})e_{i,k}) \wedge \pi(f_i(e_{i,j})e_{i,j}) - 4 \pi(\gamma_i) \wedge \pi(\delta_i) \right).\end{align} $$

Since we have bases $\epsilon _1,\ldots ,\epsilon _g$ for N and $\pi (\gamma _1),\ldots ,\pi (\gamma _g)$ for $\Lambda $ , Lemma 2.1 tells us that

$$ \begin{align*} H_{1,2}(\operatorname{\mathrm{Jac}}(G)) = \mathbb{Z}\langle{\epsilon_i \otimes ((0,\pi(\gamma_j),\pi(\gamma_j+\gamma_k)) - (0,\pi(\gamma_k),\pi(\gamma_j+\gamma_k))) \; | \; i,j,k \in [g],\, j < k}\rangle. \end{align*} $$

Applying $\Theta $ , we find that

(19) $$ \begin{align} \mathscr{P} = \mathbb{Z}\langle{2\epsilon_i \wedge \pi(\gamma_j) \wedge \pi(\gamma_k) \; | \; i,j,k \in [g],\, j < k}\rangle. \end{align} $$

Since $\pi (\gamma _j) = \sum _{m=1}^g [\gamma _j,\gamma _m] \epsilon _m$ , we may rewrite

(20) $$ \begin{align} 2\epsilon_i \wedge \pi(\gamma_j) \wedge \pi(\gamma_k) &= 2\sum_{m=1}^g \sum_{n=1}^g [\gamma_j,\gamma_m][\gamma_k,\gamma_n] \epsilon_i \wedge \epsilon_m \wedge \epsilon_n \nonumber\\ &= 2\sum_{m<n} \left([\gamma_j,\gamma_m][\gamma_k,\gamma_n]-[\gamma_j,\gamma_n][\gamma_k,\gamma_m]\right) \epsilon_i \wedge \epsilon_m \wedge \epsilon_n. \end{align} $$

In other words, the $\epsilon _i \wedge \epsilon _m \wedge \epsilon _n$ -coefficient of $2\epsilon _i \wedge \pi (\gamma _j) \wedge \pi (\gamma _k)$ is twice the $(j,k)\times (m,n)$ -minor of the Gram matrix of the pairing $[\cdot ,\cdot ]$ .

3.3 Basepoint independence of the Ceresa period

Proof. Since G is connected, it suffices to show that $\Theta (\Upsilon )$ remains unchanged whether computed using v as the basepoint or some vertex $v^{\prime }$ separated from v by an edge a. Without loss of generality, assume that a is oriented from $v^{\prime }$ to v.

We observe that, in the computations in §3.1, as well as in deriving (18), we do not use any properties of $\delta _i$ other than that it is a path from v to some vertex $p_{i,0}$ in the cycle $\gamma _i$ . In particular, if we construct $\Upsilon ^{\prime }$ as in §3.1 using the basepoint $v^{\prime }$ and the paths , then the same argument shows that $\partial \Upsilon ^{\prime } = \operatorname {\mathrm {Cer}}_{v^{\prime }}(G)$ . Moreover, the points $p_{i,0}$ have not changed, nor have the edge labels $e_{i,j}$ , so it follows from (18) that

$$ \begin{align*} \Theta(\Upsilon^{\prime}) &= \sum_{i=1}^g\epsilon_i \wedge \left( 2\sum_{0 \leq k < j \leq n_i-1} \pi(f_i(e_{i,k})e_{i,k}) \wedge \pi(f_i(e_{i,j})e_{i,j}) - 4 \pi(\gamma_i) \wedge \pi(\delta_i + a) \right)\\ &= \Theta(\Upsilon) - 4\sum_{i=1}^g \epsilon_i \wedge \pi(\gamma_i) \wedge \pi(a)\\ &= \Theta(\Upsilon) - 4\sum_{i=1}^g \epsilon_i \wedge \sum_{j=1}^g [\gamma_i,\gamma_j]\epsilon_j \wedge \pi(a)\\ &= \Theta(\Upsilon) - 4\sum_{i,j} [\gamma_i,\gamma_j] \epsilon_i \wedge \epsilon_j \wedge \pi(a). \end{align*} $$

Ordering so that $i < j$ and using the fact that $\epsilon _j \wedge \epsilon _i = -\epsilon _i \wedge \epsilon _j$ , we obtain

$$ \begin{align*} = \Theta(\Upsilon) - 4\sum_{i < j} \left([\gamma_i,\gamma_j] - [\gamma_j,\gamma_i]\right) \epsilon_i \wedge \epsilon_j \wedge \pi(a). \end{align*} $$

Finally, $[\cdot ,\cdot ]$ is symmetric, so the summation vanishes and we are left with $\Theta (\Upsilon ^{\prime }) = \Theta (\Upsilon )$ , as desired.

