## 1. Introduction

A *(separated) orbispace* [Reference HaefligerHae90, Reference BehrendBeh14] is a topological stack $X$ which admits a cover by open substacks of the form $Y/\Gamma$ (where $\Gamma \curvearrowright Y$ is a continuous action of a finite group on a topological space) and whose diagonal $X\to X\times X$ is proper (see § 3 for background on topological stacks). Familiar examples of orbispaces include orbifolds, graphs of groups, complexes of groups, and (the analytifications of) separated Deligne–Mumford algebraic stacks over $\mathbb {C}$.

An interesting question to ask about a given orbispace $X$ is whether there exists a global presentation $X=Y/G$ for $G$ a compact Lie group; let us call such an orbispace a *global quotient*. There are a number of known conditions which imply an orbispace is a global quotient. If $X=Y/\Gamma$ for a (possibly infinite) discrete group $\Gamma$, then $X$ is a global quotient by Lück and Oliver [Reference Lück and OliverLO01, Corollary 2.7]. Every (paracompact) smooth *effective* $n$-dimensional orbifold is a global quotient (of its orthonormal frame bundle by $O(n)$), and it is an old question whether every (not necessarily effective) smooth orbifold is a global quotient. A sufficient criterion for an orbifold to be a global quotient was given by Henriques and Metzler [Reference Henriques and MetzlerHM04]. Henriques [Reference HenriquesHen05] conjectured that every compact orbispace is a global quotient, however other experts have expressed skepticism that such a general result would be true [Reference SiebertSie98, § 6.4]. The analogous question for algebraic stacks has been studied by Edidin *et al.* [Reference Edidin, Hassett, Kresch and VistoliEHKV01] and Totaro [Reference TotaroTot04]. It is a result of Kresch and Vistoli [Reference Kresch and VistoliKV04, Theorem 2], [Reference KreschKre09, Theorem 4.4] (using a result of Gabber [Reference de JongdJ03], [Reference Colliot-Thélène and SkorobogatovCTS21, Chapter 4]) that smooth separated Deligne–Mumford stacks over $\mathbb {C}$ with quasi-projective coarse moduli space are global quotients.

Our main result implies that all orbispaces satisfying very mild hypotheses are global quotients. In particular, all compact orbispaces are global quotients.

Theorem 1.1 Let $X$ be an orbispace, with isotropy groups of bounded order, whose coarse space $|X|$ is coarsely finite-dimensional (every open cover has a locally finite refinement with finite-dimensional nerve). Then there exists a complex vector bundle $V$ of rank $n>0$ over $X$, whose fiber over every $x\in X$ is isomorphic to a direct sum of copies of the regular representation of the isotropy group $G_x$. We may take $n=n(d,m)$ if $|X|$ is $d$-dimensional (every open cover has a locally finite refinement with nerve of dimension $\leqslant d$) and has isotropy groups of order $\leqslant m$.

(Note that the ‘real’ and ‘complex’ versions of Theorem 1.1 are equivalent, by tensoring from $\mathbb {R}$ to $\mathbb {C}$ and by forgetting from $\mathbb {C}$ to $\mathbb {R}$.)

The proof of Theorem 1.1 is split into two parts. In § 2, we prove Theorem 1.1 for any orbispace presented by a simplicial complex of groups; this special case carries the essential topological content of the result. Due to one particular step in this proof, we have no explicit bound on the rank $n(d,m)$ of the vector bundles proven to exist. In § 4, we deduce Theorem 1.1 in general by showing that every paracompact orbispace admits a representable map to a simplicial complex of groups. This result (Proposition 4.9) is likely of independent interest; it is the analogue of mapping a paracompact Hausdorff space to the nerve of an open covering using a partition of unity (it is, thus, worthy of note that its proof is not trivial).

The following immediate corollaries of Theorem 1.1 are derived in § 5.

Corollary 1.2 For $X$ as in Theorem 1.1, we have $X=P/U(n)$ for a space $P$.

Corollary 1.3 Every paracompact smooth orbifold $X$ of dimension $\leqslant d$ with isotropy groups of order $\leqslant m$ is the quotient $X=P/U(n)$ of a smooth manifold $P$ by a smooth action of the compact Lie group $U(n)$, where $n=n(d,m)$.

It seems that Theorem 1.1 does not help resolve the question of whether every separated Deligne–Mumford stack of finite type over $\mathbb {C}$ is a global quotient; the question of whether the vector bundles produced by Theorem 1.1 on its analytification are algebraic (or even analytic) seems difficult.

In § 6, we prove the following corollary of Theorem 1.1, to which the titular phrase ‘having enough vector bundles’ refers.

Corollary 1.4 Let $X\to Y$ be a representable map of orbispaces satisfying the hypothesis of Theorem 1.1. Every vector bundle on $X$ of bounded rank embeds into the pullback of a vector bundle of bounded rank on $Y$.

It is well known that Corollary 1.4 is the key statement needed to show that the $K$-theory of finite rank vector bundles on orbispaces satisfies excision and exactness and is, thus, a cohomology theory (see [Reference Lück and OliverLO01, § 3] and [Reference HenriquesHen05, § 6.3]). We elaborate on this assertion in § 6, using the suggestive reformulation of Corollary 1.4 as the statement that pullback of vector bundles is *cofinal*. The $K$-theory of finite-rank vector bundles should agree (for reasonable orbispaces) with the other standard models of $K$-theory for orbispaces, such as using bundles of Fredholm operators [Reference SegalSeg70, Reference MatumotoMat71, Reference AtiyahAti69, Reference Atiyah and SingerAS69] or using orthogonal spectra [Reference SchwedeSch18, §§ 6.3–6.4] (compare with Remark 1.6).

Remark 1.5 Another known (to experts) consequence of Corollary 1.4 (which we do not explain in detail) is that every paracompact quasi-smooth derived smooth orbifold with tangent and obstruction spaces of dimension $\leqslant d$ and isotropy groups of order $\leqslant m$ is the derived zero set of a smooth section of a vector bundle of rank $\leqslant n$ over a smooth orbifold of dimension $\leqslant n=n(d,m)$. (‘Quasi-smooth’ means locally isomorphic to the derived zero set of a smooth section of a smooth vector bundle over a smooth orbifold.)

Remark 1.6 Combined work of Schwede [Reference SchwedeSch20] and Gepner and Henriques [Reference Henriques and GepnerHG07] established an equivalence between orthogonal spaces up to global equivalence (with respect to the ‘global family’ of all finite groups) and certain categories of ‘cellular’ topological stacks (which include, but are more general than, what we call orbispaces here). The vector bundles produced by Theorem 1.1 allow for a concrete description of the functor from orbispaces to orthogonal spaces (compare [Reference SchwedeSch18, Definition 1.1.27]). Let $X$ be an orbispace, and let $E$ be any *faithful* vector bundle over $X$ (meaning the fibers of $E$ are faithful representations of the isotropy groups of $X$). The orthogonal space corresponding to $X$ is given by

where $\operatorname {Emb}_X(E,\underline {V})$ denotes the total space of the bundle of embeddings of $E$ into $V$ (note that $\operatorname {Emb}_X(E,\underline {V})$ is a space because $E$ is faithful).

Schwede [Reference SchwedeSch18] also associates to every orthogonal spectrum a cohomology theory on orthogonal spaces, hence on orbispaces. Given an orthogonal spectrum $A$ and an orbispace $X$ which admits faithful vector bundles, the degree-zero $A$-cohomology of $X$ is (in view of the above) given by the direct limit over vector bundles $E$ over $X$ of the set of homotopy classes of sections of the fibration $\Omega ^EA(E)\to X$. More generally, we may consider the mapping spectrum $F(X,A)$ defined by

whose stable homotopy groups are the $A$-cohomology groups of $X$. If $A$ is a global $\Omega$-spectrum [Reference SchwedeSch18, Definition 4.3.8], then this direct limit is achieved at any faithful $E$, and the above definition of $F(X,A)$ is an $\Omega$-spectrum. Let us also propose a possible definition of the $A$-homology groups of $X$ as the stable homotopy groups of the spectrum

where $|\cdot |$ indicates taking the coarse space of the total space of $\Omega ^EA(E\oplus \mathbb {R}^n)$ over $X$.

Remark 1.7 It is natural to ask to what extent Theorem 1.1 may be generalized to the case of ‘Lie orbispaces’ (topological stacks locally modelled on $Y/G$ for $G$ a compact Lie group). The naive generalization is simply false: there are purely ineffective Lie orbispaces with isotropy group $S^1$ and coarse space $S^3$ which have no finite rank faithful vector bundles [Reference TotaroTot04, § 2]. It is, however, reasonable to conjecture that the proof of Theorem 1.1 could be generalized to prove that the existence of enough vector bundles on a Lie orbispace is a purely cohomological question.

## 2. The main construction

This section is devoted to proving Theorem 1.1 for orbispaces which are presented by a simplicial complex of groups. This special case (stated as Theorem 2.13) carries the essential topological content of Theorem 1.1. For general background on orbispaces and topological stacks, we refer the reader to § 3.

