2.1 Admissible G-covers

Let G be a finite group, and let $g,n\ge 0$ be integers such that ${2g-2+n>0}$ . We recall the notion of an admissible G-cover of nodal curves of type $(g,n)$ over an arbitrary base scheme T ([Reference Jarvis, Kaufmann and Kimura36, Definition 2.1], [Reference Abramovich, Corti and Vistoli2, Definition 4.3.1]). It is the data of an n-marked, stable genus g curve $(C,p_1,\ldots ,p_n)$ over T, and a covering of nodal curves $\phi \colon P\to C$ with an action of G on P leaving $\phi $ invariant, such that

1. 1. $\phi $ is a principal G-bundle away from the nodes and markings of C,

2. 2. The analytic local equations for $P\to C\to T$ at a point $p\in P$ over a node of C are

   $$\begin{align*}\operatorname{Spec} A[z,w]/(zw-t) \to \operatorname{Spec} A[x,y]/(xy-t^r)\to \operatorname{Spec} A,\end{align*}$$
   where $t\in A$ , $x = z^r$ and $y=w^r$ for some integer $r>0$ .

3. 3. The analytic local equations for $P\to C\to T$ at a point $p\in P$ over a marked point of C are

   $$\begin{align*}\operatorname{Spec} A[z]\to \operatorname{Spec} A[x] \to \operatorname{Spec} A,\end{align*}$$
   where $x=z^s$ for some integer $s>0$ .

4. 4. If $x\in P$ is a geometric node, then the action of the stabilizer $G_x$ of x on the tangent spaces of the two analytic branches at x is balanced: the characters of these two one-dimensional representations of $G_x$ are inverse to each other.

Admissible G-covers of type $(g,n)$ form a Deligne-Mumford stack, denoted $\mathrm {Adm}_{g,n}(G)$ ; this is a consequence of the identification of $\mathrm {Adm}_{g,n}(G)$ with the space $\mathcal {B}^{\mathrm {bal}}_{g,n}(G)$ of balanced twisted G-covers of type $(g,n)$ which is proven in [Reference Abramovich and Vistoli3] to be a Deligne-Mumford stack. We may write G-cover rather than admissible G-cover for short.

2.2 Admissible covers of smooth curves

Let $\mathrm {Adm}_{g,n}^\circ (G)$ denote the open substack of G-covers in which the target curve (and hence also the source curve) is smooth. In this section, we will determine the connected components of $\mathrm {Adm}_{0,n}^\circ (G)$ (Proposition 2.1). We will use this result later when determining the connected components of the corresponding space of pointed admissible G-covers.

There is a forgetful map

$$\begin{align*}\pi \colon \mathrm{Adm}_{g,n}^\circ(G)\to \mathcal{M}_{g,n}\end{align*}$$

sending a G-cover $P\to (C,p_1,\ldots ,p_n)$ to the n-pointed curve $(C,p_1,\ldots ,p_n)$ . The morphism $\pi $ is étale; this property can be deduced from [Reference Bertin and Romagny9, Theorem 5.1.5], as explained in Proposition 6.5.2 of op. cit. Working over $\mathbb {C}$ , the fiber over $(C,p_1,\ldots ,p_n)$ is identified with the set

(2.1) $$ \begin{align}\operatorname{Hom}(\pi_1(C-\{p_1,\ldots,p_n\},p_0), G)/G,\end{align} $$

where G acts by conjugation, and $p_0 \in C-\{p_1,\ldots ,p_n\}$ is any choice of basepoint. There are no other restrictions on the set (2.1); in particular, the source curves P are not required to be connected. An element of the set (2.1) specifies a G-cover of the punctured curve $C-\{p_1,\ldots ,p_n\}$ , which can be extended uniquely over the punctures. Then the data of the morphism $\pi $ is equivalent to the data of the action of $\pi _1$ of the base space $\mathcal {M}_{g,n}$ on the fiber (2.1) above. We shall now consider this action in the case $g=0$ , when the action may be understood via the classical Hurwitz theory of $\mathbb {P}^1$ . We denote by

$$\begin{align*}\varepsilon^{\mathrm{ni}}_n(G) := \{(g_1,\ldots,g_n)\in G^n: g_1\cdots g_n = 1\}\end{align*}$$

the set of Nielsen classes. We do not impose that $g_1,\ldots ,g_n$ generate G; correspondingly, our source curves are not required to be connected. The group G acts by conjugation on $\varepsilon ^{\mathrm {ni}}_n(G)$ , and the elements of $\varepsilon ^{\mathrm {ni}}_n(G)/G$ are called inner Nielsen classes. Recall the following relationship between the set (2.1) to the set of inner Nielsen classes: choose loops $\rho _1,\ldots ,\rho _n$ around $p_1,\ldots ,p_n$ , respectively, based at $p_0$ , such that $\rho _1,\ldots ,\rho _n$ generate $\pi _1(C-\{p_1,\ldots ,p_n\},p_0)$ subject only to the relation

$$\begin{align*}\rho_1\cdot\cdots\cdot \rho_n = 1.\end{align*}$$

Such a choice identifies the set (2.1) with the inner Nielsen classes.

Now the following diagram of pullback squares relates $\mathrm {Adm}_{0,n}^\circ (G)$ to Hurwitz spaces of G-covers.

The spaces above are defined as follows. The configuration spaces (ordered and unordered) of n points in $\mathbb {P}^1$ are denoted $\operatorname {Conf}_n(\mathbb {P}^1)$ and $\mathrm {UConf}_n(\mathbb {P}^1)$ , respectively. The space $U\mathcal {H}^G_{\mathbb {P}^1,n}$ is the moduli space parametrizing sets $S\subset \mathbb {P}^1$ of n points, together with a ramified G-cover $f\colon P\to \mathbb {P}^1$ whose branch locus is contained in S. The space $\mathcal {H}^G_{\mathbb {P}^1,n}$ is the ordered version of this space, obtained by pullback. The map $\mathcal {M}_{0,n}\to \operatorname {Conf}_n(\mathbb {P}^1)$ fixes $(p_1,p_2,p_3)$ to be $(0,1,\infty )$ , for instance.

Proof. For an arbitrary finite group G, the way in which $U\mathcal {H}_{\mathbb {P}^1,n}^G$ is a covering space over $\mathrm {UConf}_n(\mathbb {P}^1)$ is classically understood, essentially going back to Hurwitz [Reference Hurwitz35]; see [Reference Fulton27, p. 547]. The following is a complete description. Let $S = \{s_1,\ldots ,s_n\}$ . For an appropriate choice of basis, $\operatorname {Hom}(\pi _1(\mathbb {P}^1-S,p_0),G)$ is identified with $\varepsilon ^{\mathrm {ni}}_n(G)$ . And $\pi _1(\mathrm {UConf}_n(\mathbb {P}^1))$ has a presentation with generators $\gamma _1,\ldots ,\gamma _{n-1}$ , where $\gamma _i$ interchanges points i and $i+1$ . Furthermore, the generators $\gamma _i$ act on $\varepsilon ^{\mathrm {ni}}_n(G)$ via

$$\begin{align*}\gamma_i \cdot (g_1,\ldots,g_n) = (g_1,\ldots,g_{i-1}, g_i g_{i+1} g_i^{-1}, g_{i}, g_{i+2},\ldots, g_n). \end{align*}$$

In the case that G is abelian, the action is

$$\begin{align*}\gamma_i \cdot (g_1,\ldots,g_n) = (g_1,\ldots,g_{i-1}, g_{i+1}, g_{i}, g_{i+2},\ldots, g_n). \end{align*}$$

In other words, the action of $\pi _1(\mathrm {UConf}_n(\mathbb {P}^1))$ on $\varepsilon ^{\mathrm {ni}}_n(G)$ factors through $\pi _1(\mathrm {UConf}_n(\mathbb {P}^1))\to S_n$ . Passing to the ordered configuration space, we therefore obtain a trivial action of the spherical braid group $\pi _1(\mathrm {Conf}_n(\mathbb {P}^1))$ on $\varepsilon _n^{\mathrm {ni}}(G)$ , proving the claim.

Remark 2.2. Stack-theoretically, we have

(2.2) $$ \begin{align} \mathrm{Adm}_{0,n}^\circ(G) \cong \mathcal{M}_{0,n} \times [\varepsilon^{\mathrm{ni}}(G)/G]\end{align} $$

if G is abelian, where G acts trivially on $\varepsilon ^{\mathrm {ni}}(G)$ . Under this identification, write

(2.3) $$ \begin{align} \mathrm{Adm}_{0,n}^\circ(G;g_1,\ldots,g_n) \end{align} $$

for the connected component of $\mathrm {Adm}_{0,n}^\circ (G)$ corresponding to the Nielsen class $(g_1,\ldots ,g_n)$ ; it is isomorphic to $\mathcal {M}_{0,n}\times BG$ .