3.4 Rewriting $\Theta (\Upsilon )$ in terms of edge pairs

It is not hard to see from (18) that $\Theta (\Upsilon )$ is homogeneous of degree 2 in the variables $x_e$ . We now develop a characterization for when a pair of edges e and $e^{\prime }$ contributes a monomial $x_ex_{e^{\prime }}\, \epsilon _i \wedge \epsilon _j \wedge \epsilon _k$ to $\Theta (\Upsilon )$ . Recall from §3.1 that

Likewise, we define

Given distinct edges e and $e^{\prime }$ in $\gamma _i$ , say that $e <_i e^{\prime }$ if e occurs before $e^{\prime }$ in the ordering determined by the endpoint of $\delta _i$ and the cyclic orientation of $\gamma _i$ . Equivalently, in the notation of Figure 1, we declare $e_{i,j} <_i e_{i,j+1}$ for $0 \leq j \leq n_i - 2$ . Then let

Fix a subset $S \subset E(G) \times E(G)$ that contains exactly one element from $\{(e,e^{\prime }),(e^{\prime },e)\}$ for each pair of distinct edges e and $e^{\prime }$ . Likewise, in lieu of requiring that $i < j < k$ , we instead fix a subset $S^{\prime } \subset [g]^3$ that contains exactly one permutation of each tuple $(i,j,k) \in [g]^3$ of all distinct indices.

Proposition 3.6. Fix a graph G and let $\Upsilon $ be the $(1,2)$ -chain defined by (17). Then we may write

$$ \begin{align*} \Theta(\Upsilon) = \sum_{\substack{(e,e^{\prime}) \in S \\ (i,j,k) \in S^{\prime}}} a_{i,j,k}(e,e^{\prime}) x_ex_{e^{\prime}}\,\epsilon_i \wedge \epsilon_j \wedge \epsilon_k, \end{align*} $$

where $a_{i,j,k}(e,e^{\prime })$ takes values in $\{0,\pm 2\}$ and is nonzero precisely when, up to relabeling $(i,j,k)$ , $\gamma _i$ contains both e and $e^{\prime }$ , $\gamma _j$ contains e but not $e^{\prime }$ , and $\gamma _k$ contains $e^{\prime }$ but not e. In particular,

(21) $$ \begin{align} a_{i,j,k}(e,e^{\prime}) = 2f_j(e)f_k(e^{\prime}) \left( f_i(e)f_i(e^{\prime})h_i(e,e^{\prime}) + 2f_i(e)g_j(e^{\prime}) - 2f_i(e^{\prime})g_k(e) \right), \end{align} $$

and reversing the cyclic orientation of any one of the cycles $\gamma _i$ , $\gamma _j$ , or $\gamma _k$ flips the sign.

Proof. From (18), we may write

$$ \begin{align*} \Theta(\Upsilon) &= \sum_{i=1}^g\epsilon_i \wedge \left( 2\sum_{0 \leq k < j \leq n_i-1} \pi(f_i(e_{i,k})e_{i,k}) \wedge \pi(f_i(e_{i,j})e_{i,j}) - 4 \pi(\gamma_i) \wedge \pi(\delta_i) \right)\\ &= 2\sum_{i=1}^g \epsilon_i \wedge \left( \sum_{\substack{e,e^{\prime} \in \operatorname{\mathrm{supp}}\gamma_i \\ e <_i e^{\prime}}} f_i(e)\pi(e) \wedge f_i(e^{\prime})\pi(e^{\prime}) - 2\pi(\gamma_i) \wedge \pi(\delta_i) \right)\\ &= 2\sum_{i=1}^g \epsilon_i \wedge \left( \sum_{(e,e^{\prime}) \in S} f_i(e)f_i(e^{\prime})h_i(e,e^{\prime}) \pi(e) \wedge \pi(e^{\prime}) - 2\pi(\gamma_i) \wedge \pi(\delta_i) \right) \end{align*} $$

We may write $\gamma _i = \sum _{e \in E(G)} f_i(e)e$ and $\delta _i = \sum _{e \in E(G)} g_i(e)e$ , so that

$$ \begin{align*} &= 2\sum_{i=1}^g \epsilon_i \wedge \left( \sum_{(e,e^{\prime}) \in S} f_i(e)f_i(e^{\prime})h_i(e,e^{\prime}) \pi(e) \wedge \pi(e^{\prime}) - 2\sum_{e,e^{\prime} \in E(G)} f_i(e)g_i(e^{\prime}) \pi(e) \wedge \pi(e^{\prime}) \right)\\ &= \sum_{(e,e^{\prime}) \in S} \sum_{i=1}^g 2\left( f_i(e)f_i(e^{\prime})h_i(e,e^{\prime}) - 2f_i(e)g_i(e^{\prime}) + 2f_i(e^{\prime})g_i(e) \right) \epsilon_i \wedge \pi(e) \wedge \pi(e^{\prime}). \end{align*} $$