We begin by describing the basic idea of the proof, which we then implement in detail. Our orbispace $X$ comes with a filtration by skeleta

where $X_k$ is obtained from $X_{k-1}$ by attaching cells of the form $(D^k,\partial D^k)\times \mathbb {B} G$ for finite groups $G$ (here $\mathbb {B} G=*/G$ is the orbispace quotient). We construct the desired vector bundle by induction on skeleta.

A direct implementation of this strategy runs immediately into the following obstruction. A (complex) vector bundle $V$ on $\partial D^k\times \mathbb {B} G$ decomposes canonically as $V=\bigoplus _{\rho \in \hat G}V_\rho \otimes \rho$ for vector bundles $V_\rho$ on $\partial D^k$ indexed by the complex irreducible representations $\rho$ of $G$. Thus, $V$ is classified by an element of $\prod _{\rho \in \hat G}\pi _{k-1}(\coprod _nBU(n))$, which vanishes if and only if $V$ extends to $D^k\times \mathbb {B} G$. We seek to detect these obstructions using Chern characters. According to Bott periodicity, the homotopy groups of $BU$ are given by

and the Chern character of a generator of $\pi _{2i}(BU)$ is nonzero. Now the Chern character of $V=\bigoplus _{\rho \in \hat G}V_\rho \otimes \rho$ is given by $\operatorname {ch}(V)=\sum _{\rho \in \hat G}\operatorname {ch}(V_\rho )\dim \rho$, so triviality of $\operatorname {ch}(V)$ does not imply triviality of each $\operatorname {ch}(V_\rho )$ or of the aforementioned obstructions to extending $V$.

To capture the information we need about $V$, we consider the more refined characteristic class which we call the *inertial Chern character* Footnote ^{1} $\operatorname {ch}^I$ (studied previously by Adem and Ruan [Reference Adem and RuanAR03]), which is a cohomology class on the *inertia stack*

In local coordinates $X=Y/G$, we have $IX=\bigl (\bigsqcup _{g\in G}Y^g\bigr )/G$. In such coordinates, the inertial Chern character is defined by recalling the Chern–Weil description of the usual Chern character $\operatorname {ch}(V)=\operatorname {tr}\exp (\Omega /(-2\pi i))$, for $\Omega$ the curvature of a hermitian connection on $V$, and writing

Now the inertia stack of $\partial D^k\times \mathbb {B} G$ is $\partial D^k\times G/G$ (quotient by the conjugation action), and the inertial Chern character of $V=\bigoplus _{\rho \in \hat G}V_\rho \otimes \rho$ is given by $\operatorname {ch}^I(V)=\sum _{\rho \in \hat G}\operatorname {ch}(V_\rho )\chi _\rho$. Thus, if $\operatorname {ch}^I(V)$ is trivial, then so is each $\operatorname {ch}(V_\rho )$ by linear independence of characters, and hence $V$ extends to $D^k\times \mathbb {B} G$ as desired.

We are thus led to consider the modified problem of constructing a vector bundle on $X$ with the desired fibers and with trivial inertial Chern character. It suffices to show that if $X_{k-1}$ admits such a vector bundle, then so does $X_k$. The vanishing of $\operatorname {ch}^I(V)\in H^*(IX_{k-1})$ guarantees that the obstructions to extending $V$ to $X_k$ vanish, by the above discussion. It thus remains to show that this extension can be taken to have trivial inertial Chern character. The inertial Chern character of any extension is an element of $\ker (H^*(IX_k)\to H^*(IX_{k-1}))$. By the long exact sequence

it is thus in the image of $H^*(IX_k,IX_{k-1})$. The inertial Chern character is an even degree class, so we may assume that $k$ is even. By modifying how we extend $V$ from $X_{k-1}$ to $X_k$ by an element of $\prod _{\rho \in \hat G}\pi _k(BU)$ over a given $k$-simplex, we can shift its inertial Chern character by any integral linear combination of characters in $\operatorname {Hom}(G/G,\mathbb {C})\subseteq H^*(IX_k,IX_{k-1})$ (for $G$ the isotropy group of that given $k$-simplex). By replacing $V$ with $V^{\oplus a}$, we can multiply its inertial Chern character by any positive integer $a$. A combination of these two operations suffice to kill the inertial Chern character, provided that it is $\chi$*-rational* (rational with respect to a certain $\mathbb {Q}$-structure on $H^*(IX)$ differing from the usual one). This rationality is not at all obvious given the transcendental definition we give of the inertial Chern character, but it is true, and thus the proof is complete.

The remainder of this section is devoted to making the above outline precise. We begin by recalling the definition of a simplicial complex of groups (see also Haefliger [Reference HaefligerHae91], Corson [Reference CorsonCor92], or Bridson and Haefliger [Reference Bridson and HaefligerBH99]).

Definition 2.1 A *simplicial complex of groups* is a pair $(Z,G)$ consisting of a simplicial complex $Z$ together with the following data.

• For every simplex $\sigma \subseteq Z$, a group $G_\sigma$.

• For every pair of simplices $\sigma \subseteq \tau$, an

*injective*group homomorphism $G_\tau \hookrightarrow G_\sigma$.• For every triple of simplices $\rho \subseteq \sigma \subseteq \tau$, an element of $G_\rho$ conjugating the inclusion $G_\tau \hookrightarrow G_\rho$ to the composition of inclusions $G_\tau \hookrightarrow G_\sigma \hookrightarrow G_\rho$.

• For every quadruple of simplices $\pi \subseteq \rho \subseteq \sigma \subseteq \tau$, the resulting product of elements of $G_\pi$ conjugating $G_\tau \hookrightarrow G_\pi$ to $G_\tau \hookrightarrow G_\rho \hookrightarrow G_\pi$ to $G_\tau \hookrightarrow G_\sigma \hookrightarrow G_\rho \hookrightarrow G_\pi$ to $G_\tau \hookrightarrow G_\sigma \hookrightarrow G_\pi$ and back to $G_\tau \hookrightarrow G_\pi$ must be the identity element of $G_\pi$.

Injectivity of each map $G_\tau \hookrightarrow G_\sigma$ will ensure that the geometric realization of $(Z,G)$ is an orbispace, rather than some sort of more exotic topological stack.

A simplicial complex of groups $(Z,G)$ presents an orbispace $\|(Z,G)\|$ called its *geometric realization*. A precise definition of this geometric realization is given in § 4. For now, it will suffice to know that the coarse space of $\|(Z,G)\|$ is (the geometric realization of) $Z$ itself, and that over the open star $\operatorname {st}(\sigma )\subseteq Z$ of a simplex $\sigma \subseteq Z$, the geometric realization $\|(Z,G)\|$ is given by the orbispace quotient

where the pieces for $\tau \supseteq \tau '\supseteq \sigma$ are glued together via the map $G_\tau \setminus G_\sigma \to G_{\tau '}\setminus G_\sigma$ induced by the map $G_\tau \to G_{\tau '}$ and the element of $G_\sigma$ conjugating the composition $G_\tau \to G_{\tau '}\to G_\sigma$ to the map $G_\tau \to G_\sigma$.

We often abuse terminology and say ‘simplicial complex of groups’ when we really mean its geometric realization. Thus, we refer to $X=\|(Z,G)\|$ as a simplicial complex of groups. Its coarse space $|X|$ is the geometric realization of $Z$.

To apply the methods of differential topology to a given simplicial complex of groups, we fix a family of (germs of) smooth retractions $\tau \to \sigma$ for every pair of simplices $\sigma \subseteq \tau$ such that the maps $\tau \to \sigma \to \rho$ and $\tau \to \rho$ agree for $\rho \subseteq \sigma \subseteq \tau$ (such a family of smooth retractions may be constructed by induction). Given this data, objects of differential topology on $\tau$ (functions, differential forms, bundles, connections, etc.) are required to be pulled back under $\tau \to \sigma$ in a(n unspecified) neighborhood of every $\sigma \subseteq \tau$. For bundles, this requirement consists of the *data* of a compatible family of isomorphisms with the pullback bundles (one could equivalently consider only vector bundles built out of transition functions which satisfy the given pullback conditions).

In particular, the de Rham complex $\Omega ^*(X)$ of a simplicial complex of groups $X$ is defined, and it coincides with the de Rham complex of its coarse space $\Omega ^*(|X|)$. There is a natural map $\Omega ^*(|X|)\to C^*(|X|;\mathbb {R})=C^*(X;\mathbb {R})$ from the de Rham complex to the simplicial cochain complex over $\mathbb {R}$, given by integrating differential forms over oriented simplices. It is a standard fact that this map is a quasi-isomorphism (proof: filter $X$ by skeleta and invoke the five lemma to reduce to showing that $\Omega ^*(\Delta ^k,\partial \Delta ^k)\to C^*(\Delta ^k,\partial \Delta ^k;\mathbb {R})$ is a quasi-isomorphism, which follows from the Poincaré lemma).