2.3 Pointed admissible covers

We study spaces of pointed admissible covers and determine the connected components of these spaces in Proposition 2.4. This is an important calculation toward the computation of the connected boundary strata in Theorem 3.5 since the boundary strata of spaces of pointed admissible covers are quotients of products of smaller spaces of pointed admissible covers.

Let G be any group, not necessarily abelian. Let $\overline {\mathcal {M}}_{g,n}^G$ denote the space of n-marked pointed admissible G-covers of genus g [Reference Jarvis, Kaufmann and Kimura36]. It is a moduli space for nodal admissible G-covers $P\to (C,p_1,\ldots ,p_n)$ , together with a choice of a lift $\widetilde {p_i}$ on P of each $p_i$ . The open substack $\mathcal {M}_{g,n}^G$ is the moduli space of pointed admissible G-covers in which source and target are smooth. Summarizing, we have a Cartesian square

which lays out the unfortunate lack of parallelism in the notation for these spaces. The notation comes from the literature, however.

For easy reference, we prove Proposition 2.3 below. We note, however, that the argument appears as part of the proof in [Reference Jarvis, Kaufmann and Kimura36, Theorem 2.4] of the fact that $\overline {\mathcal {M}}_{g,n}^G$ is a smooth Deligne-Mumford stack, flat, proper and quasi-finite over $\overline {\mathcal {M}}_{g,n}$ .

Proof. We verify the second statement, which implies the first. Recall the construction of $\overline {\mathcal {M}}_{g,n}^G$ , which we summarize following [Reference Jarvis, Kaufmann and Kimura36]. Let $E\to \mathcal {C}=[E/G]$ denote the universal source curve and stacky target curves, respectively, over $\mathrm {Adm}_{g,n}(G)$ , and let C denote the coarse space of $\mathcal {C}$ . For $i=1,\ldots ,n$ , let $\mathcal {S}_i \to \mathcal {C}$ denote the closed substack of $\mathcal {C}^{\mathrm {sm}}$ whose image in C is the universal $i^{\mathrm {th}}$ marked point; $\mathcal {S}_i$ is an étale gerbe over $\mathrm {Adm}_{g,n}(G)$ . Let $E_i = E\times _{\mathcal {C}} \mathcal {S}_i$ . We have the following diagram, whose top square is Cartesian and where the morphisms known to be étale are labeled:

The morphism $E_i\to \mathrm {Adm}_{g,n}(G)$ is étale since it is a composition of $E_i\to \mathcal {S}_i$ , which is a pullback of an étale morphism and hence étale, and the étale gerbe $\mathcal {S}_i\to \mathrm {Adm}_{g,n}(G)$ . Therefore,

$$\begin{align*}\overline{\mathcal{M}}_{g,n}^G = E_1\times_{\mathrm{Adm}_{g,n}(G)} \cdots \times_{\mathrm{Adm}_{g,n}(G)} E_n\end{align*}$$

is also étale over $\mathrm {Adm}_{g,n}(G)$ .

The spaces $\mathcal {M}_{g,n}^G$ and $\mathrm {Adm}^\circ _{g,n}(G)$ need not be connected, as observed in Remark 2.2. Given $g_1,\ldots ,g_n\in G$ , write $\mathcal {M}_{g,n}^G(g_1,\ldots ,g_n)$ for the open and closed substack of $\mathcal {M}_{g,n}^G$ in which the monodromy at the marking $\widetilde {p_i}$ in the source curve is $g_i$ . We recall the notion of monodromy at a point in the source curve, following [Reference Jarvis, Kaufmann and Kimura36, §2.1]: pick a small oriented loop around the point $p_i$ in the target curve, say based at a point $q_i$ near $p_i$ . Then the loop lifts to d possible paths between the d preimages of $q_i$ near $\widetilde p_i$ , where d temporarily denotes the number of sheets of P meeting at $\widetilde p_i$ . Each of these d paths starts and ends at points x and $gx$ , respectively, for some well-defined $g\in G$ . Indeed, this g is independent of choice of one of those d paths since they are each of the form $g^ix$ to $g^{i+1}x$ for $i=0,\ldots ,d-1$ . The monodromy at $\widetilde p_i$ is then defined to be g. (Note that g can depend, a priori, on choice of lift $\widetilde p_i$ of $p_i$ if G is not abelian. Indeed, the action of any $h\in G$ moves the previously mentioned path near $\widetilde p_i$ from x to $gx$ to a path near $h\widetilde p_i$ from $hx$ to $hgx = (hgh^{-1})hx$ , so the monodromy at $h \widetilde p_i$ is $hgh^{-1}$ .)

Proposition 2.4. Let G be an abelian group. Suppose $g_1\cdots g_n = 1$ , so that $\mathcal {M}_{0,n}^G(g_1,\ldots ,g_n)$ is nonempty. The connected components of $\mathcal {M}_{0,n}^G(g_1,\ldots ,g_n)$ are in bijection with orbits of functions

$$\begin{align*}\{1,\ldots,n\}\to G/\langle g_1,\ldots, g_n\rangle \end{align*}$$

under left G-translation.

Proof. The restriction of the map $\mathcal {M}_{0,n}^G\xrightarrow {\pi } \mathrm {Adm}^\circ _{0,n}(G)$ to $\mathcal {M}^G_{0,n}(g_1,\ldots ,g_n)$ becomes a surjection

$$\begin{align*}\mathcal{M}^G_{0,n}(g_1,\ldots,g_n)\xrightarrow{\pi} \mathrm{Adm}^\circ_{0,n}(G;g_1,\ldots,g_n) \cong \mathcal{M}_{0,n}\times BG,\end{align*}$$

where the last isomorphism was established in Proposition 2.1. This morphism is étale by Proposition 2.3.

Now let $P\to (C, p_1,\ldots ,p_n)$ be any unpointed admissible cover; the fiber of $\pi $ over it is the action groupoid on all lifts $\widetilde {p}_1,\ldots , \widetilde {p}_n$ of $p_1,\ldots ,p_n$ respectively, with the group G acting by simultaneous translation of the $\widetilde {p}_i$ . The connected components of $\mathcal {M}_{0,n}^G(g_1,\ldots ,g_n)$ are in bijection with the orbits of this category under the further action of pure mapping class group $\mathrm {Mod}_{0,n}$ . Those orbits are in bijection with orbits of functions $\{1,\ldots ,n\}\to \pi _0(P)$ under left G-translation, and $\pi _0(P)\cong G/\langle g_1,\ldots ,g_n\rangle $ .

It will be convenient to work with pointed curves labeled by arbitrary finite sets. Thus, let G be a finite group, S a finite set, and $\rho \colon S\to G$ any function. For $g\ge 0$ with $2g-2+|S|>0$ , let

$$\begin{align*}\overline{\mathcal{M}}^G_{g,S}(\rho)\end{align*}$$

denote the space of pointed admissible G-covers of genus g curves with specified monodromy $\rho $ . Let ${\mathcal {M}}^G_{g,S}(\rho )$ denote the open subset parametrizing admissible G-covers in which the target curve is smooth.

3.1 Categories of covers of graphs

Throughout Section 3, let G be a finite abelian group. In this section, we describe the boundary strata of the compactification

$$\begin{align*}\mathcal{M}_{0, S}^{G}(\rho) \hookrightarrow \overline{\mathcal{M}}_{0, S}^G(\rho),\end{align*}$$

showing in Theorem 3.5 that they are in correspondence with graph-theoretic admissible G-covers, which we will now define.

A graph $C = (V,H,i_C,r_C)$ is the data of two finite sets of vertices $V=V(C)$ , and half-edges $H=H(C)$ , together with maps

$$\begin{align*}i_C \colon H\to H,\qquad r_C \colon H\to V\end{align*}$$

such that $i_C$ is an involution. We abbreviate $i= i_C$ and $r = r_C$ . We permit i to have fixed points, and let $L = L(C)$ denote the set of fixed elements of i, called legs. View $r_C$ as the map taking a half-edge to its incident vertex. The edge set $E = E(C)$ is the set of pairs $\{h, i(h)\}$ for $i(h)\ne h$ ; view $i_C$ as the ‘other half’ map on the half-edges.