Expanding $\pi (e) = x_e\sum _{j=1}^gf_j(e)\epsilon _j$ and $\pi (e^{\prime }) = x_{e^{\prime }}\sum _{k=1}^gf_k(e^{\prime })\epsilon _k$ , we obtain

$$ \begin{align*} &= \sum_{(e,e^{\prime}) \in S} \sum_{i,j,k} 2\left( f_i(e)f_i(e^{\prime})h_i(e,e^{\prime}) - 2f_i(e)g_i(e^{\prime}) + 2f_i(e^{\prime})g_i(e) \right) f_j(e) f_k(e^{\prime}) x_ex_{e^{\prime}}\, \epsilon_i \wedge \epsilon_j \wedge \epsilon_k. \end{align*} $$

We reindex by $S^{\prime }$ to get

(22) $$ \begin{align} = \sum_{(e,e^{\prime}) \in S} \sum_{(i,j,k) \in S^{\prime}} a_{i,j,k}(e,e^{\prime}) x_ex_{e^{\prime}}\, \epsilon_i \wedge \epsilon_j \wedge \epsilon_k, \end{align} $$

where

As one might expect, reordering the tuple $(e,e^{\prime })$ does not affect $a_{i,j,k}(e,e^{\prime })$ , while permuting $(i,j,k)$ changes it by the sign of the permutation. We record here the possible combinations of values of ${{\lvert {f_{\cdot }(\cdot )} \rvert }}$ up to such reordering:

$$ \begin{gather*} \begin{array}{ c | c c c } \text{(a)} & i & j & k \\ \hline e & 1 & 1 & * \\ e^{\prime} & 1 & 1 & * \end{array} \qquad \begin{array}{ c | c c c } \text{(b)} & i & j & k \\ \hline e & 1 & 1 & 0 \\ e' & 1 & 0 & 1 \end{array} \qquad \begin{array}{ c | c c c } \text{(c)} & i & j & k \\ \hline e & * & 1 & 1 \\ e' & * & 0 & 0 \end{array} \qquad \begin{array}{ c | c c c } \text{(d)} & i & j & k \\ \hline e & * & * & 0 \\ e' & * & * & 0 \end{array}. \end{gather*} $$

A $*$ indicates that the corresponding entry can have value either $0$ or $1$ . While these cases are not all pairwise disjoint, they do cover all the possibilities. Indeed, up to permuting the rows and columns, if zero or one of the six values is $0$ , then we are in case (a). If exactly two values are $0$ , then we fall into one of (b), (c) or (d). If three are $0$ , then we are in cases (c) or (d). Finally, if at least four values are $0$ , then case (d) applies.

We claim that the $a_{i,j,k}(e,e^{\prime }) \neq 0$ only in case (b). As a shortcut, we may consider only edges e and $e^{\prime }$ in T; by Lemma 3.5, the remaining edges do not contribute terms. Then $T \setminus \{e,e^{\prime }\}$ has three connected components; let $G^{\prime }$ be the graph obtained from G by contracting each of these components to a vertex and removing the additional edges $e_t$ for $t \not \in \{i,j,k\}$ . We depict in Figure 3 the resulting graph for each of the cases (a), (b) and (c) with the edges $e_i,e_j,e_k$ drawn as necessary. The basepoint v descends to one of the three vertices of $G^{\prime }$ , but by Proposition 3.2, we may fix it arbitrarily. We remark that the values of $f_t$ , $g_t$ and $h_t$ on e and $e^{\prime }$ remain unchanged for $t \in \{i,j,k\}$ when passing from G to $G^{\prime }$ .

The following observations will be helpful in further narrowing down the cases. By Remark 3.1, $\Theta (\Upsilon )$ is independent of the orientations on the edges of G. Then without loss of generality, we may orient the edges of $G^{\prime }$ as shown in Figure 3. Meanwhile, replacing $\gamma _i$ with the cycle having the opposite cyclic orientation of edges (i.e., the reverse of the partial order $<_i$ ) changes the sign of $f_i$ and $h_i$ . Hence, only $A_i$ , $B_{i,k}$ , and $B_{i,j}$ change sign, thereby causing $a_{i,j,k}(e,e^{\prime })$ also to change sign. The same applies to the indices j and k. Notably, this means that the choice for each t of $\gamma _t$ versus $\bar \gamma _t$ does not affect whether or not $a_{i,j,k}(e,e^{\prime }) = 0$ , so in Figure 3, we orient $\gamma _t$ (not depicted) in the same direction as $e_t$ . Finally, recall from §3.1 that the path $\delta _t$ is defined uniquely by the fact that it lies in T and has disjoint support from the cycle $\gamma _t$ .