### Definition 2.2 (Chern character)

Let $X$ be a simplicial complex of groups, and let $V\to X$ be a complex vector bundle. Given a hermitian metric $\mu$ and hermitian connection $\theta$ on $V$, the curvature of $\theta$ is a $2$-form $\Omega (V,\theta )$ valued in $\mathfrak u(V,\mu )\subseteq \mathfrak {gl}(V)=\operatorname {End}(V)$, and the Chern character form

is closed. Its class in cohomology $\operatorname {ch}(V)\in H^\mathrm {even}(X;\mathbb {R})$ is called the Chern character of $V$. This class is independent of $\mu$ and $\theta$ by an interpolation argument (interpolate on $X\times [0,1]$ between any two metric/connection pairs on $X\times 0$ and $X\times 1$, and use the fact that $H^*(X\times [0,1])=H^*(X)$).

For any stack $X$, its *inertia stack* is the stack

When $X=Y/G$, we have $IX=\bigl (\bigsqcup _{g\in G}Y^g\bigr )/G$. In particular, the local description of the geometric realization of a simplicial complex of groups provides a local description of its inertia stack as well. In fact, if $X$ is a simplicial complex of groups then so is $IX$: a simplex of $IX$ is a pair $(\sigma,[g])$ where $\sigma \subseteq X$ is a simplex and $[g]\subseteq G_\sigma$ is a conjugacy class, and $G_{(\sigma,[g])}$ is the centralizer of any element of the conjugacy class $[g]\subseteq G_\sigma$, etc.

### Definition 2.3 (Inertial Chern character)

Let $X$ be a simplicial complex of groups, and let $V\to X$ be a complex vector bundle. The inertial Chern character $\operatorname {ch}^I(V)\in H^\mathrm {even}(IX;\mathbb {C})$ is represented by the closed form

for any choice of hermitian metric and connection $\theta$ on $V$. Closedness can be seen by splitting the pullback of $V$ to $IX$ into isotypic pieces for the action of the cyclic group generated by $g$ and observing that the contribution of each such piece is a (usual) Chern character form. Independence of the metric and connection follows from the same interpolation argument as before.

Example 2.4 If $X=\mathbb {B} G$, a vector bundle over $X$ is simply a representation of $G$, the inertia stack $IX=G/G$ is the stack quotient of the conjugation action of $G$ on itself, and the inertial Chern character $\operatorname {ch}^I(V):G/G\to \mathbb {C}$ is the character $g\mapsto \operatorname {tr}(g|V)$ of $V$ regarded as a representation of $G$.

Example 2.5 If $X=Y\times \mathbb {B} G$ for a smooth manifold $Y$ and $V=\bigoplus _{\rho \in \hat G}V_\rho \otimes \rho$, then $IX=Y\times (G/G)$ and the inertial Chern character $\operatorname {ch}^I(V):G/G\to H^*(Y;\mathbb {C})$ is given by $\operatorname {ch}^I(V)=\sum _\rho \operatorname {ch}(V_\rho )\chi _\rho$. As the characters $\chi _\rho$ of the irreducible representations $\rho$ of $G$ form a basis for the space of maps $G/G\to \mathbb {C}$, we see that for $X=Y\times \mathbb {B} G$, the inertial Chern character determines (and is determined by) the Chern characters of each of the associated bundles $V_\rho$.

Example 2.6 The degree-zero part of the inertial Chern character $\operatorname {ch}_0^I(V)\in H^0(IX;\mathbb {C})$ records the characters of the fibers of $V$, regarded as representations of the isotropy groups $G_x$ of the points of $X$.

Despite admitting the aforementioned transcendental definition in terms of differential forms, the (usual) Chern character $\operatorname {ch}(V)\in H^*(X;\mathbb {R})$ is well known be rational, i.e. it lies in the subspace $H^*(X;\mathbb {Q})\subseteq H^*(X;\mathbb {R})$. The inertial Chern character does not always lie in the subspace $H^*(IX;\mathbb {Q})\subseteq H^*(IX;\mathbb {C})$, rather it is rational with respect to a different $\mathbb {Q}$-structure, which we now introduce.

### Definition 2.7 ($\chi$-integral cohomology of $IX$)

Regard the simplicial cochain group $C^*(IX;\mathbb {C})$ as the group of simplicial cochains on $X$ with coefficients in $\operatorname {Hom}(G_x/G_x,\mathbb {C})$. In this description, let us replace $\operatorname {Hom}(G_x/G_x,\mathbb {C})$ with its subspace of integral (respectively, rational) linear combinations of characters of $G_x$. Equivalently, we consider the complex of simplicial cochains on $IX$ which act on $\sigma \times (G_\sigma /G_\sigma )\subseteq IX$ (for any simplex $\sigma \subseteq X$) by an integral (respectively, rational) linear combination of characters of $G_\sigma$. We denote the resulting cohomology groups by $H^*(IX;\mathbb {Z}_\chi )$ (respectively, $H^*(IX;\mathbb {Q}_\chi )$), which are evidently functorial in $X$. We say that an element of $H^*(IX;\mathbb {C})$ is $\chi$*-integral* (respectively, $\chi$*-rational*) to mean that it lies in the image of $H^*(IX;\mathbb {Z}_\chi )$ (respectively, in the subspace $H^*(IX;\mathbb {Q}_\chi )$).

Example 2.8 If $X=Y\times \mathbb {B} G$, then $IX=Y\times (G/G)$ and $H^*(IX;\mathbb {Z}_\chi )=H^*(X;\mathbb {Z}[\hat G])$. Thus, $\operatorname {ch}^I(V)=\sum _\rho \operatorname {ch}(V_\rho )\chi _\rho$ is $\chi$-rational because each $\operatorname {ch}(V_\rho )$ is rational.

As the coefficient systems on $X$ appearing in Definition 2.7 are finite free modules over $\mathbb {Z}$, we may dualize to define homology groups $H_*(IX;\mathbb {Z}_\chi )$ as well.

Lemma 2.9 There are canonical isomorphisms

and a canonical short exact sequence

Proof. The complex $C_*(IX;\mathbb {Z}_\chi )$ is degreewise free, and the complexes $C^*(IX;\mathbb {Z}_\chi )$, $C^*(IX;\mathbb {Q}_\chi )$, and $C^*(IX;\mathbb {C})$ are obtained from it by applying the functors $\operatorname {Hom}(-,\mathbb {Z})$, $\operatorname {Hom}(-,\mathbb {Q})$, and $\operatorname {Hom}(-,\mathbb {C})$, respectively.

### Proposition 2.10 ($\chi$-rationality of $\operatorname {ch}^I$)

Let $X$ be a simplicial complex of groups of order $\leqslant m$. The inertial Chern character of any vector bundle $V/X$ is $\chi$-rational. In fact, there exists a positive integer $N=N(m,d,\dim V)$ such that $N\cdot \operatorname {ch}^I_d(V)$ is $\chi$-integral.

It is worth remarking that the proof we give does not provide an explicit expression for $N(m,d,n)$ other than for $n=1$.

Proof. We first show that the inertial Chern character of any line bundle $L/X$ is $\chi$-rational. We have $\operatorname {tr}(ab)=\operatorname {tr}(a)\operatorname {tr}(b)$ for endomorphisms $a,b$ of $\mathbb {C}$, so the inertial Chern character of a line bundle $L$ splits as the product

where $\operatorname {ch}_0^I(L)\in H^0(IX;\mathbb {Z}_\chi )$ is the fiberwise character of $L$ and $\operatorname {ch}_1(L)\in H^2(X;\mathbb {C})$ is the Chern character in degree one. It thus suffices to show that $\operatorname {ch}_1(L)$ is rational. We have $k\operatorname {ch}_1(L)=\operatorname {ch}_1(L^{\otimes k})$, so it suffices to show that $\operatorname {ch}_1(L^{\otimes k})$ is integral for some positive integer $k$. As all isotropy groups of $X$ have order $\leqslant m$, the tensor power $L^{\otimes m!}$ descends to a line bundle $M$ on the coarse space. As $M$ is an ordinary line bundle over a space (rather than an orbispace), its first Chern class $c_1(M)=\operatorname {ch}_1(M)$ is integral. We have thus shown that $\operatorname {ch}^I(L)$ is $\chi$-rational (in fact, we have shown that $d!(m!)^d\operatorname {ch}^I_d(L)$ is $\chi$-integral).

To treat the case of vector bundles of dimension greater than one, we use the *splitting principle*. Given a vector bundle $V/X$ with hermitian metric, let $\mathbb {F}(V)\to X$ denote the fibration whose fiber over $x\in X$ is the space of decompositions of $V_x$ into ordered orthogonal one-dimensional subspaces. In other words, $\mathbb {F}(V)=P/U(1)^n$ where $P\to X$ is the principal $U(n)$-bundle associated to $V/X$ with its chosen metric. We claim that the pullback map

is injective. The fiber of $I\mathbb {F}(V)\to IX$ over a given point $(x,g)\in IX$ is the space of $g$-invariant ordered decompositions of $V_x$ into one-dimensional subspaces. As the group generated by $g$ is abelian, its irreducible representations are one-dimensional, and hence there are plenty of such decompositions which are $g$-invariant, namely when each one-dimensional subspace is contained in some $g$-isotypic piece of $V_x$. Thus, $I\mathbb {F}(V)\to IX$ is a disjoint union of iterated projective space bundles, from which the desired injectivity statement follows (for any projective space bundle $\mathbb {P}(W)\to Z$ over an orbispace $Z$, the pullback map $H^*(Z)\to H^*(\mathbb {P}(W))$ is split by the map $\alpha \mapsto \int (c_1(L)^{\dim W-1}\cup \alpha )$ where $L/\mathbb {P}(W)$ denotes the tautological line bundle and $\int :H^{*+2(\dim W-1)}(\mathbb {P}(W))\to H^*(Z)$ denotes fiberwise integration).