A morphism of graphs $f\colon C\to C'$ is given by set maps $f_V \colon V\to V'$ and $f_H\colon H\to H'$ such that the relevant squares commute:

For a finite set S, an S-marking of C is an injection $m = m_C\colon S\to L(C)$ . It will be convenient not to require that m is a bijection. A morphism of S-marked graphs $(C,m_C) \to (C', m_{C'})$ is a morphism of graphs $f\colon C\to C'$ that preserves the S-marking (i.e., $f_H \circ m_C = m_{C'}$ ).

It is clear from condition (c) that the genus function g is determined by the monodromy marking $\mu $ as well as the morphism $P \to C$ . Moreover, since C is a tree, the data of C and $\mu $ , without the S-marking, actually determine P and $\Phi $ up to isomorphism. However, the S-marking on P is not in general determined by the S-marking on C.

Figure 1

A G-cover of $5$ -marked graphs, for $G=\mathbb {Z}/4\mathbb {Z} = \{0,1,2,3\}$ . The labels of legs are boxed to avoid confusion with the monodromy marking $\mu : H(C) \to \mathbb {Z}/4\mathbb {Z}$ .

If $\mathcal {P}\to \mathcal {C}$ is an S-marked admissible G-cover of nodal curves, with $\mathcal {C}$ a stable S-marked curve of genus  $0$ , then we obtain a corresponding S-marked admissible G-cover of dual graphs $\mathbf {P}\to \mathbf {C}$ . The meaning of condition (a) is that the subgroup of G stabilizing the generic point of an irreducible component of $\mathcal {P}$ above a given irreducible component $\mathcal {C}_v$ of $\mathcal {C}$ is exactly the subgroup of G generated by the monodromy elements around the special points (nodes and marked points) on $\mathcal {C}_v$ . Thus, each irreducible component of $\mathcal {P}$ above $\mathcal {C}_v$ maps to $\mathcal {C}_v$ with degree $|\langle \mu (r_C^{-1}(v)) \rangle |$ . The content here is that since $\mathcal {C}_v$ is rational, $\pi _1(\mathcal {C}_v)$ is generated by keyhole loops around the special points. Similarly, the data of a homomorphism $\pi _1(\mathcal {C}_v)\to G$ , for appropriately chosen keyhole loops, are the data of an ordered tuple of elements of G whose product is the identity. Condition (b) is similar.

3.2 The dual complex of the boundary

We now state Theorem 3.5 on the boundary complex of the space of pointed admissible covers. Recall the category of symmetric $\Delta $ -complexes (see [Reference Chan, Galatius and Payne21]) (i.e., the category $\mathrm {Fun}(\mathsf {FI}^{\mathrm {op}}, \mathsf {Set})$ , where $\mathsf {FI}$ is the category of finite sets with injections). For $q\ge -1$ an integer, we henceforth write

$$\begin{align*}[q] = \{0,\ldots,q\}.\end{align*}$$

This notational convention includes the special case $[-1] = \emptyset $ . Given $X\colon \mathsf {FI}^{\mathrm {op}}\to \mathsf {Set}$ and an integer $q\ge -1$ , write

$$\begin{align*}X_q = X([q])\end{align*}$$

for the set of q-simplices of X.

Definition 3.4. Fix $g=0$ and G abelian. For data $G,S$ and $\rho $ as above, we define a symmetric $\Delta $ -complex

$$\begin{align*}\Delta_{0,S}^G(\rho)\colon \mathsf{FI}^{\mathrm{op}}\to\mathsf{Set}\end{align*}$$

as follows.

For each $q\ge -1$ , the set $\Delta _{0,S}^G(\rho )_q$ is the set of isomorphism classes of pairs $(\mathbf {P} \to \mathbf {C}, \omega )$ , where

1. 1. $\mathbf {P} \to \mathbf {C}$ is an object of $\Gamma _{0, S}^{G}(\rho )$

2. 2. $\omega \colon [q]\to E(C)$ is a bijection, called an edge-labeling.

An isomorphism of pairs $(\mathbf {P} \to \mathbf {C}, \omega ) \to (\mathbf {P}' \to \mathbf {C}', \omega ')$ is an isomorphism $(\mathbf {P} \to \mathbf {C}) \to (\mathbf {P}' \to \mathbf {C}')$ such that if $\psi : \mathbf {C} \to \mathbf {C}'$ is the induced isomorphism on targets, we have $\omega ' = \psi \circ \omega $ as maps $[q] \to E(\mathbf {C}')$ .

For morphisms, given $i\colon [q']\hookrightarrow [q]$ , and given a graph-theoretic admissible cover $\mathbf {P}\to \mathbf {C}$ as above, contract the edges $E(C) - \omega (i([q']))$ to obtain a new object of $\Gamma _{0, S}^{G}(\rho )$ and take the unique edge-labeling by $[q']$ which preserves the order of the remaining edges.

Theorem 3.5. Let G be an abelian group, and S a finite set. There is an isomorphism of symmetric $\Delta $ -complexes

$$\begin{align*}\Delta_{0,S}^G(\rho)\cong \Delta({\mathcal{M}}^G_{0,S}(\rho)\subset \overline{\mathcal{M}}^G_{0,S}(\rho)).\end{align*}$$

Proof. Let us start with the stratification of the boundary of $\mathrm {Adm}_{0,S}(G;\rho )$ . The space $\mathrm {Adm}_{0,S}(G;\rho )$ is nonempty if and only if $\prod _{s\in S} \rho (s) = 1_G$ . The boundary complex of $\mathrm {Adm}^\circ _{0,S}(G;\rho )\subset \mathrm {Adm}_{0,S}(G;\rho )$ is the complex of trees C with a bijective S-marking $m\colon S\to L(C)$ , together with a monodromy marking $\mu \colon H(C)\to G$ extending $\rho $ , which must satisfy, for every vertex $v\in V(C)$ and $e = \{h_1,h_2\}\in E(C)$ ,

$$\begin{align*}\prod_{h\in r^{-1}(v)} \mu(h) = 1, \qquad \mu(h_1)\mu(h_2)=1.\end{align*}$$

The stratum of the boundary indexed by such a triple $(C,m,\mu )$ is indeed connected since it is, up to finite quotient, isomorphic to a product $\prod _{v\in V(C)} \mathrm {Adm}_{0,n_v}(G;\mu _v)$ of varieties that are themselves connected; see Equation (2.3). More formally, as a symmetric $\Delta $ -complex, the boundary complex has a q-simplex for every such datum $(C, m, \mu )$ together with an arbitrary bijective edge-labeling $\omega \colon [q] \to E(C)$ , one for each isomorphism class of $(C,m,\mu , \omega )$ .

Suppose

$$\begin{align*}(C, m\colon S\to L(C), \mu\colon H(C)\to G)\end{align*}$$

is a stable S-marked tree with monodromy marking $\mu $ as above. For $v\in C$ , write $n_v = r^{-1}(v)$ for the set of half-edges (including legs) at v, and write

$$\begin{align*}G_v = \langle \mu(h)\colon h\in n_v\rangle.\end{align*}$$

Let $\mu _v$ be the restriction of $\mu $ to $n_v$ . As noted above, $(C,m,\mu )$ indexes a boundary stratum of $\mathrm {Adm}_{0,S}(G;\rho )$ . The preimage in $\overline {\mathcal {M}}_{0,S}^G(\rho )$ of this stratum is isomorphic to the variety

(3.2) $$ \begin{align} \!\! \prod_{v\in V(C)} \!\! \left( \mathcal{M}_{0,n_v}^G(\mu_v) \,/\,G^{E(C)} \right ),\end{align} $$

for example, by [Reference Petersen41, §2]. Let us explain the action of $G^{E(C)}$ in (3.2). For a given edge $e = \{h,h'\}$ , incident to vertices v and $v'$ , the copy of G indexed by e acts by translating the lifted marked point indexed by h, respectively $h'$ , in the moduli space $\mathcal {M}_{0,n_v}^G(\mu _v)$ , respectively $\mathcal {M}_{0,n_{v'}}^G(\mu _{v'}).$ (In general, G would also change the values of the marking functions $\mu _v(h)$ and $\mu _{v'}(h')$ , respectively, by conjugation, but G is abelian here.)