Figure 3 Cases for the contracted graph $G'$ .

To reiterate, all of the choices that went into drawing Figure 3 were made without loss of generality up to a sign. Therefore, one may read off the values of $f_t$ , $g_t$ , and $h_t$ on e and $e^{\prime }$ for each of the relevant indices of t from $\{i,j,k\}$ ; one finds that $a_{i,j,k}(e,e^{\prime })$ vanishes in cases (a) and (c) and equals $-2$ in case (b). That $a_{i,j,k}(e,e^{\prime })$ vanishes in case (d) is not hard to see, since $f_k(e) = f_k(e^{\prime }) = 0$ forces $B_{j,k} = B_{i,k} = A_k = 0$ . The particular expression for the coefficient given by (21) in the statement of the result follows from case (b) by plugging into $a_{i,j,k}(e,e^{\prime })$ only the values of the indicator functions that vanish.

4.1 Base cases

Example 4.1. Let be the graph with basepoint v and oriented edges $e_i$ as depicted in Figure 4(a), with $T = \{e_4,e_5,e_6\}$ , paths $\delta _i = 0$ for all i, and the cyclic orientation of $\gamma _i$ chosen to match the orientation of $e_i$ . Let . Then $[\cdot ,\cdot ]$ has Gram matrix

$$ \begin{align*} \begin{pmatrix} x_1+x_5+x_6 & -x_6 & -x_5 \\ -x_6 & x_2+x_4+x_6 & -x_4 \\ -x_5 & -x_4 & x_3+x_4+x_5. \end{pmatrix}. \end{align*} $$

One can show using either (18) or Proposition 3.6 that

$$ \begin{align*} \Theta(\Upsilon) = 2(x_4x_5+x_4x_6+x_5x_6)\,\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3. \end{align*} $$

Meanwhile, (19) and (20) imply that $\mathscr {P}$ is generated by the elements

$$ \begin{align*} 2(x_{2} x_{5} + x_{4} x_{5} + x_{4} x_{6} + x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3, \\ 2(-x_{4} x_{5} - x_{3} x_{6} - x_{4} x_{6} - x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3, \\ 2(x_{2} x_{3} + x_{2} x_{4} + x_{3} x_{4} + x_{2} x_{5} + x_{4} x_{5} + x_{3} x_{6} + x_{4} x_{6} + x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 \\ 2(x_{1} x_{4} + x_{4} x_{5} + x_{4} x_{6} + x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3, \\ 2(-x_{1} x_{3} - x_{1} x_{4} - x_{1} x_{5} - x_{3} x_{5} - x_{4} x_{5} - x_{3} x_{6} - x_{4} x_{6} - x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3, \\ 2(x_{4} x_{5} + x_{3} x_{6} + x_{4} x_{6} + x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3, \\ 2(x_{1} x_{2} + x_{1} x_{4} + x_{2} x_{5} + x_{4} x_{5} + x_{1} x_{6} + x_{2} x_{6} + x_{4} x_{6} + x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3, \\ 2(-x_{1} x_{4} - x_{4} x_{5} - x_{4} x_{6} - x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3, \\ 2(x_{2} x_{5} + x_{4} x_{5} + x_{4} x_{6} + x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3. \end{align*} $$

Simplifying, we are left with six generators:

$$ \begin{align*} 2(x_{1} x_{4} - x_{3} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 \\ 2(x_{2} x_{5} - x_{3} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 \\ 2(x_{1} x_{2} + x_{2} x_{5} + x_{1} x_{6} + x_{2} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 \\ 2(x_{1} x_{3} + x_{1} x_{5} + x_{3} x_{5} + x_{3} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 \\ 2(x_{2} x_{3} + x_{2} x_{4} + x_{3} x_{4} + x_{3} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 \\ 2(x_{4} x_{5} + x_{3} x_{6} + x_{4} x_{6} + x_{5} x_{6})\,&\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3. \end{align*} $$

Suppose that we could obtain $\Theta (\Upsilon )$ as some $\mathbb {Z}$ -linear combination of these generators. Since $x_1x_4$ , $x_1x_2$ , $x_1x_3$ , and $x_2x_3$ each appear in only one generator and not in $\Theta (\Upsilon )$ , those generators cannot contribute. This leaves the second and the sixth. Continuing along the same lines, the second generator uniquely contains the term $x_2x_5$ , so it must be trivial. Now the sixth generator uniquely contains $x_3x_6$ , so it also vanishes, a contradiction. In other words, $\alpha (K_4) \neq 0$ , with

$$ \begin{align*} \alpha(K_4) = 2(x_4x_5+x_4x_6+x_5x_6)\,\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3\quad \pmod{\mathscr{P}}. \end{align*} $$