As $H^*(IX;\mathbb {C})\to H^*(I\mathbb {F}(V);\mathbb {C})$ is injective and $\operatorname {Hom}_\mathbb {Z}(-,\mathbb {C})$ is exact, we conclude from Lemma 2.9 that $\operatorname {Hom}(H_*(IX;\mathbb {Z}_\chi )/H_*(I\mathbb {F}(V);\mathbb {Z}_\chi ),\mathbb {C})=0$, and hence that the domain is torsion. From this and Lemma 2.9 again, it follows that a class in $H^*(IX;\mathbb {C})$ is $\chi$-rational if and only if its pullback to $H^*(I\mathbb {F}(V);\mathbb {C})$ is $\chi$-rational. Now consider the pullback of $V$ to $\mathbb {F}(V)$. This pullback splits as a direct sum of line bundles, so because the inertial Chern character is additive under direct sum, we conclude that the pullback of $\operatorname {ch}^I(V)$ to $I\mathbb {F}(V)$ is $\chi$-rational, and hence that $\operatorname {ch}^I(V)$ is itself $\chi$-rational.

It remains to produce an integer $N=N(m,d,n)$ such that $N\cdot \operatorname {ch}^I_d(V)$ is $\chi$-integral for $\dim V=n$. We claim that there exists a *finite* orbi-complex $\mathbb {B}_{m,d}U(n)$ carrying a principal $U(n)$-bundle

with the property that every principal $U(n)$-bundle over a simplicial complex of groups $X$ of dimension $\leqslant d$ and isotropy $\leqslant m$ is a pullback of $\mathbb {E}_{m,d}U(n)\to \mathbb {B}_{m,d}U(n)$. As the inertial Chern character of $\mathbb {E}_{m,d}U(n)\to \mathbb {B}_{m,d}U(n)$ is $\chi$-rational and $\mathbb {B}_{m,d}U(n)$ is finite, there exists an integer $N=N(m,d,n)$ such that $N$ times this inertial Chern character is $\chi$-integral in cohomological degree $d$. By pullback, the same integer $N$ works for any vector bundle of rank $n$ over a simplicial complex of groups of order $\leqslant m$. Note that this argument gives no explicit bound on the integer $N$.

It remains to construct $\mathbb {E}_{m,d}U(n)\to \mathbb {B}_{m,d}U(n)$. We seek $U(n)$-spaces $\mathbb {E}_{m,d}U(n)$ such that

where $G\subseteq U(n)$ ranges over all finite subgroups of $U(n)$ of order $\leqslant m$. Note that there are finitely many conjugacy classes of such subgroups, and it suffices to consider just one representative of each conjugacy class. To show that $\mathbb {E}_{m,d}U(n)$ exists given $\mathbb {E}_{m,d-1}U(n)$, argue as follows. We have $\pi _{d-1}((\mathbb {E}_{m,d-1}U(n))^G)=H_{d-1}((\mathbb {E}_{m,d-1}U(n))^G)$ by Hurewicz, and the latter group is finitely generated because $\mathbb {E}_{m,d-1}U(n)$ is made up of finitely many cells. We define $\mathbb {E}_{m,d}U(n)$ by attaching cells $(D^d,\partial D^d)\times U(n)/G$ along a choice of finitely many generators of $\pi _{d-1}((\mathbb {E}_{m,d-1}U(n))^G)$, for each of the finitely many subgroups $G\subseteq U(n)$ on our fixed set of representatives. Note that $(D^d\times U(n)/H)^G=D^d\times \{a\in U(n)\,|\,a^{-1}Ga\subseteq H\}$, so any map $S^r\to (\mathbb {E}_{m,d}U(n))^G$ for $r< d$ can be homotoped to land inside $(\mathbb {E}_{m,d-1}U(n))^G$. It follows that the homotopy groups of $(\mathbb {E}_{m,d}U(n))^G$ satisfy the desired vanishing property.

Let us show that any principal $U(n)$-bundle $P\to X$ is pulled back from $\mathbb {E}_{m,d}U(n)\to \mathbb {B}_{m,d}U(n)$ when $\dim X\leqslant d$ with isotropy groups of order $\leqslant m$. That is, we should construct an equivariant map $P\to \mathbb {E}_{m,d}U(n)$ (note that since the target is a space, this is the same as an equivariant map from the coarse space $|P|$). By induction on the cells of $X$, it suffices to solve the extension problem for equivariant maps from $(D^r,\partial D^r)\times U(n)/G$ to $\mathbb {E}_{m,d}U(n)$ for $r\leqslant d$ and $|G|\leqslant m$. This is equivalent to the extension problem for maps $(D^r,\partial D^r)\to (\mathbb {E}_{m,d}U(n))^G$, whose positive solution for $r\leqslant d$ is one of the defining properties of $\mathbb {E}_{m,d}U(n)$.

Example 2.11 A previous version of this text claimed a version of Proposition 2.10 with $N$ independent of $m$. This stronger result is false; here is a counterexample. Begin with $\mathbb {B}(\mathbb {Z}/m)$ with a line bundle given by multiplication by $e^{2\pi i/m}$ on $\mathbb {C}$. Glue on a disk $D^2$ using an attaching map $\partial D^2=S^1\to \mathbb {B}(\mathbb {Z}/m)$ to a generator of $\mathbb {Z}/m$. The line bundle extends because all complex line bundles on a circle are trivial. The inertial Chern character of this line bundle has denominator $m$.

### Lemma 2.12 (Bott)

The image of the Chern character map

is precisely $H^{2n}(S^{2n};\mathbb {Z})$.

Proof. This is an immediate corollary of Bott periodicity; for completeness, we include the proof from [Reference MathewMat11]. Let $\eta /S^2$ denote the line bundle with $c_1(\eta )=1$, and let $1$ denote the trivial line bundle. According to Bott periodicity, multiplication with $\eta -1\in \tilde K^0(S^2)$ defines an isomorphism $\tilde K^0(X)\to \tilde K^0(\Sigma ^2X)$. In particular, $(\eta -1)^{\otimes n}\in \tilde K^0(S^{2n})$ is a free generator. Multiplicativity of the Chern character gives $\operatorname {ch}_n((\eta -1)^{\otimes n})=\operatorname {ch}_1(\eta -1)^n=1^n=1$.

We now have all the ingredients we need to prove the main result of this section, namely Theorem 1.1 for simplicial complexes of groups.

Theorem 2.13 Let $X$ be a $d$-dimensional simplicial complex of groups of order $\leqslant m$. There exists a complex vector bundle $V$ of rank $n=n(d,m)>0$ over $X$, whose fiber over $x\in X$ is isomorphic to a direct sum of copies of the regular representation of $G_x$.

Proof. The condition on the fibers of $V$ amounts to the assertion that $\operatorname {ch}^I_0(V)=n\chi _1$ where $\chi _1\in H^0(IX;\mathbb {Q}_\chi )$ denotes ‘the characteristic function of the identity’ as a function $G_x/G_x\to \mathbb {C}$. We construct $V$ satisfying

(meaning all higher inertial Chern characters vanish). Certainly such a vector bundle exists over the $0$-skeleton $X_0$ of $X$, with rank $m!$, because all isotropy groups have order $\leqslant m$. It therefore suffices to show that any such vector bundle on $X_{k-1}$ extends to $X_k$.

To extend $V$ as a vector bundle from $X_{k-1}$ to $X_k$ amounts to doing an extension from $\partial D^k\times \mathbb {B} G$ to $D^k\times \mathbb {B} G$ for each $k$-simplex. The obstruction to doing this lies in $\prod _{\rho \in \hat G}\pi _{k-1}(BU(n_\rho ))$, where $n_\rho =(\dim V)(\dim \rho )/|G|$. The map $BU(n_\rho )\to BU$ is an isomorphism on $\pi _{k-1}$ provided $k-1<2n_\rho +2$, which is guaranteed by taking say $\dim V>dm$. The Chern character detects $\pi _*(BU)$ by Lemma 2.12, so vanishing of the inertial Chern character and linear independence of characters means that these obstructions all vanish. Thus, $V$ extends to $X_k$.

It remains to show that the extension of $V$ to $X_k$ can be taken to have trivial inertial Chern character. The space of extensions over a given $k$-simplex is a torsor for $\prod _{\rho \in \hat G}\pi _k(BU(n_\rho ))$, which is again $\prod _{\rho \in \hat G}\pi _k(BU)$ once we take $\dim V>dm$. We use this freedom to ensure that the inertial Chern character of $V$ on $X_k$ vanishes. Consider the long exact sequence

As the inertial Chern character in $H^*(IX_k;\mathbb {Q}_\chi )$ maps to zero in $H^*(IX_{k-1};\mathbb {Q}_\chi )$, it is in the image of $H^*(IX_k,IX_{k-1};\mathbb {Q}_\chi )$. In particular, it may be nonzero only in degree $k$. Let us now replace $V$ with $V^{\oplus a}$ (which multiplies its inertial Chern character by $a$) for an appropriate positive integer $a$ so that by Proposition 2.10 its inertial Chern character is $\chi$-integral. Modifying our bundle by an element of (the product over the $k$-simplices of) $\prod _{\rho \in \hat G}\pi _k(BU)$ shifts its inertial Chern character by anything in $H^k(IX_k,IX_{k-1};\mathbb {Z}_\chi )$ by Lemma 2.12. We are done because $H^k(IX_k,IX_{k-1};\mathbb {Z}_\chi )\to H^k(IX_k;\mathbb {Z}_\chi )$ is surjective by the long exact sequence.