The variety (3.2) may not be connected, and it remains to describe its connected components. For each $v\in V(C)$ , let

$$\begin{align*}X_v = \{\mathrm{Fun}(n_v, G/G_v)\}/G,\end{align*}$$

where the quotient is with respect to the G-action on $G/G_v$ . From Proposition 2.4, the connected components of (3.2) are in bijection with

(3.3) $$ \begin{align} \left(\prod_{v\in V(C)} X_v \right)\!/ G^{E(C)}.\end{align} $$

The last step is a combinatorial identification of (3.3) with the set of isomorphism classes of graph-theoretic S-pointed admissible G-covers. Let us begin by considering local data at a single vertex $v\in V(C)$ . Consider an element $f_v\in X_v$ , together with the data of $\mu |_{n_v} \colon n_v \to G$ . From $f_v$ and $\mu |_{n_v}$ we can extract a graph-theoretic $n_v$ -pointed admissible cover involving graphs with legs but no edges. $C_v$ is a single vertex, with legs $n_v$ ; $V(P_v) = G/G_v$ as a left G-set, and above each leg $h\in n_v$ of C is a set of legs in $P_v$ isomorphic to $G/\langle \mu (h)\rangle ,$ with root map compatible with the map $G/\langle \mu (h)\rangle \to G/G_v$ . Finally, $P_v$ has S-marking given by $f_v$ .

Continue to fix a stable S-marked tree C and monodromy marking $\mu $ on C. Now, given $(f_v)_v \in \prod X_v$ , we assemble the local picture above into an admissible cover of graphs. For every edge $e = \{h,h'\}$ of C, with root vertices $v=r(h)$ and $v'=r(h')$ , the half-edges of $P_v$ above h and the half-edges of $P_{v'} $ above $h'$ are each isomorphic to $G/\langle \mu (h) \rangle = G/\langle \mu (h') \rangle $ as G-sets. There is a unique G-equivariant bijection between these two sets that sends the chosen lift of h to the chosen lift of $h'$ , and another choice of lifts of h and $h'$ produce the same bijection if they are related to the original choices by the same element of G. Therefore, these identifications glue the half-edges above h and $h'$ into edges above e, obtaining a graph-theoretic admissible cover $P\to C$ which was independent of the action of $G^{E(C)}$ . It is straightforward to reverse this process, giving an element of the set (3.3) starting from a graph-theoretic admissible cover.

Remark 3.6. Theorem 3.5 furnishes an explicit description of the symmetric $\Delta $ -complex

(3.4) $$ \begin{align}\Delta(\mathcal{M}_{g,n}^G \subset \overline{\mathcal{M}}_{g,n}^G) \end{align} $$

when $g=0$ and G is abelian. It is sufficiently explicit that it can be programmed, and indeed, we carry out computer calculations for the results in Appendix A. Without restrictions on G and g, it is still possible to give a general description of (3.4) using the framework of graphs of groups, roughly, decorating vertices of graphs with fundamental groups of punctured curves. This idea will appear in future work by M. Talpo, M. Ulirsch and D. Zakharov; we thank Ulirsch for bringing it to our attention. This general description is not explicit in the above sense. It involves the very interesting sub-question of determining the connected components of the spaces $\mathcal {M}_{g,n}^G$ in general; compare with Proposition 2.1. We also refer to forthcoming work of P. Souza that constructs (3.4) in the case of G cyclic with g arbitrary and identifies it as the nonarchimedean skeleton of the toroidal pair. Moreover, that work is a precursor to further work by Y. El Maazouz, P. Helminck, F. Röhrle, P. Souza and C. Yun studying the homotopy type of boundary complexes of unramified $\mathbb {Z}/p\mathbb {Z}$ covers for $g=2$ .

4.1 The complex $\widetilde {\Theta }_{g, n}$

First, let $\widetilde {\mathcal {H}}_{g,n}$ denote the moduli stack of hyperelliptic curves of genus g with n distinct marked points and $2g+2$ labeled Weierstrass points. The symmetric group on $2g+2$ letters permutes the labels on Weierstrass points, and

$$\begin{align*}\mathcal{H}_{g,n} \cong [\widetilde{\mathcal{H}}_{g,n}/S_{2g+2}].\end{align*}$$

In this subsection, we will provide a normal crossings compactification of $\widetilde {\mathcal {H}}_{g,n}$ and give the corresponding dual complex. Then we will quotient out by $S_{2g+2}$ to give a normal crossings compactification of $\mathcal {H}_{g,n}$ .

In $\widetilde {\mathcal {H}}_{g,n}$ , a marked point is allowed to coincide with a Weierstrass point, and two marked points are allowed to form a conjugate pair under the hyperelliptic involution. Because of this, two types of graphs will require special attention.

Proposition 4.2. There is an open inclusion

$$\begin{align*}\widetilde{\mathcal{H}}_{g,n} \hookrightarrow \overline{\mathcal{M}}^{\mathbb{Z}/2\mathbb{Z}}_{0,S}(\rho)\end{align*}$$

which is a normal crossings compactification, and whose boundary complex $\widetilde {\Theta }_{g, n}$ is isomorphic to the subcomplex of

$$\begin{align*}\Delta_{0,S}^{\mathbb{Z}/2\mathbb{Z}}(\rho)\end{align*}$$

on simplices whose vertices are not of type (1) or (2) in Definition 4.1.

Proof. Let $\widetilde {\mathcal {H}}^\circ _{g,n}$ denote the open substack of $\widetilde {\mathcal {H}}_{g,n}$ in which a marked point may not collide with a Weierstrass point, and two marked points may not form a conjugate pair. Then

$$\begin{align*}\widetilde{\mathcal{H}}^\circ_{g,n} \cong \mathcal{M}_{0,S}^{\mathbb{Z}/2\mathbb{Z}}(\rho),\end{align*}$$

where $\mathcal {M}_{0,S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ denotes the interior of the moduli space $\overline {\mathcal {M}}_{0,S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ of pointed admissible covers. We define a partial compactification $\mathcal {H}_{g, n}^*$ of $\widetilde {\mathcal {H}}_{g, n}^{\circ }$ , such that

$$\begin{align*}\widetilde{\mathcal{H}}_{g, n}^{\circ} \subset \mathcal{H}_{g, n}^* \subset \overline{\mathcal{M}}^{\mathbb{Z}/2\mathbb{Z}}_{0,S}(\rho), \end{align*}$$

and the second inclusion is normal crossings. In $\overline {\mathcal {M}}^{\mathbb {Z}/2\mathbb {Z}}_{0,S}(\rho )$ , define $\mathcal {H}_{g,n}^*$ to be the open complement of all boundary divisors except for those corresponding to dual graphs of type (1) or (2) (see Definition 4.1). Since $\mathcal {H}_{g, n}^*$ is the complement of a subset of the boundary divisors, the divisor

$$\begin{align*}\overline{\mathcal{M}}_{0, S}^{\mathbb{Z}/2\mathbb{Z}}(\rho) \smallsetminus \mathcal{H}_{g, n}^* \end{align*}$$

still has normal crossings. Stabilization gives a canonical isomorphism $\mathcal {H}_{g,n}^* \cong \widetilde {\mathcal {H}}_{g,n}$ which is equivariant with respect to the action of $S_n$ , thus giving the first part of the result.

We now turn our attention to the boundary complex. Denote by $\Delta _{0,S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ the dual complex of the compactification

$$\begin{align*}\widetilde{\mathcal{H}}^\circ_{g,n} \cong \mathcal{M}_{0,S}^{\mathbb{Z}/2\mathbb{Z}}(\rho)\subset \overline{\mathcal{M}}_{0,S}^{\mathbb{Z}/2\mathbb{Z}}(\rho).\end{align*}$$

The target graphs of type (1) and (2) in Definition 4.1 have one edge and correspond to vertices in $\Delta _{0,S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ . Then the boundary complex $\widetilde {\Theta }_{g, n}$ of the inclusion

$$\begin{align*}\widetilde{\mathcal{H}}_{g,n} \subset \overline{\mathcal{M}}^{\mathbb{Z}/2\mathbb{Z}}_{0,S}(\rho)\end{align*}$$

is the subcomplex of $\Delta _{0,S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ determined by those simplices which have no vertices of type (1) or (2) in Definition 4.1.

Let us now describe the complex $\Delta _{0,S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ in more detail. Its q-simplices are given by isomorphism classes of pairs $(\mathbf {P} \to \mathbf {C}, \omega )$ , where $\mathbf {P} \to \mathbf {C}$ is an object of the category $\Gamma _{0, S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ (Definition 3.3), and $\omega \colon [q]\to E(C)$ is an edge-labeling. Moreover, on $L(C)$ , the monodromy marking $\mu $ satisfies $\mu (m_C(j)) = 0$ if $j \in \{1, \ldots , n\}$ , and $\mu (m_C(j)) = 1$ if $j \in \{w_1, \ldots , w_{2g + 2}\}$ . We will call the elements of

$$\begin{align*}m_C(\{w_1, \ldots, w_{2g + 2} \}) \subset L(C) \end{align*}$$

the branch legs of C.

