Enough vector bundles on orbispaces
John Pardon

TL;DR
This paper proves that under mild conditions, orbispaces possess sufficient vector bundles, enabling their K-theory to be viewed as a cohomology theory and providing global presentation results for related geometric structures.
Contribution
It establishes the existence of enough vector bundles on orbispaces and derives consequences for K-theory and global presentations of orbifold structures.
Findings
Orbispaces with mild hypotheses have enough vector bundles.
K-theory of these orbispaces forms a cohomology theory.
Global presentation results for smooth and derived orbifolds are obtained.
Abstract
We show that every orbispace satisfying certain mild hypotheses has 'enough' vector bundles. It follows that the K-theory of finite rank vector bundles on such orbispaces is a cohomology theory. Global presentation results for smooth orbifolds and derived smooth orbifolds also follow.
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TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory · Advanced Topics in Algebra
Enough vector bundles on orbispaces
John Pardon
(10 December 2021)
Abstract
We show that every orbispace satisfying certain mild hypotheses has ‘enough’ vector bundles. It follows that the -theory of finite rank vector bundles on such orbispaces is a cohomology theory. Global presentation results for smooth orbifolds and derived smooth orbifolds also follow.
1 Introduction
A (separated) orbispace [15, 6] is a topological stack which admits a cover by open substacks of the form (where is a continuous action of a finite group on a topological space) and whose diagonal 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 .
An interesting question to ask about a given orbispace is whether there exists a global presentation for 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 for a (possibly infinite) discrete group , then is a global quotient by Lück–Oliver [24, Corollary 2.7]. Every (paracompact) smooth effective -dimensional orbifold is a global quotient (of its orthonormal frame bundle by ), 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–Metzler [19]. Henriques [20] conjectured that every compact orbispace is a global quotient, however other experts have expressed skepticism that such a general result would be true [35, §6.4]. The analogous question for algebraic stacks has been studied by Edidin–Hassett–Kresch–Vistoli [14] and Totaro [36]. It is a result of Kresch–Vistoli [22, Theorem 2] [21, Theorem 4.4] (using a result of Gabber [12] [10, Chapter 4]) that smooth separated Deligne–Mumford stacks over 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 be an orbispace, with isotropy groups of bounded order, whose coarse space is coarsely finite-dimensional (every open cover has a locally finite refinement with finite-dimensional nerve). Then there exists a complex vector bundle of rank over , whose fiber over every is isomorphic to a direct sum of copies of the regular representation of the isotropy group . We may take if is -dimensional (every open cover has a locally finite refinement with nerve of dimension ) and has isotropy groups of order .
(Note that the ‘real’ and ‘complex’ versions of Theorem 1.1 are equivalent, by tensoring from to and by forgetting from to .)
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 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 as in Theorem 1.1, we have for a space .
Corollary 1.3**.**
Every paracompact smooth orbifold of dimension with isotropy groups of order is the quotient of a smooth manifold by a smooth action of the compact Lie group , where .
It seems that Theorem 1.1 does not help resolve the question of whether every separated Deligne–Mumford stack of finite type over 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 be a representable map of orbispaces satisfying the hypothesis of Theorem 1.1. Every vector bundle on of bounded rank embeds into the pullback of a vector bundle of bounded rank on .
It is well known that Corollary 1.4 is the key statement needed to show that the -theory of finite rank vector bundles on orbispaces satisfies excision and exactness and is thus a cohomology theory (see [24, §3] and [20, §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 -theory of finite rank vector bundles should agree (for reasonable orbispaces) with the other standard models of -theory for orbispaces, such as using bundles of Fredholm operators [34, 26, 3, 4] or using orthogonal spectra [31, §§6.3–6.4] (compare Remark 1.6 below).
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 and isotropy groups of order is the derived zero set of a smooth section of a vector bundle of rank over a smooth orbifold of dimension . (‘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 [32] and Gepner–Henriques [18] establishes 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 [31, Definition 1.1.27]). Let be an orbispace, and let be any faithful vector bundle over (meaning the fibers of are faithful representations of the isotropy groups of ). The orthogonal space corresponding to is given by
[TABLE]
where denotes the total space of the bundle of embeddings of into (note that is a space since is faithful).
Schwede [31] also associates to every orthogonal spectrum a cohomology theory on orthogonal spaces, hence on orbispaces. Given an orthogonal spectrum and an orbispace which admits faithful vector bundles, the degree zero -cohomology of is (in view of the above) given by the direct limit over vector bundles over of the set of homotopy classes of sections of the fibration . More generally, we may consider the mapping spectrum defined by
[TABLE]
whose stable homotopy groups are the -cohomology groups of . If is a global -spectrum [31, Definition 4.3.8], then this direct limit is achieved at any faithful , and the above definition of is an -spectrum. Let us also propose a possible definition of the -homology groups of as the stable homotopy groups of the spectrum
[TABLE]
where indicates taking the coarse space of the total space of over .
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 for a compact Lie group). The naive generalization is simply false: there are purely ineffective Lie orbispaces with isotropy group and coarse space which have no finite rank faithful vector bundles [36, §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.
1.1 Acknowledgements
I would like to thank André Henriques for discussions about orbispaces and topological stacks, David Treumann for conversations about the algebraic case, and the referees for their remarks. I also thank the Department of Mathematics at Cambridge University for their hospitality during the time when this paper was conceived. This research was conducted during the period the author served as a Clay Research Fellow and was partially supported by a Packard Fellowship and by the National Science Foundation under the Alan T. Waterman Award, Grant No. 1747553.