Remark 2.14 One may interpret the proof of Theorem 2.13 in homotopy theoretic terms as follows. Vector bundles are classified by maps to a classifying space $\mathbb {B} U(n)$, and because $\mathbb {B} U(n)$ is not contractible, the extension problem for vector bundles has nontrivial obstructions. Vector bundles with rationally trivialized inertial Chern character are classified by maps to the total space of a fibration over $\mathbb {B} U(n)$ whose fiber classifies odd-dimensional rational cohomology classes on the inertia stack. Our observation that the obstructions to this new problem are torsion is essentially the observation that this total space is rationally contractible. The definition of this fibration over $\mathbb {B} U(n)$ depends on the additivity of the Chern character (note that $\mathbb {E} U(n)\to \mathbb {B} U(n)$ is not suitable for this argument because $\mathbb {E} U(n)$ is a space, hence every bundle pulled back from $\mathbb {E} U(n)$ has trivial isotropy representations). It may prove interesting to interpret this argument within Schwede's framework of global homotopy theory [Reference SchwedeSch18].

## 3. Topological stacks

We review some basic facts about stacks (on the category of topological spaces), we give a precise definition of what we mean by an ‘orbispace’, and we establish some of their basic properties. For further background, the reader may wish to consult Noohi [Reference NoohiNoo05], Gepner and Henriques [Reference Henriques and GepnerHG07], Behrend [Reference BehrendBeh14], Metzler [Reference MetzlerMet03], Behrend and Noohi [Reference Behrend and NoohiBN06], Heinloth [Reference HeinlothHei05], or Laumon and Moret-Bailly [Reference Laumon and Moret-BaillyLMB00].

Let $\mathsf {Top}$ denote the category of topological spaces and continuous maps, and let $\mathsf {Grpd}$ denote the 2-category of (essentially) small groupoids. A *stack* is a functor $F:\mathsf {Top}^\mathsf {op}\to \mathsf {Grpd}$ which satisfies *descent*, i.e. such that for every topological space $U$ and every open cover $\{U_i\to U\}_i$, the natural functor

is an equivalence. Stacks form a 2-category, with morphisms given by natural transformations of functors. The 2-category of stacks is complete, meaning all (small) limits exist; furthermore, these limits may be calculated pointwise in the sense that $(\lim _\alpha F_\alpha )(U)=\lim _\alpha (F_\alpha (U))$. Note that, as we are working in a 2-categorical context, all functors are 2-functors, all diagrams are 2-diagrams, all limits are 2-limits, etc. (though we usually omit the prefix ‘2-’).

The Yoneda lemma implies that the Yoneda functor $X\mapsto \operatorname {Hom}(-,X)$ embeds the category of topological spaces fully faithfully into the 2-category of stacks, and moreover that the natural map from $F(X)$ to the groupoid of maps of stacks $\operatorname {Hom}(-,X)\to F(-)$ is an equivalence. The category of topological spaces is complete, and the Yoneda embedding is continuous (commutes with limits). Hence, we make no distinction between a topological space $X$ and the associated stack $\operatorname {Hom}(-,X)$ of maps to $X$, nor between objects of $F(X)$ and maps $\operatorname {Hom}(-,X)\to F(-)$ (which we simply write as $X\to F$).

Every stack $X$ has a *coarse space* $|X|$ (a topological space) which is initial in the category of maps from $X$ to topological spaces. Concretely, the points of $|X|$ are the isomorphism classes of maps $*\to X$, and a subset $U\subseteq |X|$ is open if and only if for every map $Y\to X$ from a topological space $Y$, the inverse image of $U$ is an open subset of $Y$.

A stack is called *representable* if and only if it is in the essential image of the Yoneda embedding (i.e. it is isomorphic to a topological space). A morphism of stacks $F\to G$ is called representable if and only if for every map $X\to G$ from a topological space $X$, the fiber product $F\times _GX$ is representable.

For any property $\mathcal {P}$ of morphisms of topological spaces which is preserved under pullback, a representable morphism of stacks $F\to G$ is said to have property $\mathcal {P}$ if and only if the pullback $F\times _GX\to X$ has $\mathcal {P}$ for every map $X\to G$ from a topological space $X$. The following are examples of properties $\mathcal {P}$ of morphisms $f:X\to Y$ which are preserved under pullback:

• $f$ is

*injective*;• $f$ is

*surjective*;• $f$ is

*open*, meaning that the image of any open set is open. In contrast, the property of being*closed*, meaning that the image of any closed set is closed, is*not*preserved under pullback;• $f$ is an

*embedding*, meaning that it is a homeomorphism onto its image;• $f$ is a

*closed embedding*;• $f$ is

*étale*, meaning that for every $x\in X$ there exists an open neighborhood $x\in U\subseteq X$ such that $f|_U:U\to Y$ is an open embedding;• $f$ is

*separated*, meaning that for every distinct pair $x_1,x_2\in X$ with $f(x_1)=f(x_2)$, there exist open neighborhoods $x_i\in U_i\subseteq X$ which are disjoint $U_1\cap U_2=\varnothing$ (this is equivalent to the relative diagonal $X\to X\times _YX$ being a closed embedding);• $f$ is

*universally closed*, meaning that $X\times _YZ\to Z$ is closed for every $Z\to Y$; this is equivalent to the assertion that for every $y\in Y$ and every collection of open sets $\{U_i\subseteq X\}_i$ covering $f^{-1}(y)$, there exists a finite subcollection which covers $f^{-1}(V)$ for some open neighborhood $y\in V\subseteq Y$ (one proof of this equivalence goes via yet a third equivalent condition, namely that every net $\{x_\alpha \in X\}_\alpha$ with $f(x_\alpha )\to y$ has a subnet converging to some $x\in f^{-1}(y)$);• $f$ is

*proper*, meaning that it is separated and universally closed;• $f$

*admits local sections*, meaning that there is an open cover $\{U_i\subseteq Y\}_i$ such that every restriction $f|_{f^{-1}(U_i)}:f^{-1}(U_i)\to U_i$ admits a section;• $f$ is a

*finite covering space*, meaning that there is an open cover $\{U_i\subseteq Y\}_i$ such that every restriction $f|_{f^{-1}(U_i)}:f^{-1}(U_i)\to U_i$ is isomorphic to $U_i^{\sqcup n_i}\to U_i$ for some integer $n_i\geqslant 0$.

Each of these properties $\mathcal {P}$ is also closed under composition, and thus also under fiber products, meaning that for maps $X\to X'$ and $Y\to Y'$ over $Z$, if both $X\to X'$ and $Y\to Y'$ have $\mathcal {P}$, then $X\times _ZY\to X'\times _ZY'$ also has $\mathcal {P}$ (indeed, $X\times _ZY\to X\times _ZY'$ is a pullback of $Y\to Y'$).

For any stack $X$, open (respectively, closed) embeddings $Y\hookrightarrow X$ are in natural bijection with open (respectively, closed) subsets of $|X|$.

To check that a given map of spaces satisfies one of the properties $\mathcal {P}$ above, it is often helpful to make use of the fact that these $\mathcal {P}$ are all *local on the target*, meaning that for every open cover $\{U_i\subseteq Y\}_i$, if $X\times _YU_i\to U_i$ has $\mathcal {P}$ for every $i$, then so does $X\to Y$. This leads to the following generalization for maps of stacks: if $F\to G$ is a representable morphism of stacks and $G'\to G$ is a representable morphism of stacks admitting local sections, then $F\to G$ has $\mathcal {P}$ if and only if $F\times _GG'\to G'$ has $\mathcal {P}$. In fact, in this statement we need not assume that $G'\to G$ is representable, just that it admit local sections in the generalized sense that for every map $X\to G$ from a topological space $X$, there exists an open cover $\{U_i\subseteq X\}_i$ such that each $G'\times _GU_i\to U_i$ admits a section. We thus say that ‘$\mathcal {P}$ descends along maps admitting local sections’. The same descent property holds for representability itself.

### Lemma 3.1 (Representability descends under maps admitting local sections)

Let $F\to G$ be a map of stacks, and let $G'\to G$ be a map of stacks admitting local sections. If $F\times _GG'\to G'$ is representable, then so is $F\to G$.

Proof. By replacing $F\to G$ and $G'\to G$ with their pullbacks under $U\to G$ for a topological space $U$, we may assume without loss of generality that $G$ is representable. As $G'\to G$ admits local sections, we may replace $G'\to G$ with the composition $\bigsqcup _iU_i\to G'\to G$ where $\{U_i\subseteq G\}_i$ is an open cover. Now each $F\times _GU_i$ is representable by assumption, and gluing these spaces together on their common overlaps $F\times _G(U_i\cap U_j)$ gives a topological space representing $F$.