Notice that the above conditions on $\mu |_{L(C)}$ suffice to determine $\mu $ on all other half-edges of C, by condition (1) of Definition 3.4. Call a vertex $v \in V(C)$ a leaf vertex if it is incident to only one edge. If a leaf vertex $v \in V(C)$ supports an odd number of branch legs, then the non-leg half edge h incident to v must satisfy $\mu (h) = 1$ . However, if a leaf vertex v supports an even number of branch legs, then the non-leg half edge h incident to v must satisfy $\mu (h) = 0$ . Proceeding inductively, this determines $\mu $ on all half-edges incident to non-leaf vertices of C as well.

This discussion implies that given the monodromy data $\rho $ and an S-marked stable tree C, the only additional data required to determine an object of the category $\Gamma _{0, S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ is a lift of the marking function $m_C\colon S\to L(C)$ to a function $m_P\colon S\to L(P)$ such that the diagram

commutes. (Note that the morphism of graphs $P \to C$ , without the marking function on P, is already determined by C and $\mu $ .) Moreover, since each branch leg in C has a unique preimage in P, one only needs to choose, for each $i \in \{1, \ldots , n\}$ , a leg in the preimage of $m(i) \in L(C)$ . Two such choices are equivalent if they differ by the $\mathbb {Z}/2\mathbb {Z}$ -action on P. See Figure 3 for an example.

Figure 3

A $\{1, 2\} \cup \{w_1, \ldots , w_8\}$ -marked stable tree C, together with the two lifts of $m_C$ to a marking $m_P$ . These non-isomorphic lifts are determined by a choice of element in the fiber over each leg marked by $\{1, 2\}$ on C, and two such choices define the same graph-theoretic admissible $\mathbb {Z}/2\mathbb {Z}$ -cover if they differ by the $\mathbb {Z}/2\mathbb {Z}$ -action on P.

4.2 The complex $\Theta _{g, n}$

We now construct a normal crossings compactification of $\mathcal {H}_{g,n}$ and the corresponding dual complex  $\Theta _{g, n}$ .

By Proposition 4.2, in order to pass from $\Delta _{0, S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ to $\widetilde {\Theta }_{g, n}$ , we remove all edge-labeled pairs $(\mathbf {P} \to \mathbf {C}, \omega )$ such that $\mathbf {P} \to \mathbf {C}$ admits a contraction to covers of type (1) or (2) in Definition 4.1. To that end, let

$$\begin{align*}\Gamma_{0, S}^{\mathbb{Z}/2\mathbb{Z}, *}(\rho)\end{align*}$$

be the full subcategory of $\Gamma _{0, S}^{\mathbb {Z}/2\mathbb {Z}}(\rho )$ on those covers which do not admit a contraction to covers of type (1) or (2).

Proof. Since the action of $S_{2g+2}$ on $\widetilde {\mathcal {H}}_{g,n} \subset \overline {\mathcal {M}}^{\mathbb {Z}/2\mathbb {Z}}_{0,S}(\rho )$ preserves $\widetilde {\mathcal {H}}_{g,n}$ and sends strata isomorphically to strata, we have that

$$ \begin{align*}\mathcal{H}_{g,n} \cong [\widetilde{\mathcal{H}}_{g,n} / S_{2g+2}] \subset [\overline{\mathcal{M}}^{\mathbb{Z}/2\mathbb{Z}}_{0,S}(\rho) / S_{2g+2}]\end{align*} $$

is a normal crossings compactification with boundary complex equal to

$$\begin{align*}\Delta(\widetilde{\mathcal{H}}_{g,n} \subset \overline{\mathcal{M}}^{\mathbb{Z}/2\mathbb{Z}}_{0,S}(\rho)) / S_{2g+2} = \widetilde{\Theta}_{g, n}/S_{2g + 2},\end{align*}$$

and the described symmetric $\Delta $ -complex is precisely the quotient of $\widetilde {\Theta }_{g, n}$ by $S_{2g + 2}$ .

As a direct result of Proposition 4.4, we have the following corollary identifying the weight zero compactly supported cohomology of $\mathcal {H}_{g, n}$ with the reduced cohomology of $\Theta _{g, n}$ ; see [Reference Chan, Galatius and Payne21, Theorem 5.8].

Corollary 4.5. For each i, there are canonical $S_n$ -equivariant isomorphisms

$$\begin{align*}W_0 H^i_c(\mathcal{H}_{g, n};\mathbb{Q}) \cong \widetilde{H}^{i - 1}(\Theta_{g, n};\mathbb{Q}) \cong \widetilde{H}_{i - 1}(\Theta_{g, n};\mathbb{Q})^{\vee}, \end{align*}$$

where $\widetilde {H}^*$ and $\widetilde {H}_*$ denote reduced cohomology and homology, respectively.

We now establish some conventions for working with objects of the category $\Gamma ^{\mathcal {H}}_{g, n}$ .

The total weight of the vertices of C is $2g+2$ . The weight in this sense should not be confused with the notion of vertex weights corresponding to genera of irreducible curves. The two notions of vertex weight are related by the Riemann-Hurwitz formula.

When depicting objects of $\Gamma _{g, n}^{\mathcal {H}}$ , we adopt the following conventions. Instead of drawing the unmarked legs of $\mathbf {C}$ , we will label each vertex of $\mathbf {C}$ with its weight. To avoid confusion with the genera of vertices in the source graph, we will depict the weight of a vertex in $\mathbf {C}$ with the color grey and genera of vertices with blue. Since each unmarked leg of C has a unique preimage in P, we will not draw those legs of P. When a leg of C has two preimages in P, so only one is marked, we will suppress the other leg. See Figure 4 for the images of the $\Gamma _{0, S}^{\mathbb {Z}/2\mathbb {Z}, *}(\rho )$ objects from Figure 3 under the functor to $\Gamma ^{\mathcal {H}}_{g, n}$ . See Figure 5 for a complete list of isomorphism classes of $\Gamma _{g, n}^{\mathcal {H}}$ -objects when $g = 2$ and $n = 0$ .

Figure 4

The images of the graph-theoretic admissible $\mathbb {Z}/2\mathbb {Z}$ -covers in $\Gamma _{0, S}^{\mathbb {Z}/2\mathbb {Z}, *}(\rho )$ from Figure 3, under the functor $\Gamma _{0, S}^{\mathbb {Z}/2\mathbb {Z}, *}(\rho ) \to \Gamma _{g, n}^{\mathcal {H}}$ . The number of unmarked legs at a vertex of a target tree is indicated by the weight function. We do not depict any unmarked legs of the source graph since they are determined by the legs of the target.

Figure 5

The set of isomorphism classes of $\Gamma ^{\mathcal {H}}_{g, n}$ -objects for $g = 2$ and $n = 0$ .

Remark 4.7. We remark on the case $n=0$ . In this case, the symmetric $\Delta $ -complex $\Theta _{g,0}$ is isomorphic to the quotient of the dual complex

(4.1) $$ \begin{align} \Delta_{0, 2g+2}:= \Delta\left(\mathcal{M}_{0, 2g+ 2} \subset\overline{\mathcal{M}}_{0, 2g + 2}\right)\end{align} $$

by the $S_{2g + 2}$ -action permuting the marked points. The dual complex (4.1) is the moduli space of $(2g+2)$ -marked tropical curves of genus zero and volume one [Reference Chan, Galatius and Payne22], also known as the space of phylogenetic trees [Reference Ardila and Klivans4, Reference Billera, Holmes and Vogtmann12, Reference Robinson and Whitehouse42]. The identification

$$\begin{align*}\Theta_{g,0} = \Delta_{0,2g + 2} /S_{2g + 2} \end{align*}$$

can be seen directly from our description of the category $\Gamma _{g}^{\mathcal {H}}$ and holds despite the fact that the morphism

$$\begin{align*}[\overline{\mathcal{M}}_{0, 2g+2}^{\mathbb{Z}/2\mathbb{Z}}(\rho)/S_{2g + 2}] \to [\overline{\mathcal{M}}_{0, 2g + 2}/S_{2g + 2}] \end{align*}$$

is not an isomorphism or even a $\mathbb {Z}/2\mathbb {Z}$ -gerbe, due to the possible presence of extra automorphisms, more than $\mathbb {Z}/2\mathbb {Z}$ , in the source curves of $\mathbb {Z}/2\mathbb {Z}$ -admissible covers.