Example 4.2. Consider the graph with basepoint v and oriented edges $e_i$ as shown in Figure 4(b), with $T = \{e_5,e_6\}$ , paths $\delta _i = 0$ for all i, and the cyclic orientation of $\gamma _i$ matching $e_i$ . We again abbreviate . Then $[\cdot ,\cdot ]$ has Gram matrix

$$ \begin{align*} \begin{pmatrix} x_1+x_6 & 0 & x_6 & x_6 \\ 0 & x_2+x_5 & x_5 & x_5 \\ x_6 & x_5 & x_3+x_5+x_6 & x_5+x_6 \\ x_6 & x_5 & x_5+x_6 & x_4+x_5+x_6 \end{pmatrix}\, . \end{align*} $$

One can show that

$$ \begin{align*} \Theta(\Upsilon) = 2x_5x_6(\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_4 + \epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3). \end{align*} $$

By Corollary 2.6, to prove that $\alpha (L_3) \neq 0$ , it suffices to show that some tropical curve overlying $L_3$ has nontrivial Ceresa period. Indeed, let C be the tropical curve obtained by evaluating every edge length $x_i$ to $1$ , with the corresponding map . Then $\mathscr {P}_C$ is generated by the four elements

$$\begin{align*}\begin{aligned}2\epsilon_2 \wedge \epsilon_3 \wedge \epsilon_4 && &+ 2\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_4 &\!\!\!\! +\ 2\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 &, \\&& \phantom{{}+{}} 2\epsilon_1 \wedge \epsilon_3 \wedge \epsilon_4 &+ 2\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_4 &\!\!\!\! +\ 2\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 &, \\&&& \quad 4\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_4 & &,\\& && & 4\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3 &.\end{aligned} \end{align*}$$

Let . Then $\Theta (\Upsilon _C) = 2\epsilon _1 \wedge \epsilon _2 \wedge \epsilon _4 + 2\epsilon _1 \wedge \epsilon _2 \wedge \epsilon _3$ , which is clearly not in $\mathscr {P}_C$ , as desired. Therefore, $\alpha (L_3) \neq 0$ , with

$$ \begin{align*} \alpha(L_3) = 2x_5x_6(\epsilon_1 \wedge \epsilon_2 \wedge \epsilon_4 + \epsilon_1 \wedge \epsilon_2 \wedge \epsilon_3)\quad \pmod {\mathscr{P}}. \end{align*} $$

4.2 Contraction

Given a graph G with a non-loop edge a, we let $G / a$ denote the graph obtained from G by contracting a. We identify $E(G) = E(G / a) \sqcup \{a\}$ in the usual way.

Although the explicit $(1,2)$ -chain $\Upsilon $ is not strictly necessary for proving the first part of this result, it is nonetheless helpful in simplifying the argument.

Proof. Let . We shall declare to be the basepoint of both G and $G^{\prime }$ . Fix a spanning tree T of G containing a and let be the corresponding spanning tree in $G^{\prime }$ . Let $\gamma _1,\ldots ,\gamma _g$ be the basis of $H_1(G,\mathbb {Z}) \cong H_1(G^{\prime },\mathbb {Z})$ determined by T in G and $T^{\prime }$ in $G^{\prime }$ . Finally, choose orientations of the edges and cycles in G; these descend to $G^{\prime }$ , allowing us to define both $\Upsilon $ and $\Upsilon ^{\prime }$ as in §3.1.

Let be the natural inclusion, which misses $x_a$ , and define a projection map via $x_a \mapsto 0$ and $x_e \mapsto x_e$ for $e \neq a$ . Notice that $\rho \circ \iota = \mathrm {id}$ . Then the induced maps and making the natural identification $N \cong N^{\prime }$ also satisfy $\rho \circ \iota = \mathrm {id}$ . We write for the homomorphism sending $a \mapsto 0$ and $e \mapsto e$ for $e \neq a$ .

It is straightforward to check that $\rho ([e,e^{\prime }]) = [c(e),c(e^{\prime })]^{\prime }$ for all $e,e^{\prime } \in E(G)$ , and hence, $\rho \circ \pi = \pi ^{\prime } \circ c$ . In particular,

$$ \begin{align*} \rho(\epsilon_i \wedge \pi(\gamma_j) \wedge \pi(\gamma_k)) = \epsilon_i \wedge \pi^{\prime}(\gamma_j) \wedge \pi^{\prime}(\gamma_k); \end{align*} $$

by (19), $\rho $ induces a group isomorphism $\mathscr {P} \cong \mathscr {P}^{\prime }$ . Moreover, it is clear from the edge pair characterization in Proposition 3.6 that, for edges $e,e^{\prime } \in E(G^{\prime })$ , the indicator functions $f_t$ , $g_t$ and $h_t$ are the same in $G^{\prime }$ as they are in G. In other words,

$$ \begin{align*} \Theta(\Upsilon) = \iota(\Theta^{\prime}(\Upsilon^{\prime})) + \sum_{\substack{e \in E(G) \\ (i,j,k) \in S^{\prime}}} a_{i,j,k}(e,a) x_ex_a \epsilon_i \wedge \epsilon_j \wedge \epsilon_k. \end{align*} $$