A complex vector bundle over a stack $X$ is a representable map $V\to X$ together with maps $V\times _XV\to V$ and $\mathbb {C}\times V\to V$ (both over $X$) such that for every map $U\to X$ from a topological space $U$, there exists an open cover $\{U_i\subseteq U\}_i$ and integers $n_i\geqslant 0$ such that $V\times _XU_i\to U_i$ is isomorphic to $\mathbb {C}^{n_i}\times U_i\to U_i$ equipped with its fiberwise addition and scaling maps. A pullback of a complex vector bundle is naturally a complex vector bundle.

The class of so called *topological stacks* (those which admit a presentation via a topological groupoid) are somewhat better behaved than general stacks. A *topological groupoid* $M\rightrightarrows O$ consists of a pair of topological spaces $O$ (‘objects’) and $M$ (‘morphisms’), two maps $M\rightrightarrows O$ (‘source’ and ‘target’), a map $O\to M$ (‘identity’), an involution $M\to M$ (‘inverse’), and a map $M\times _OM\to M$ (‘composition’) satisfying the axioms of a groupoid. A topological groupoid $M\rightrightarrows O$ presents a stack $[M\rightrightarrows O]$ defined as follows. An object of $[M\rightrightarrows O](X)$ is an open cover $\{U_i\subseteq X\}_i$ together with maps $U_i\to O$ and $U_i\cap U_j\to M$ satisfying a compatibility condition, and an isomorphism in $[M\rightrightarrows O](X)$ consists of maps $U_i\cap U_{i'}'\to M$ satisfying a compatibility condition. There is a natural map $O\to [M\rightrightarrows O]$ (take the trivial open cover $\{X\subseteq X\}$) and the fiber product $O\times _{[M\rightrightarrows O]}O$ is naturally identified with $M$. The morphism $O\to [M\rightrightarrows O]$ admits local sections (by definition), so because $O\times _{[M\rightrightarrows O]}O\to O$ is representable, it follows by descent that $O\to [M\rightrightarrows O]$ is representable. Conversely, a representable map $U\to X$ admitting local sections from a topological space $U$ to a stack $X$ determines a topological groupoid $U\times _XU\rightrightarrows U$ presenting $X$. Indeed, the fiber product $U\times _XU$ is representable (because $U$ and $U\to X$ are), it admits two maps to $U$ (the two projections), an involution (exchanging the two factors), and a composition map $(U\times _XU)\times _U(U\times _XU)=U\times _XU\times _XU\to U\times _XU$ (forgetting the middle factor), and one can check using the stack property that the natural map $[U\times _XU\rightrightarrows U]\to X$ is an equivalence. A stack $X$ for which there exists a representable map $U\to X$ admitting local sections from a space $U$ is called *topological*, and such $U\to X$ is called an *atlas* for $X$.

For a topological stack $X$ with atlas $U\to X$, for any property $\mathcal {P}$ which descends along maps admitting local sections, the map $U\to X$ has $\mathcal {P}$ if and only if both maps $U\times _XU\to U$ have $\mathcal {P}$, and the diagonal $X\to X\times X$ has $\mathcal {P}$ if and only if the map $U\times _XU\to U\times U$ has $\mathcal {P}$. In particular, for any topological stack $X$, the diagonal $X\to X\times X$ is representable, and thus every map $Z\to X$ from a topological space $Z$ is representable. More generally, for any map of topological stacks $X\to Y$, the relative diagonal $X\to X\times _YX$ is representable (by descent from its pullback $V\times _XV\to V\times _YV$ for an atlas $V\to X$). If $X\to Y$ is representable and $Y$ is topological, then so is $X$ (if $U\to Y$ is an atlas for $Y$, then its pullback $V=U\times _YX\to X$ is an atlas for $X$), and the relative diagonal $X\to X\times _YX$ has $\mathcal {P}$ if and only if the relative diagonal $V\to V\times _UV$ of atlases has $\mathcal {P}$ (the latter is the pullback of the former under the map $V\times _UV\to X\times _YX$, which is representable and admits local sections because it is a pullback of $U\to Y$).

For a topological space $V$, a topological group $G$, and a continuous group action $G\curvearrowright V$, we may consider the action groupoid $G\times V\rightrightarrows V$ with source and target maps $(g,x)\mapsto x$ and $(g,x)\mapsto gx$. The stack associated to this groupoid is denoted $V/G$ and is called the stack quotient of the action $G\curvearrowright V$. If $G$ is discrete, the two maps $G\times V\to V$ are étale, so $V\to V/G$ is étale. If $G$ is compact and $V$ is Hausdorff, the map $G\times V\to V\times V$ is universally closed (factor as $G\times V\to G\times V\times V$ which is a closed embedding because $V$ is Hausdorff and $G\times V\times V\to V\times V$ which is universally closed because $G$ is compact), so $V/G$ has universally closed diagonal. If $G$ is Hausdorff, then the map $G\times V\to V\times V$ is separated, so $V/G$ has separated diagonal. Thus, if $G$ is compact Hausdorff and $V$ is Hausdorff, then $V/G$ has proper diagonal.

A groupoid presentation of a topological stack $X$ also gives a description of its coarse space $|X|$ as follows. For an atlas $U\to X$, consider the equivalence relation $\sim _X$ on $U$ given by the image of $U\times _XU\to U\times U$. There is a map $X\to U/{\sim _X}$ (which is tautological once we regard $X$ as $[U\times _XU\rightrightarrows U]$), inducing a map $|X|\to U/{\sim _X}$, which is a bijection, essentially by definition. As an open substack of $X$ pulls back to an open subset of $U$ invariant under $\sim _X$, it follows that $|X|\xrightarrow \sim U/{\sim _X}$ is open and is, thus, a homeomorphism. In particular, it follows that the coarse space $\big |V/G\big |$ of the stack quotient $V/G$ is the usual topological quotient of $V$ by $G$.

An action $G\curvearrowright V$ is called *locally trivial* if and only if $V$ admits a cover by $G$-invariant open sets $\{G\times Z_i\subseteq V\}_i$ where $G\curvearrowright G\times Z_i$ acts by left multiplication on $G$ and trivially on $Z_i$. If $G\curvearrowright V$ is locally trivial, then the natural map from the stack quotient to the topological quotient $V/G\to |V/G|$ is an equivalence. Indeed, this assertion is local on $|V/G|$, so it suffices to consider the case of $G\curvearrowright G\times Z$, where it holds by inspection.

Definition 3.2 A *(separated) orbispace* is a stack $X$ for which there exists a representable étale surjection $U\to X$ from a topological space $U$ (an ‘étale atlas’), and the diagonal $X\to X\times X$ is proper.

Proposition 3.3 A stack $X$ is an orbispace if and only if $|X|$ is Hausdorff and there exists an open cover $\{V_i/G_i\subseteq X\}_i$ where $G_i$ are finite discrete groups acting on Hausdorff spaces $V_i$.

(Similar results include [Reference BehrendBeh14, Theorem 1.108] and [Reference NoohiNoo05, Proposition 14.10].)

Proof. Let $X$ be an orbispace, and let us show that there is an open cover $\{V_i/G_i\subseteq X\}_i$. Fix an étale atlas $U\to X$, and let $u\in U$. The automorphism group $G_u:=\{u\}\times _X\{u\}\subseteq U\times _XU$ is finite and discrete because $X\to X\times X$ is proper. As the two projections $U\times _XU\rightrightarrows U$ are étale, for every $g\in G$ there exists an open neighborhood $g\in U_g\subseteq U\times _XU$ such that each projection restricted to $U_g$ is an open embedding (this gives another proof that $G$ has the discrete topology). As $G$ is finite and $U\times _XU\to U\times U$ is separated, we may take these $U_g$ to be disjoint. Now the complement of $\bigsqcup _gU_g$ is closed and disjoint from $G$, so it projects to a closed (by properness) subset of $U\times U$ disjoint from $(u,u)$. Hence, there is an open neighborhood $u\in V\subseteq U$ such that $V\times _XV\subseteq \bigsqcup _gU_g$. Thus, $V\times _XV$ is a disjoint union of pieces indexed by $G$, and each piece maps homeomorphically to $V$ under each projection. By further shrinking $V$, we may assume that the map $V\times _XV\to G$ respects composition (this is possible because composition is continuous). It follows that $V\times _XV\rightrightarrows V$ is an action groupoid $G\times V\rightrightarrows V$ for an action $G\curvearrowright V$. As $V=\{1\}\times V\hookrightarrow G\times V=V\times _XV\to V\times V$ expresses the diagonal of $V$ as a composition of proper maps, we conclude that $V$ is Hausdorff. Now the map of groupoids $(G\times V\rightrightarrows V)=(V\times _XV\rightrightarrows V)\to (U\times _XU\rightrightarrows U)$ induces a map of stacks $V/G\to X$. To see that this is an open embedding, let $V^+\subseteq U$ denote the orbit of $V$ under the morphisms $U\times _XU\rightrightarrows U$. As the projections $U\times _XU\rightrightarrows U$ are étale, it follows that $V^+\subseteq U$ is open, and hence $[V^+\times _XV^+\rightrightarrows V^+]\to [U\times _XU\rightrightarrows U]$ is an open embedding of stacks; denote this open substack by $Z\subseteq X$, so $V\to Z$ is an étale atlas. Thus, the topological groupoid $V\times _XV=V\times _ZV\rightrightarrows V$ presents $Z$, so $V/G\to X$ is an open embedding as desired.