5.1 The cellular chain complex of $\Theta _{g, n}$

Following [Reference Chan, Galatius and Payne21, §3], the reduced rational homology of $\Theta _{g, n}$ is computed by the graph complex $\mathcal {C}_*^{(g, n)}$ described as follows. In degree p, $\mathcal {C}_p^{(g, n)}$ is spanned by pairs $(\mathbf {P} \to \mathbf {C}, \omega )$ where $\mathbf {P} \to \mathbf {C}$ is an object of $\Gamma ^{\mathcal {H}}_{g, n}$ , and $\omega \colon [p] \to E(\mathbf {C})$ is a bijective edge-labeling. These pairs are subject to the relation $(\mathbf {P} \to \mathbf {C}, \omega ) = \mathrm {sgn}(\rho ) (\mathbf {P} \to \mathbf {C}, \omega \circ \rho )$ whenever $\rho \in S_{p + 1} = \operatorname {Aut}([p])$ .

The differential $\partial \colon \mathcal {C}_p^{(g, n)} \to \mathcal {C}_{p - 1}^{(g, n)}$ is given by the signed sum of edge contractions:

$$\begin{align*}\partial (\mathbf{P} \to \mathbf{C}, \omega) = \sum_{i \in [p]} (-1)^i (\delta^{i})^{*}(\mathbf{P} \to \mathbf{C}, \omega), \end{align*}$$

where $\delta ^i \colon [p - 1] \to [p]$ is the unique order-preserving injection which misses i.

To prove Theorem 5.1, we will show that the corresponding sub-chain complexes of $\mathcal {C}^{(g, n)}_*$ are acyclic. Denote by $\mathcal {R}^{(g, n)}_*$ the sub-chain complex of $\mathcal {C}^{(g, n)}_*$ spanned by those pairs $(\mathbf {P} \to \mathbf {C}, \omega )$ such that $\mathbf {P}$ has a vertex v that has at least two markings from $\{1, \ldots , n\}$ ; this is the chain complex which computes the reduced rational homology of $\Theta _{g, n}^{\mathrm {rep}}$ . Denote by $\mathcal {Q}^{(g, n)}_*$ the sub-chain complex of $\mathcal {C}^{(g, n)}_*$ spanned by those pairs $(\mathbf {P} \to \mathbf {C}, \omega )$ where $\mathbf {C}$ has at least one vertex v with weight at least $3$ .

We will show that the chain complexes $\mathcal {R}^{(g, n)}_*$ and $\mathcal {Q}^{(g, n)}_* \cap \mathcal {R}^{(g, n)}_*$ are acyclic for all $g \geq 2$ and all $n \geq 2$ (Theorem 5.5), that the chain complex $\mathcal {Q}^{(g, n)}_*$ is acyclic for all $g \geq 2$ and all $n \geq 0$ (Theorem 5.9), and that the chain complex $\mathcal {C}^{(g, n)}_*$ is acyclic for all $g \geq 2$ and $n \leq 1$ (Theorem 5.12). Thus, Theorem 5.5 and Theorem 5.9 prove Theorem 5.1, and Theorem 5.12 gives part (1) of Proposition B.

The proofs of these theorems are informed by previous work of Chan–Galatius–Payne on contractibility criteria for symmetric $\Delta $ -complexes [Reference Chan, Galatius and Payne22], as well as work of Conant–Gerlits–Vogtmann [Reference Conant, Gerlits and Vogtmann23] on the acyclicity of the subcomplex of Kontsevich’s graph complex spanned by graphs with cut vertices.

5.2 The homology of $\Theta _{g, n}^{\mathrm {rep}}$

It will be useful to isolate specific types of edges of covers with repeated markings.

Definition 5.3. For a $\Gamma ^{\mathcal {H}}_{g, n}$ -object $\mathbf {P} \to \mathbf {C}$ with repeated markings, we say an edge $e \in E(\mathbf {C})$ is a supporting edge, with support equal to $S \subseteq [n]$ , if, upon contracting all edges of $\mathbf {C}$ which are not equal to e, as well as their preimages in $\mathbf {P}$ , we obtain the cover $\mathbf {B}_S \to \mathbf {E}_S$ depicted in Figure 6. If $|S| = i$ , we will call e an i-supporting edge. Note that, necessarily, $i\ge 2$ .

Figure 6

The cover $\mathbf {B}_S \to \mathbf {E}_S$ .

Proof. We will prove the theorem only for $\mathcal {R}^{(g, n)}_*$ , as the same argument works for $\mathcal {R}^{(g, n)}_* \cap \mathcal {Q}^{(g, n)}_*$ . For ease of notation, fix $g, n \ge 2$ and put

$$\begin{align*}\mathcal{R}_* := \mathcal{R}^{(g,n)}_*. \end{align*}$$

First, filter $\mathcal {R}_*$ as follows: let

$$\begin{align*}\mathcal{R}^{\geq i}_* \hookrightarrow \mathcal{R}_* \end{align*}$$

be the subcomplex generated by covers which have a k-supporting edge for some $k \geq i$ . More precisely, we mean that $\mathcal {R}^{\geq i}_*$ is spanned by covers obtained by edge-contraction from covers with supporting edges of this type. We apply this definition even when $i=n+1$ , in which case $\mathcal {R}^{\ge {n+1}}_* = 0$ . Then we have a filtration

$$\begin{align*}0 = \mathcal{R}^{\ge n+1}_* \hookrightarrow \mathcal{R}^{\geq n}_* \hookrightarrow \cdots \hookrightarrow \mathcal{R}^{\geq 2}_* = \mathcal{R}_*. \end{align*}$$

Passing to the associated spectral sequence, it suffices to show that for each $i=2,\ldots ,n$ , the successive quotient chain complexes

$$\begin{align*}\mathcal{R}^{i}_* := \mathcal{R}^{\geq i}_* / \mathcal{R}^{\geq i + 1}_* \end{align*}$$

are acyclic. These quotient chain complexes are spanned by covers with i-supporting edges and their edge-contractions but do not include any covers with k-supporting edges or their edge contractions for any $k> i$ . Now we filter $\mathcal {R}^i_*$ . Define

$$\begin{align*}F_p\mathcal{R}^i_* \hookrightarrow \mathcal{R}^i_* \end{align*}$$

to be the sub-chain complex spanned by graphs with at most p non-supporting edges. The number of non-supporting edges cannot increase under edge contraction, so $F_p \mathcal {R}^i_*$ really is a subcomplex. We obtain an ascending filtration

$$\begin{align*}0 = F_{-1}\mathcal{R}^i_* \hookrightarrow F_{0}\mathcal{R}^i_* \hookrightarrow \cdots\hookrightarrow \mathcal{R}^i_*,\end{align*}$$

and again by considering the associated spectral sequence, it suffices to show that successive quotients

$$\begin{align*}G_p \mathcal{R}^i_* := F_p \mathcal{R}^i_* / F_{p -1}\mathcal{R}^i_* \end{align*}$$

are acyclic, in order to conclude that $\mathcal {R}^i_*$ and hence $\mathcal {R}_*$ is acyclic. For fixed i and p, let $A_{i, p}$ denote the set of isomorphism classes of $\Gamma ^{\mathcal {H}}_{g, n}$ -objects $\mathbf {P} \to \mathbf {C}$ where $|E(\mathbf {C})| = p$ and which (1) do not have any supporting edges, (2) admit a contraction from a cover with an i-supporting edge and (3) do not admit a contraction from any covers with k-supporting edges for $k> i$ . Then we have a direct sum decomposition

$$\begin{align*}G_p \mathcal{R}^i_* = \bigoplus_{\mathbf{P} \to \mathbf{C} \in A_{i,p}} \mathcal{L}^{\mathbf{P} \to \mathbf{C}}_*, \end{align*}$$

where $\mathcal {L}^{\mathbf {P} \to \mathbf {C}}_*$ is the sub-chain complex consisting of those covers whose supporting edge retraction is equal to $\mathbf {P} \to \mathbf {C}$ . This direct sum decomposition holds because the differential on $G_p \mathcal {R}^i_*$ is given by a signed sum of supporting edge contractions and hence preserves the supporting edge retraction of a given cover. Next, given $\mathbf {P} \to \mathbf {C} \in A_{i, p}$ , we have a tensor product decomposition