Then $\rho (\Theta (\Upsilon )) = \Theta ^{\prime }(\Upsilon ^{\prime })$ , proving the first part of the statement. If a is a separating edge, then a is not part of any cycle. This implies that $x_a$ does not appear in $\pi (\gamma _t)$ for any t, so $\iota $ induces $\mathscr {P}^{\prime } \cong \mathscr {P}$ . Likewise, $a_{i,j,k}(e,a) = 0$ for all $e \in E(G)$ , so $\iota (\Theta ^{\prime }(\Upsilon ^{\prime })) = \Theta (\Upsilon )$ . This proves the remaining part of the statement.

4.3 Deletion

Fix a graph G and let $\Upsilon $ be the explicit $(1,2)$ -chain defined in §3.1. If $\alpha (G) = 0$ , then by definition, we can write $\Theta (\Upsilon )$ as some $\mathbb {Z}$ -linear combination of the generators in (19). In fact, not all of the generators are necessary.

Lemma 4.4. If $\alpha (G) = 0$ , then $\Theta (\Upsilon )$ is a $\mathbb {Z}$ -linear combination of the generators

$$ \begin{align*} \{2\epsilon_i \wedge \pi(\gamma_i) \wedge \pi(\gamma_j) \; |\; i, j \in [g],\, i\neq j\}. \end{align*} $$

Proof. Fix indices i, j, and k all distinct with $j < k$ . By (20), the generator $2\epsilon _i \wedge \pi (\gamma _j) \wedge \pi (\gamma _k)$ contributes a term

$$ \begin{align*} 2 \left( [\gamma_j,\gamma_j][\gamma_k,\gamma_k] - [\gamma_j,\gamma_k]^2 \right)\epsilon_i \wedge \epsilon_j \wedge \epsilon_k, \end{align*} $$

which contains $2x_{e_j}x_{e_k}\, \epsilon _i \wedge \epsilon _j \wedge \epsilon _k$ . Since $e_t$ appears only in $\gamma _t$ , it is straightforward to show that this is in fact the only generator that contains a nonzero multiple of $2x_{e_j}x_{e_k} \epsilon _i \wedge \epsilon _j \wedge \epsilon _k$ . Moreover, by Lemma 3.5, such a term also does not appear in $\Theta (\Upsilon )$ ; as there is no way to cancel this term out with a different generator, we conclude that $\epsilon _i \wedge \pi (\gamma _j) \wedge \pi (\gamma _k)$ cannot contribute to a combination equaling $\Theta (\Upsilon )$ . In other words, we must have either $i = j$ or $i = k$ .

Proof. Let . In either case (a) or (b), a is part of some cycle. Without loss of generality, we may choose the spanning tree T and the labeling on the edges of $G \setminus T$ so that $e_g = a$ . We also fix a basepoint v and orientations on the edges and cycles. These choices descend to $G^{\prime }$ . Identifying edges of $G^{\prime }$ with the corresponding edges of G, there is a natural inclusion that misses $x_{e_g}$ . Likewise, identifying the cycles of $G^{\prime }$ with the corresponding cycles in G, we let denote the induced inclusion on homology and the projection that kills $\gamma _g$ . Define

In coordinates, we may identify $N^{\prime }_{R^{\prime }} \cong R^{\prime }\langle {\epsilon _1,\ldots ,\epsilon _{g-1}}\rangle $ and $N_R \cong R\langle {\epsilon _1,\ldots ,\epsilon _g}\rangle $ ; then $\nu $ is the inclusion of $R^{\prime }$ -modules sending $\epsilon _i \mapsto \epsilon _i$ , while $\xi $ is the projection of R-modules sending $\epsilon _i \mapsto \epsilon _i$ for $i < g$ and $\epsilon _g \mapsto 0$ .

Because $x_{e_g}$ appears in $[e,\gamma _i]$ only for $e = e_g$ and $i = g$ , it is straightforward to check that