Let us now show that for any orbispace $X$, its coarse space $|X|$ is Hausdorff. We saw earlier that $|X|$ is the quotient of $U$ by the image of $U\times _XU\to U\times U$ (which is an equivalence relation). This equivalence relation is closed because $U\times _XU\to U\times U$ is proper, so $|X|$ is Hausdorff provided the quotient map $U\to |X|$ is open. Now openness of $U\to |X|$ does not depend on which atlas $U\to X$ we take: there are étale maps $U\leftarrow U\times _XU'\to U'$ over $|X|$, which means that $U\to |X|$ is open if and only if $U'\to |X|$ is open. Moreover, openness of $U\to |X|$ can be checked locally on $|X|$, so we may assume without loss of generality that $X=V/G$. Hence, it is enough to note that the quotient map $V\to |V/G|$ (induced by the canonical étale atlas $V\to V/G$) is open. Thus, $|X|$ is Hausdorff.

Finally, let us show that if $|X|$ is Hausdorff and there is an open cover $\{V_i/G_i\subseteq X\}_i$, then $X$ is an orbispace. The maps $V_i\to V_i/G_i$ are representable étale, so $U:=\bigsqcup _iV_i\to X$ is an étale atlas. To show that the diagonal $X\to X\times X$ is proper, it is equivalent to show that $U\times _XU\to U\times U$ is proper. This reduces to showing that $V\times _XV'\to V\times V'$ is proper for any pair $V/G\hookrightarrow X\hookleftarrow V'/G'$. As $|X|$ is Hausdorff, the map $V\times _{|X|}V'\to V\times V'$ a closed embedding (as it is a pullback of the diagonal of $|X|$), so it follows that $V\times _XV'\to V\times V'$ is proper if and only if $V\times _XV'\to V\times _{|X|}V'$ is proper. Now for the purposes of studying the latter map $V\times _XV'\to V\times _{|X|}V'$, we may as well shrink $V$, $V'$, and $X$ so that $V/G=X=V'/G'$. Now the diagonal of $V/G=V'/G'=X$ is proper, hence so is $V\times _XV'\to V\times V'$.

Corollary 3.4 Every orbispace $X$ has an étale atlas $U\to X$ with $U$ Hausdorff; equivalently, every étale atlas $U\to X$ has $U$ locally Hausdorff.

Proof. By Proposition 3.3 there is an open cover $\{V_i/G_i\subseteq X\}_i$ with $V_i$ Hausdorff, so $U:=\bigsqcup _iV_i\to X$ is an étale atlas with $U$ Hausdorff. Now for any two étale atlases $U,U'\to X$, consideration of the surjective étale maps $U\leftarrow U\times _XU'\to U'$ shows that $U$ is locally Hausdorff if and only if $U'$ is locally Hausdorff. Given any étale atlas $U\to X$ with $U$ locally Hausdorff and any open cover $\{U_i\subseteq U\}_i$ with $U_i$ Hausdorff, the disjoint union $U':=\bigsqcup _iU_i\to X$ is an étale atlas with $U'$ Hausdorff.

Corollary 3.5 For any étale atlas $U\to X$ on an orbispace, there exists an open cover $\{V_i/G_i\subseteq X\}_i$ as in Proposition 3.3 such that each map $V_i\to X$ factors through an open embedding $V_i\to U$.

Proof. Let $U\to X$ be given and fix any open cover $\{V_i/G_i\subseteq X\}_i$ as in Proposition 3.3. As $U\times _XV_i\to V_i$ is étale, it admits local sections, and hence by replacing each $V_i$ with an open cover of itself, we may assume that each projection $U\times _XV_i\to V_i$ admits a section (which is thus an open embedding). Now the resulting maps $V_i\to U$ are étale by the factorization $V_i\to U\times _XV_i\to U$, so by again replacing each $V_i$ with an open cover, we may assume they are open embeddings.

Alternatively, we could note that the open cover $\{V_i/G_i\subseteq X\}_i$ produced by the proof of Proposition 3.3 is in fact of the desired form.

Corollary 3.6 A map of orbispaces $X\to Y$ is representable if and only if it is injective on isotropy groups. In particular, an orbispace $X$ is a space if and only if it has trivial isotropy.

Proof. Any representable morphism of stacks is injective on isotropy groups (just test against points). Thus, we are left with showing that a map of orbispaces $X\to Y$ which is injective on isotropy groups is representable.

As representability descends under maps admitting local sections, it suffices to show that the fiber product $X'=X\times _YY'$ is representable for some étale atlas $Y'\to Y$. Note that $X'$ has trivial isotropy because $Y'$ has trivial isotropy and $X'\to Y'$ is injective on isotropy groups (being a pullback of $X\to Y$).

We claim that $X'$ is an orbispace, provided we take $Y'$ to be Hausdorff (which we can by Corollary 3.4). The pullback of an étale atlas $U\to X$ is an étale atlas $U'\to X'$ (because $X'\to X$ is representable, being a pullback of $Y'\to Y$). The diagonal of $X'$ is the composition $X'\to X'\times _XX'\to X'\times X'$. The second map $X'\times _XX'\to X'\times X'$ is proper as it is a pullback of $X\to X\times X$. To analyze the first map $X'\to X'\times _XX'$, note that it pulls back to $U'\to U'\times _UU'$ under the map $U'\times _UU'\to X'\times _XX'$ which admits local sections (being a pullback of $U\to X$). As $Y\to Y\times Y$ is separated, its pullback $Y'\times _YY'\to Y'\times Y'$ is also separated, which implies each projection $Y'\times _YY'\rightrightarrows Y'$ is separated because $Y'$ is Hausdorff, which implies $Y'\to Y$ is separated. Hence, its pullback $U'\to U$ is separated, so $U'\to U'\times _UU'$ is a closed embedding, hence proper. Thus, $X'\to X'\times _XX'$ is proper, and we conclude that $X'$ is an orbispace.

We are thus reduced to showing that an orbispace $Z$ with trivial isotropy is a space. By Proposition 3.3, we know that $Z$ is given locally by $V/G$ for $V$ Hausdorff and $G$ finite discrete acting freely (because $Z$ has trivial isotropy). Free actions $G\curvearrowright V$ with $V$ Hausdorff and $G$ finite are locally trivial, so we conclude that the map $Z\to |Z|$ is an equivalence. Alternatively, we could note that the chart $V/G$ near a given $x\in X$ constructed in the proof of Proposition 3.3 in fact satisfies $G=G_x$ by definition.

## 4. Coverings and nerves

We show how Theorem 2.13 implies Theorem 1.1. It is enough to show that a given orbispace admits a representable map to a simplicial complex of groups; this is Proposition 4.9. This simplicial complex of groups is basically just the nerve of a suitable open cover, however its construction is somewhat more delicate than one might initially expect.

A *sieve* on a topological space $X$ is a subset $S\subseteq 2^X$ consisting of open sets such that $U'\subseteq U\in S$ implies $U'\in S$. A *covering sieve* on $X$ is a sieve $S$ such that $\bigcup _{U\in S}U=X$. An open cover $\{U_i\subseteq X\}_i$ is said to *generate* the covering sieve on $X$ consisting of those open sets which are contained in some $U_i$.

Definition 4.1 A *connection sieve* on a map of spaces $f:X\to Y$ is a covering sieve $S$ on $X$ such that (1) for $U\in S$, the composition $U\to X\to Y$ is an open embedding, and (2) for $U,V\in S$ with $f(U)=f(V)$, either $U=V$ or $U\cap V=\varnothing$.

Note that for sieves satisfying condition (1), condition (2) is equivalent to condition (2$'$) for $U,V\in S$ either $U\cap V=\varnothing$ or $f(U\cap V)=f(U)\cap f(V)$. If $S'\subseteq S$ is an inclusion of covering sieves and $S$ is a connection sieve on $X\to Y$, then so is $S'$. In particular, if $S$ and $S'$ are connection sieves on $X\to Y$, then so is $S\cap S'$. To check that an open cover $\{U_i\subseteq X\}_i$ generates a connection sieve, it is enough to check axioms (1) and (2$'$) for the open sets $U_i$.

Definition 4.2 A map $X\to Y$ is called *strongly étale* if and only if it admits a connection sieve.

Open embeddings are strongly étale, and strongly étale maps are separated and étale (however, the converse is false by Example 4.3). Being strongly étale is preserved under pullback (take the connection sieve generated by the pullback of the original connection sieve), and the class of strongly étale maps is closed under composition (a connection sieve $S_{X/Z}$ for a composition $X\to Y\to Z$ is given by those elements of a fixed connection sieve $S_{X/Y}$ for $X\to Y$ whose image lies in a fixed connection sieve $S_{Y/Z}$ for $Y\to Z$). A disjoint union $\bigsqcup _iX_i\to \bigsqcup _iY_i$ of strongly étale maps $X_i\to Y_i$ is strongly étale (take disjoint union of connection sieves), and the projection $A\times Y\to Y$ is strongly étale for any discrete space $A$. Being strongly étale is *not* local on the target, as shown by the following example.