$$\begin{align*}\mathcal{L}_*^{\mathbf{P} \to \mathbf{C}} \cong \left(\bigotimes_{v \in V^{\mathrm{rep}}_i(\mathbf{P})} (\mathbb{Q} \xrightarrow{\sim} \mathbb{Q})\right)[1 - p], \end{align*}$$

where $V^{\mathrm {rep}}_i(\mathbf {P})$ denotes the set of vertices of $\mathbf {P}$ which contain exactly i markings, and the first copy of $\mathbb {Q}$ is in degree 1. This tensor product decomposition holds because a generator of $\mathcal {L}_*^{\mathbf {P} \to \mathbf {C}}$ is determined by a choice of subset of those vertices of $\mathbf {P}$ which contain i markings: the corresponding generator is determined by expanding a single i-supporting edge from the image of each chosen vertex in $\mathbf {C}$ . (Since $i\ge 2$ , such an expansion is indeed possible, producing a stable S-marked target tree.) The degree shift is required to account for the p edges of $\mathbf {P} \to \mathbf {C}$ . Altogether, this shows that $\mathcal {L}^{\mathbf {P} \to \mathbf {C}}_*$ is a tensor product of acyclic chain complexes, so $\mathcal {L}^{\mathbf {P} \to \mathbf {C}}_*$ is itself acyclic, and the proof is complete.

5.3 The homology of $\Theta _{g, n}^{\geq 3}$

We will now show that the chain complex $\mathcal {Q}^{(g, n)}_*$ is acyclic. It will again be convenient to name particular types of edges.

It is straightforward to see that for any cover $\mathbf {P} \to \mathbf {C}$ , the poset of $3$ -end expansions of $\mathbf {P} \to \mathbf {C}$ has a maximal element, as in the following lemma. We omit the proof; see Figure 8 for an example of how this expansion is constructed.

Figure 8

A cover in $\Gamma ^{\mathcal {H}}_{5, 2}$ and its maximal expansion by $3$ -ends.

Given a $\Gamma ^{\mathcal {H}}_{g, n}$ -object $\mathbf {P} \to \mathbf {C}$ , let $A(\mathbf {P} \to \mathbf {C})$ be the set of isomorphism classes of covers obtained from $\mathbf {P} \to \mathbf {C}$ by contracting $3$ -ends. We define a chain complex $\mathcal {Q}^{\mathbf {P} \to \mathbf {C}}_*$ as follows: the vector space $\mathcal {Q}^{\mathbf {P} \to \mathbf {C}}_p$ is spanned by pairs $(\mathbf {H} \to \mathbf {K}, \omega )$ , where $\mathbf {H} \to \mathbf {K}$ is an element of $A(\mathbf {P} \to \mathbf {C})$ with $|E(\mathbf {K})| = p + 1$ , and $\omega \colon [p] \to E(\mathbf {K})$ is an edge-labeling. These generators are subject to the usual relation

$$\begin{align*}(\mathbf{H} \to \mathbf{K}, \omega \circ \rho ) = \mathrm{sgn}(\rho) (\mathbf{H} \to \mathbf{K}, \omega)\end{align*}$$

for $\rho \in \operatorname {Aut}([p])$ . The differential on $\mathcal {Q}^{\mathbf {P} \to \mathbf {C}}_*$ is given by the signed sum of $3$ -end contractions; we set it equal to $0$ on any generators which do not have any $3$ -ends.

Proof. First, consider the case where $\mathbf {C}$ has no automorphisms. This implies that all $3$ -end contractions of $\mathbf {C}$ have no automorphisms, since any automorphism of the target tree of a $\Gamma ^{\mathcal {H}}_{g, n}$ -object must lift to an automorphism of its maximal $3$ -end expansion. Let $q + 1$ be the number of distinct $3$ -ends of $\mathbf {C}$ . We can understand $\mathcal {Q}^{\mathbf {P} \to \mathbf {C}}_*$ as a shift of the augmented cellular chain complex of the standard q-simplex $\sigma ^{q}$ , viewed as the space parameterizing assignments of nonnegative lengths to the $q + 1$ distinct $3$ -ends of $\mathbf {C}$ , such that the lengths sum to one. Note that $q\ge 0$ by assumption, so that $\sigma ^q$ is nonempty. So in the automorphism-free case, $\mathcal {Q}^{\mathbf {P} \to \mathbf {C}}_*$ is acyclic.

For the general case, when $\mathbf {C}$ and its contractions may have automorphisms, fix a labeling of the edges of $\mathbf {C}$ , and denote the resulting object by $\mathbf {C}^\dagger $ . This induces a labeling of the edges of each contraction of $\mathbf {C}$ . Let $A( \mathbf {C}^\dagger )$ be the set consisting of $\mathbf {C}^\dagger $ and all of its $3$ -end contractions. We can make a chain complex $\mathcal {Q}^{\mathbf {C}, \dagger }_{*}$ which in degree p is spanned by pairs $[\mathbf {K}, \omega ]$ where $\mathbf {K}$ is an element of $A(\mathbf {C}^{\dagger })$ with $|\mathbf {K}| = p + 1$ , and $\omega \colon [p] \to E(\mathbf {K})$ is a bijection, subject to the usual relations under the action of $\operatorname {Aut}([p])$ . Observe that there is a canonical action of $\operatorname {Aut}(\mathbf {C})$ on the chain complex $\mathcal {Q}^{\mathbf {C}, \dagger }_{*}$ , and $\mathcal {Q}^{\mathbf {P} \to \mathbf {C}}_*$ is identified with the $\operatorname {Aut}(\mathbf {C})$ -coinvariants of the complex $\mathcal {Q}^{\mathbf {C}, \dagger }_{*}$ , by the second part of Lemma 5.7. Since $\operatorname {Aut}(\mathbf {C})$ is finite, it has no homology over the rationals. Moreover, $\mathcal {Q}^{\mathbf {C}, \dagger }_*$ is acyclic by the first part of the proof. We conclude that

$$\begin{align*}H_*(\mathcal{Q}^{\mathbf{P} \to \mathbf{C}}_*) = H_*((\mathcal{Q}^{\mathbf{C}, \dagger}_{*})_{\operatorname{Aut}(\mathbf{C})}) = (H_*(\mathcal{Q}^{\mathbf{C}, \dagger}_{*}))_{\operatorname{Aut}(\mathbf{C})} = 0, \end{align*}$$

as desired.

We now prove that $\mathcal {Q}^{(g, n)}_*$ is acyclic.

Proof. Let $F_p \mathcal {Q}^{(g, n)}_*$ denote the subspace spanned by those covers whose target tree has at most p edges which are not $3$ -ends. This defines a bounded, increasing filtration of $\mathcal {Q}^{(g, n)}_*$ . The $E^0$ page

$$\begin{align*}E^{0}_{p, q} = F_p \mathcal{Q}^{(g, n)}_{p + q}/ F_{p - 1}\mathcal{Q}^{(g, n)}_{p + q} \end{align*}$$

of the associated spectral sequence is spanned by covers whose target tree has exactly p edges which are not $3$ -ends. The differential $\partial _0\colon E^{0}_{p, q} \to E^{0}_{p, q-1}$ is given by a signed sum of $3$ -end contractions. Therefore, by Lemma 5.7, the pth column of the $E^0$ page breaks up into a direct sum of chain complexes of the form $\mathcal {Q}^{\mathbf {P} \to \mathbf {C}}_*$ , where $\mathbf {C}$ has at least one $3$ -end, and the tree obtained from $\mathbf {C}$ by contracting all $3$ -ends has p edges. Proposition 5.8 then implies that the $E^1$ page vanishes, which completes the proof.

5.4 Calculations on $\Theta _{g, n}$ for $n \leq 2$

We conclude this section by proving Proposition B. The first part of Proposition B asserts that $\mathcal {C}^{(g, n)}_*$ is acyclic for $n \leq 1$ , and the proof is similar to the one that $\mathcal {Q}^{(g, n)}_*$ is acyclic. Once again, we isolate particular types of edges:

Definition 5.10. Let $\mathbf {P} \to \mathbf {C}$ be a $\Gamma _{g, n}^{\mathcal {H}}$ -object. An edge $e \in E(\mathbf {C})$ is called a $2$ -end if upon contracting all edges of $\mathbf {C}$ except for e, and their preimages in $\mathbf {P}$ , we obtain the cover $\mathbf {J} \to \mathbf {K}$ in Figure 9.

Figure 9

The cover $\mathbf {J} \to \mathbf {K}$ .