$$ \begin{align*} \nu(\pi^{\prime}(e)) = \xi(\pi(e)) \end{align*} $$

for all edges $e \in E(G^{\prime }) \subset E(G)$ . We claim that $\nu _{*}\Upsilon ^{\prime } = \xi _{*}\Upsilon $ . Indeed, observe first that $\Upsilon ^{\prime }$ does not have an $\epsilon _g$ -component, and that the $\epsilon _g$ -component of $\Upsilon $ is killed by $\xi _{*}$ . Fixing $i \neq g$ , we recall that the paths $\delta _i$ defined in §3.1 remain in $T = T^{\prime }$ , so $\delta _i^{\prime } = \delta _i$ . The definition of $\Upsilon ^{\prime }$ given by (17) depends only on the points $u_{i,j}^{\prime } = \pi ^{\prime }\left (\delta _i + \sum _{k=0}^{j-1} f_i(e_{i,k})e_{i,k} \right ) \in N^{\prime }_{R^{\prime }}$ . Then $\nu (u_{i,j}^{\prime }) = \xi (u_{i,j})$ for all j, so the definition of the pushforward in (10) implies that $\nu _{*}(\epsilon _i \otimes \Upsilon _i^{\prime }) = \xi _{*}(\epsilon _i \otimes \Upsilon _i)$ . The claim follows. Consequently,

$$ \begin{align*} \nu(\Theta^{\prime}(\Upsilon^{\prime})) = \Theta(\nu_{*}\Upsilon^{\prime}) = \Theta(\xi_{*}\Upsilon) = \xi(\Theta(\Upsilon)). \end{align*} $$

A similar computation on generators of $H_{1,2}(\operatorname {\mathrm {Jac}}(G))$ shows that $\nu (\mathscr {P}^{\prime }) \subset \xi (\mathscr {P})$ , but we may not have equality in general. Explicitly,

$$ \begin{align*} \nu(\mathscr{P}^{\prime}) = \mathbb{Z}\langle{2\epsilon_i \wedge \xi(\pi(\gamma_j)) \wedge \xi(\pi(\gamma_k)) \mid i,j,k \in [g-1],\, j < k}\rangle, \end{align*} $$

leaving open the possibility that $\xi $ fails to map generators of the form $2\epsilon _i \wedge \pi (\gamma _j) \wedge \pi (\gamma _g)$ into $\nu (\mathscr {P}^{\prime })$ . Here, we have restricted our attention to the case where $i < g$ , since $\xi (\epsilon _g) = 0$ . Our assumption that $\alpha (G) = 0$ means that we may write $\Theta (\Upsilon )$ explicitly as a $\mathbb {Z}$ -linear combination of the generators in Lemma 4.4. Therefore, if we can show that $2\epsilon _i \wedge \xi (\pi (\gamma _i)) \wedge \xi (\pi (\gamma _g))$ lies in $\nu (\mathscr {P}^{\prime })$ for any such generator appearing in the linear combination, then we will have shown that $\nu (\Theta ^{\prime }(\Upsilon ^{\prime })) \in \nu (\mathscr {P}^{\prime })$ ; by injectivity of $\nu $ , this would imply in turn that $\Theta ^{\prime }(\Upsilon ^{\prime }) \in \mathscr {P}^{\prime }$ , as desired.

For case (a), this is straightforward because $\pi (\gamma _g) = \pi (e_g) = x_{e_g}\epsilon _g$ , so $\xi (\pi (\gamma _g)) = 0$ . In case (b), recall that $e_g$ is parallel to an edge $a^{\prime }$ . Without loss of generality, assume that $e_g$ and $a^{\prime }$ have the same orientation. If $a^{\prime }$ is a separating edge in $G^{\prime }$ , then $\gamma _g = -a^{\prime } + e_g$ and $a^{\prime }$ is not part of any other cycle $\gamma _i$ . In particular, $\pi (\gamma _g) = (x_{a^{\prime }} + x_{e_g})\epsilon _g$ , so we again have that $\xi (\pi (\gamma _g)) = 0$ . Otherwise, $a^{\prime }$ is not separating in $G^{\prime }$ , so we may choose T so that $e_{g-1} = a^{\prime }$ . We write explicitly

$$ \begin{align*} \Theta(\Upsilon) \equiv 2 \sum_{i=1}^{g-1} a_i \epsilon_i \wedge \pi(\gamma_i) \wedge \pi(\gamma_g)\quad \pmod{\xi^{-1}(\nu(\mathscr{P}^{\prime}))} \end{align*} $$

for $a_i \in \mathbb {Z}$ , where we have omitted from the linear combination the generators of $H_{1,2}(\operatorname {\mathrm {Jac}}(G))$ that we already know map to $\nu (\mathscr {P}^{\prime })$ under $\xi $ – that is, those that do not contain $\pi (\gamma _g)$ as a factor or that have $\epsilon _g$ as the first factor.