Example 4.3 Let us construct a finite covering space which is not strongly étale. Let $Z=\{1,\frac 12,\frac 14,\frac 18,\ldots \}\cup \{0\}\subseteq \mathbb {R}$ with the subspace topology. Every double cover of $Z$ or $Z\setminus 0$ is trivial. A given double cover of $Z\setminus 0$ has, however, many distinct extensions to $Z$, indexed by functions $Z\setminus 0\to \mathbb {Z}/2$ modulo those functions which extend continuously to zero.

Now suppose given a double cover of $Z$ together with a connection sieve on it. The restriction of this data to $Z\setminus 0$ remembers the extension of the double cover to $Z$ (use the elements of the connection sieve projecting to open sets of the form $Z\cap (0,\varepsilon )$).

Let $\tilde {Z}^+_\alpha \to Z^+$ denote the double cover obtained by gluing two copies of $Z\sqcup Z\to Z$ over $Z\setminus 0$ via some map $\alpha :Z\setminus 0\to \mathbb {Z}/2$. If $\tilde {Z}^+_\alpha \to Z^+$ is strongly étale, then we can take any connection sieve on it, restrict to the common $Z\setminus 0\subseteq Z^+$, and deduce that the two extensions of the double cover of $Z\setminus 0$ to $Z$ coincide, in other words that $\alpha$ extends continuously to zero. Hence, the double cover $\tilde {Z}^+_\alpha \to Z^+$ is strongly étale if and only if $\alpha$ extends continuously to zero.

Recall that a topological space is called *paracompact* if and only if every open cover admits a locally finite refinement [Reference DieudonnéDie44]. If $X$ is paracompact and Hausdorff, then there exists a *partition of unity* subordinate to any given locally finite open cover $\{U_i\subseteq X\}_i$, namely functions $f_i:X\to \mathbb {R}_{\geqslant 0}$ with $\operatorname {supp} f_i\subseteq U_i$ such that $\sum _if_i\equiv 1$ (recall that the support $\operatorname {supp} f$ of a function $f:X\to \mathbb {R}_{\geqslant 0}$ is, by definition, the complement of the largest open set over which $f$ vanishes identically).

Lemma 4.4 If $Y$ is paracompact Hausdorff, then a map $X\to Y$ is strongly étale if and only if there exists an open cover $\{U_i\subseteq Y\}_i$ such that each map $X\times _YU_i\to U_i$ is strongly étale.

Proof. Fix an open cover $\{U_i\subseteq Y\}_i$ and connection sieves $S_i$ on $X\times _YU_i\to U_i$. As $Y$ is paracompact, we may assume that our open cover $\{U_i\subseteq Y\}_i$ is locally finite. Using a partition of unity subordinate to this open cover, we may find another open cover $\{V_I\subseteq Y\}_I$ indexed by nonempty finite subsets $I$ of the original index set, such that $V_I\subseteq \bigcap _{i\in I}U_i$ and $V_I\cap V_J=\varnothing$ unless $I\subseteq J$ or $J\subseteq I$ (explicitly, we may take $V_I$ to be the locus where $\min _{i\in I}f_i>\max _{i\notin I}f_i$). We may now define a connection sieve on $X\to Y$ as the union over $I$ of $2^{X\times _YV_I}\cap \bigcap _{i\in I}S_i$.

An orbispace will be called paracompact (respectively, coarsely finite-dimensional, $d$-dimensional) if and only if its coarse space is.

Proposition 4.5 Every paracompact orbispace $X$ has an étale atlas $U\to X$ for which the projections $U\times _XU\rightrightarrows U$ are strongly étale. In fact, there exists such $U$ of the form $U=\bigsqcup _iV_i$ for an open cover $\{V_i/G_i\subseteq X\}_i$ as in Proposition 3.3.

Proof. Fix an open cover $\{V_i/G_i\hookrightarrow X\}_i$ as in Proposition 3.3. As $|X|$ is paracompact, we may shrink the spaces $V_i$ ($G_i$-equivariantly) so as to ensure that the associated open cover $\{\big |V_i/G_i\big |\subseteq |X|\}_i$ of coarse spaces is locally finite. Choose a partition of unity $\{f_i:|X|\to \mathbb {R}_{\geqslant 0}\}_i$ subordinate to the open cover $\{\big |V_i/G_i\big |\subseteq |X|\}_i$. Let $V_i^0\subseteq V_i$ denote the open subset where $f_i>0$. We show that the étale atlas $U:=\bigsqcup _iV_i^0$ has the desired property.

It suffices to show that for any pair of open embeddings $V/G\hookrightarrow X\hookleftarrow W/H$ and pair of functions $f_V,f_W:|X|\to \mathbb {R}_{\geqslant 0}$ supported inside $|V/G|$ and $|W/H|$, respectively, the projection $W^0\times _XV^0\to V^0$ is strongly étale. We begin by considering the map $W\times _XV\to V$, which is a finite covering space over its open image $W/H\times _XV\subseteq V$ (by pullback from $W\to W/H$, which is a finite covering space by descent from its pullback $H\times W\to W$). Thus, every point of $W/H\times _XV$ has a neighborhood over which $W\times _XV\to V$ is strongly étale. Even better, because $V$ is Hausdorff and $G$ is finite, each $G$-orbit inside $W/H\times _XV$ has a neighborhood over which $W\times _XV\to V$ is strongly étale. In other words, each point of $|W/H|\cap |V/G|\subseteq |X|$ has a neighborhood over which $W\times _XV\to V$ is strongly étale.

Now because $|X|$ is paracompact, there exists a locally finite open cover of $|X|$ by $|X|\setminus (\operatorname {supp} f_V\cup \operatorname {supp} f_W)$ together with open subsets $\{A_i\subseteq |W/H|\cap |V/G|\}_i$ over which $W\times _XV\to V$ is strongly étale. Fix a partition of unity $g:|X|\to \mathbb {R}_{\geqslant 0}$ supported inside $|X|\setminus (\operatorname {supp} f_V\cup \operatorname {supp} f_W)$ and $\{g_i:|X|\to \mathbb {R}_{\geqslant 0}\}$ supported inside $A_i$, that is $g+\sum _ig_i\equiv 1$. Now the patching procedure for connection sieves from the proof of Lemma 4.4 shows that $W\times _XV\to V$ is strongly étale over the complement of $\operatorname {supp} g$. In particular, it follows by restriction that $W^0\times _XV^0\to V^0$ is strongly étale.

A *simplicial complex* is a pair $X=(V,S)$ consisting of a set $V$ (‘vertices’) and a set $S\subseteq 2^V\setminus \{\varnothing \}$ (‘simplices’) of finite subsets of $V$ such that $S$ contains all singletons and $\varnothing \ne A\subseteq B\in S$ implies $A\in S$. The *star* $\operatorname {st}(X,\sigma )\subseteq X$ of a simplex $\sigma$ in a simplicial complex $X$ is the subcomplex consisting of all simplices $\tau \subseteq X$ with $\sigma \cup \tau \in S(X)$.

A map of simplicial complexes $X\to Y$ is a map of vertex sets $V(X)\to V(Y)$ which maps simplices to simplices (the image of an element of $S(X)$ is an element of $S(Y)$). A map of simplicial complexes is called injective if and only if the map on vertex sets (hence, also the map on simplices) is injective. A map of simplicial complexes $f:X\to Y$ is called *étale* (respectively, *locally injective*) if and only if the induced maps on stars $\operatorname {st}(X,\sigma )\to \operatorname {st}(Y,f(\sigma ))$ are isomorphisms (respectively, injective). We call a map of simplicial complexes $X\to Y$ *sufficiently étale* if and only if every simplex $\sigma \subseteq Y$ (equivalently, every vertex) is the image of a simplex $\tau \subseteq X$ at which $X\to Y$ is étale (this is a useful weakening of the condition of being surjective and étale, which in the context of simplicial complexes is too strong).

The *geometric realization* $\|X\|$ of a simplicial complex $X$ is the set of tuples $t\in \mathbb {R}_{\geqslant 0}^{V(X)}$ with $\sum _vt_v=1$ such that $\{v:t_v>0\}\in S(X)$, topologized by declaring that the realization of the complete simplex on $k+1$ vertices has the usual topology and that a realization $\|X\|$ is given the strongest topology for which (the realization of) every map from a complete simplex to $X$ is continuous. The geometric realization of an étale map of simplicial complexes is an étale map of spaces.

A *locally injective simplicial complex groupoid* $M\rightrightarrows O$ consists of simplicial complexes $O$ and $M$ together with structure maps satisfying the axioms of a groupoid, where both maps $M\rightrightarrows O$ are locally injective. Local injectivity of the two maps $M\rightrightarrows O$ implies that the natural map $\big \|M\times _OM\big \|\xrightarrow \sim \|M\|\times _{\|O\|}\|M\|$ is a homeomorphism, and thus the geometric realization