The key to the proof of acyclicity of $\mathcal {C}^{(g, n)}_*$ when $n \leq 1$ is the following lemma.

Lemma 5.11. Let $\mathbf {P} \to \mathbf {C}$ be an object of $\Gamma ^{\mathcal {H}}_{g, n}$ for $n \leq 1$ . Then the poset of expansions of $\mathbf {P} \to \mathbf {C}$ by $2$ -ends has a unique maximal element $\mathbf {P}' \to \mathbf {C}'$ . Moreover, this expansion is canonical in the sense that any automorphism of $\mathbf {P} \to \mathbf {C}$ lifts to one of $\mathbf {P}' \to \mathbf {C}'$ .

Figure 10

A cover $\mathbf {P} \to \mathbf {C}$ in $\Gamma _{5, 2}^{\mathcal {H}}$ and the two distinct maximal elements of its poset of $2$ -end expansions. When $n \leq 1$ , this poset always has a unique maximal element, as explained in the proof of Lemma 5.11.

Given Lemma 5.11, the proof of the following theorem is completely analogous to the proof of Theorem 5.9; we will only outline the necessary steps.

Theorem 5.12. For $g\geq 2$ and $n \leq 1$ , the chain complex $\mathcal {C}^{(g, n)}_*$ is acyclic.

Figure 11

A cycle spanning $\widetilde {H}_{2g}(\Theta _{g, 2}; \mathbb {Q})$ .

Theorem 5.12 gives part (1) of Proposition B. Part (2) states that

$$\begin{align*}W_0 H^{2g + 1}_c(\mathcal{H}_{g, 2};\mathbb{Q}) \cong \mathbb{Q} \end{align*}$$

and that the corresponding $S_2$ -representation is trivial if g is even and given by the sign representation if g is odd. We prove this now by writing down an explicit cycle in $\mathcal {C}^{(g, 2)}_{2g}$ corresponding to this class. See Figure 11.

Proof of Proposition B, part (2).

We have an isomorphism of $S_2$ -representations

$$\begin{align*}W_0 H^{2g + 1}_c(\mathcal{H}_{g, 2};\mathbb{Q}) \cong \widetilde{H}_{2g}(\Theta_{g, 2};\mathbb{Q})^\vee\end{align*}$$

by Corollary 4.5. We have

$$\begin{align*}\widetilde{H}_{2g}(\Theta_{g, 2};\mathbb{Q}) = H_{2g}\left(\mathcal{C}^{(g, 2)}_*\right). \end{align*}$$

Observe that $2g$ is the top homological degree of $\mathcal {C}^{(g, 2)}$ : the maximal number of edges of a stable tree with $2g + 4$ legs is $2g + 1$ . Therefore, any cycle in $\mathcal {C}^{(g, 2)}_{2g}$ defines a class in homology. Any target tree for a cover in $\mathcal {C}^{(g, 2)}_{2g}$ must be trivalent, and to be a nonzero element, it cannot have any automorphisms which act by an odd permutation of the edge set. It is straightforward to conclude that such a tree must be equal to the tree depicted in Figure 11. This tree $\mathbf {C}$ has two covers, depicted in Figure 11. Therefore, $\dim \mathcal {C}^{(g, 2)}_{2g} = 2$ , where a basis is given by choosing any edge-labeling of the aforementioned tree. One can verify directly that neither one of these basis elements forms a cycle on their own, but their difference does. From this we conclude that $H_{2g}(\mathcal {C}_*^{(g,2)}) \cong \mathbb {Q}$ . To understand the $S_2$ -representation, we note that when g is even, the transposition in $S_2$ induces an even permutation on any edge-labeling of the given tree, and when g is odd, the transposition induces an odd permutation of the edge labels.

Remark 5.13. Theorem 5.5 generalizes to other spaces of admissible covers. Fix an integer $N> 0$ , and let G be an abelian group, which we now write additively to be consistent with our notation for $G=\mathbb {Z}/2\mathbb {Z}$ . Let

$$\begin{align*}\mu\colon \{w_1, \ldots, w_N\} \to G \end{align*}$$

be a function such that the image of $\mu $ generates G, which additionally satisfies

$$\begin{align*}\sum_{i = 1}^{N} \mu(w_i) = 0, \end{align*}$$

where $0 \in G$ denotes the identity element. For any integer $n \geq 0$ , we can extend $\mu $ to a function

$$\begin{align*}\{1, \ldots, n\} \cup \{w_1, \ldots, w_N\} \to G \end{align*}$$

by setting the image of each $i \in \{1, \ldots , n\}$ to be $0$ ; for ease of notation, we will also call this extension $\mu $ . We set the notation

$$\begin{align*}\overline{\mathcal{M}}_{0, n + N}^G(\mu):= \overline{\mathcal{M}}_{0, \{1, \ldots, n \} \cup \{w_1, \ldots, w_N \}}^{G}(\mu) \end{align*}$$

and define $\mathcal {M}_{0, n + N}^G(\mu )$ similarly. We now define an intermediate locus

$$\begin{align*}\mathcal{M}_{0, n + N}^G(\mu) \subset \widetilde{\mathcal{M}}_{0, n + N}^G(\mu) \subset \overline{\mathcal{M}}_{0, n + N}^G(\mu) \end{align*}$$

in analogy with the space $\widetilde {\mathcal {H}}_{g, n}$ of n-marked hyperelliptic curves of genus g together with a labeling of their Weierstrass points, considered in §4.1. Given a graph-theoretic pointed admissible G-cover $\mathbf {P} \to \mathbf {C} \in \mathrm {Ob}(\Gamma _{0, n+N}^G(\mu ))$ , where $\Gamma _{0, n+N}^{G}(\mu )$ is the category defined in Definition 3.3, we say that $\mathbf {P} \to \mathbf {C}$ is forbidden if all of the following conditions hold:

1. (a) $|E(\mathbf {C})| = 1$ ,

2. (b) If we erase all of the legs labeled by $\{1, \ldots , n\}$ from $\mathbf {C}$ , the resulting $\{w_1, \ldots , w_N\}$ -marked tree is not stable in the sense of Definition 3.1, and

3. (c) the source graph $\mathbf {P}$ has no vertices supporting repeated markings among $\{1, \ldots , n\}$ .

Each forbidden cover $\mathbf {P} \to \mathbf {C}$ corresponds uniquely to a boundary divisor of $\overline {\mathcal {M}}_{0, n + N}^G(\mu )$ , and we define $\widetilde {\mathcal {M}}_{0, n + N}^G(\mu )$ to be the complement in $\overline {\mathcal {M}}_{0, n + N}^G(\mu )$ of those boundary divisors which are not forbidden.

When $G = \mathbb {Z}/2\mathbb {Z} = \{0, 1\}$ , $N = 2g + 2$ , and $\mu (w_i) = 1$ for all i, the forbidden divisors are precisely those of type (1) and (2) in Definition 4.1, and we have

$$\begin{align*}\widetilde{\mathcal{M}}_{0, n + 2g + 2}^{\mathbb{Z}/2\mathbb{Z}}(\mu) \cong \widetilde{\mathcal{H}}_{g, n}. \end{align*}$$

For general G and $\mu $ , the space $\widetilde {\mathcal {M}}_{0, n + N}^G(\mu )$ can be identified with the moduli space of smooth N-pointed admissible G-covers of $\mathbb {P}^1$ , with monodromy specified by $\mu $ , together with n distinct marked points on the source curve. This space admits an $S_n$ -action given by permuting the n marked points on the source, and the isomorphism with $\widetilde {\mathcal {M}}_{0, n + N}^G(\mu )$ is $S_n$ -equivariant.

The dual complex $\widetilde {\Theta }_{0, n+N}^G(\mu )$ of the normal crossings compactification

$$\begin{align*}\widetilde{\mathcal{M}}_{0, n + N}^G(\mu) \subset \overline{\mathcal{M}}_{0, n + N}^G(\mu) \end{align*}$$

is the subcomplex of $\Delta _{0, n+ N}^G(\mu )$ of those simplices which have no forbidden vertices. The analogue of Theorem 5.5 holds for $\widetilde {\Theta }_{0, n+N}^G(\mu )$ : the subcomplex parameterizing graph-theoretic admissible G-covers $\mathbf {P} \to \mathbf {C}$ where $\mathbf {P}$ has a repeated marking is acyclic. Our proof of Theorem 5.5 carries through to this setting, mutatis mutandis. In Remark 6.6 below, we explain how this leads to a generalization of Theorem A for these spaces.