The fact that $\gamma _g = \gamma _{g-1} - e_{g-1} + e_g$ allows us to rewrite

(23) $$ \begin{align} \Theta(\Upsilon) \equiv - 2 x_{e_{g-1}} \sum_{i=1}^{g-1} a_i \epsilon_i \wedge \pi(\gamma_i) \wedge \epsilon_{g-1} + 2 x_{e_g} \sum_{i=1}^{g-1} a_i \epsilon_i \wedge \pi(\gamma_i) \wedge \epsilon_g\quad \pmod{\xi^{-1}(\nu(\mathscr{P}^{\prime}))}. \end{align} $$

Since $x_{e_g}$ appears as a coefficient in $\pi (\gamma _j)$ only for $j = g$ , the only place where it appears in (23) is where we have explicitly written it before the second summation. In particular, $x_{e_g}$ does not appear in any of the omitted generators of the form $2\epsilon _g \wedge \pi (\gamma _g) \wedge \pi (\gamma _j)$ because the leading factor of $\epsilon _g$ kills the $\epsilon _g$ -term of $\pi (\gamma _g)$ , nor does it appear in $\Theta (\Upsilon )$ by Lemma 3.5. We conclude that

$$ \begin{align*} 0 &= \sum_{i=1}^{g-1} a_i \epsilon_i \wedge \pi(\gamma_i) \wedge \epsilon_g\\ &= \sum_{i=1}^{g-1} a_i \epsilon_i \wedge \left(\sum_{j=1}^g[\gamma_i,\gamma_j]\epsilon_j\right) \wedge \epsilon_g\\ &= \sum_{i=1}^{g-1}\sum_{j=1}^{g-1} [\gamma_i,\gamma_j] a_i \epsilon_i \wedge \epsilon_j \wedge \epsilon_g,\\ &= \sum_{1 \leq i < j \leq g-1} [\gamma_i,\gamma_j] (a_i-a_j) \epsilon_i \wedge \epsilon_j \wedge \epsilon_g, \end{align*} $$

and hence, $[\gamma _i,\gamma _j](a_i - a_j) = 0$ for all $i,j \in [g-1]$ . Applying $\xi $ to (23) kills terms that contain $\epsilon _g$ as a factor, so we obtain

$$ \begin{align*} \xi(\Theta(\Upsilon)) &\equiv \xi\left(-2 x_{e_{g-1}} \sum_{i=1}^{g-1} a_i \epsilon_i \wedge \pi(\gamma_i) \wedge \epsilon_{g-1}\right)\quad \pmod{\nu(\mathscr{P}^{\prime})}\\ &= \xi\left(-2 x_{e_{g-1}} \sum_{i=1}^{g-1}\sum_{j=1}^g [\gamma_i,\gamma_j] a_i \epsilon_i \wedge \epsilon_j \wedge \epsilon_{g-1}\right)\\ &= \xi\left(-2 x_{e_{g-1}} \sum_{i=1}^{g-1}\sum_{j=i+1}^g [\gamma_i,\gamma_j] (a_i-a_j) \epsilon_i \wedge \epsilon_j \wedge \epsilon_{g-1}\right)\\ &= \xi\left(-2 x_{e_{g-1}} \sum_{i=1}^{g-1} [\gamma_i,\gamma_g] (a_i-a_g) \epsilon_i \wedge \epsilon_g \wedge \epsilon_{g-1}\right)\\ &= 0, \end{align*} $$

completing the proof.

4.4 Graphs of hyperelliptic type

Given a graph G, we write $G^2$ to mean the 2-edge-connectivization of G, obtained by contracting each of the separating edges of G.

Let G and $G^{\prime }$ be graphs. We say that $G^{\prime }$ is a permissible minor of G if we may obtain $G^{\prime }$ from G by deleting only loops or parallel edges and contracting only non-loop edges. Then Propositions 4.3 and 4.5 immediately imply that $\alpha (G) = 0$ only if $\alpha (G^{\prime }) = 0$ .

A tropical curve C is hyperelliptic if it admits an involution $\iota $ for which the quotient $C/\iota $ is a tree. More generally, it is of hyperelliptic type if its Jacobian is isomorphic to that of a hyperelliptic tropical curve. We say that a graph G is of hyperelliptic type if some choice of edge lengths makes it into a hyperelliptic-type tropical curve. For more details on these notions, we refer to [Reference Baker and Norine10, Reference Amini, Baker, Brugallé and Rabinoff3, Reference Amini, Baker, Brugallé and Rabinoff4, Reference Chan14]. By [Reference Corey15, Proposition 3.3], the property of being of hyperelliptic type does not in fact depend on the choice of edge lengths. Following [Reference Corey15], we say that G is strongly of hyperelliptic type if some choice of edge lengths yields a hyperelliptic tropical curve.

Figure 5 The graph $L(T)$ for the tree T marked with bold edges.

Let T be a tree of maximal valence 3 and fix a disjoint copy $T^{\prime }$ of T. Let be the involution that identifies each vertex of T with the corresponding vertex of $T^{\prime }$ . Following [Reference Chan14, Definition 4.7], we define the ladder over T to be the graph $L(T)$ obtained by adding $3 - \operatorname {\mathrm {val}}(v)$ parallel edges between v and $\iota (v)$ for each $v \in V(T)$ . We call the edges added in this way vertical edges.