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 below) 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 comes with a filtration by skeleta
[TABLE]
where is obtained from by attaching cells of the form for finite groups (here is the orbispace quotient). We will 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 on decomposes canonically as for vector bundles on indexed by the complex irreducible representations of . Thus is classified by an element of , which vanishes iff extends to . We shall seek to detect these obstructions using Chern characters. According to Bott periodicity, the homotopy groups of are given by
[TABLE]
and the Chern character of a generator of is nonzero. Now the Chern character of is given by , so triviality of does not imply triviality of each or of the aforementioned obstructions to extending .
To capture the information we need about , we consider the more refined characteristic class which we will call the inertial Chern character111It is tempting to call it the “Chern character character”. (studied previously by Adem–Ruan [1]), which is a cohomology class on the inertia stack
[TABLE]
In local coordinates , 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 , for the curvature of a hermitian connection on , and writing
[TABLE]
Now the inertia stack of is (quotient by the conjugation action), and the inertial Chern character of is given by . Thus if is trivial, then so is each by linear independence of characters, and hence extends to as desired.
We are thus led to consider the modified problem of constructing a vector bundle on with the desired fibers and with trivial inertial Chern character. It suffices to show that if admits such a vector bundle, then so does . The vanishing of guarantees that the obstructions to extending to 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 . By the long exact sequence
[TABLE]
it is thus in the image of . The inertial Chern character is an even degree class, so we may assume that is even. By modifying how we extend from to by an element of over a given -simplex, we can shift its inertial Chern character by any integral linear combination of characters in (for the isotropy group of that given -simplex). By replacing with , we can multiply its inertial Chern character by any positive integer . A combination of these two operations suffice to kill the inertial Chern character, provided that it is -rational (rational with respect to a certain -structure on 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 completed.
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 [16], Corson [11], or Bridson–Haefliger [8]).
Definition 2.1**.**
A simplicial complex of groups is a pair consisting of a simplicial complex together with the following data:
For every simplex , a group .
For every pair of simplices , an injective group homomorphism .
For every triple of simplices , an element of conjugating the inclusion to the composition of inclusions .
For every quadruple of simplices , the resulting product of elements of conjugating to to to and back to must be the identity element of .
Injectivity of each map will ensure that the geometric realization of is an orbispace, rather than some sort of more exotic topological stack.
A simplicial complex of groups presents an orbispace 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 is (the geometric realization of) itself, and that over the open star of a simplex , the geometric realization is given by the orbispace quotient
[TABLE]
where the pieces for are glued together via the map induced by the map and the element of conjugating the composition to the map .
We will often abuse terminology and say ‘simplicial complex of groups’ when we really mean its geometric realization. Thus we will refer to as a simplicial complex of groups. Its coarse space is the geometric realization of .
To apply the methods of differential topology to a given simplicial complex of groups, we fix a family of (germs of) smooth retractions for every pair of simplices such that the maps and agree for (such a family of smooth retractions may be constructed by induction). Given this data, objects of differential topology on (functions, differential forms, bundles, connections, etc.) are required to be pulled back under in a(n unspecified) neighborhood of every . 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 of a simplicial complex of groups is defined, and it coincides with the de Rham complex of its coarse space . There is a natural map from the de Rham complex to the simplicial cochain complex over , given by integrating differential forms over oriented simplices. It is a standard fact that this map is a quasi-isomorphism (proof: filter by skeleta and invoke the five-lemma to reduce to showing that is a quasi-isomorphism, which follows from the Poincaré lemma).
Definition 2.2** (Chern character).**
Let be a simplicial complex of groups, and let be a complex vector bundle. Given a hermitian metric and hermitian connection on , the curvature of is a -form valued in , and the Chern character form
[TABLE]
is closed. Its class in cohomology is called the Chern character of . This class is independent of and by an interpolation argument (interpolate on between any two metric/connection pairs on and , and use the fact that ).
For any stack , its inertia stack is the stack
[TABLE]
When , 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 is a simplicial complex of groups then so is : a simplex of is a pair where is a simplex and is a conjugacy class, and is the centralizer of any element of the conjugacy class , etc.
Definition 2.3** (Inertial Chern character).**
Let be a simplicial complex of groups, and let be a complex vector bundle. The inertial Chern character is represented by the closed form
[TABLE]
for any choice of hermitian metric and connection on . Closedness can be seen by splitting the pullback of to into isotypic pieces for the action of the cyclic group generated by 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 , a vector bundle over is simply a representation of , the inertia stack is the stack quotient of the conjugation action of on itself, and the inertial Chern character is the character of regarded as a representation of .
Example 2.5*.*
If for a smooth manifold and , then and the inertial Chern character is given by . Since the characters of the irreducible representations of form a basis for the space of maps , we see that for , the inertial Chern character determines (and is determined by) the Chern characters of each of the associated bundles .
Example 2.6*.*
The degree zero part of the inertial Chern character records the characters of the fibers of , regarded as representations of the isotropy groups of the points of .
Despite admitting the aforementioned transcendental definition in terms of differential forms, the (usual) Chern character is well known be rational, i.e. it lies in the subspace . The inertial Chern character does not always lie in the subspace , rather it is rational with respect to a different -structure, which we now introduce.
Definition 2.7** (-integral cohomology of ).**
Regard the simplicial cochain group as the group of simplicial cochains on with coefficients in . In this description, let us replace with its subspace of integral (resp. rational) linear combinations of characters of . Equivalently, we consider the complex of simplicial cochains on which act on (for any simplex ) by an integral (resp. rational) linear combination of characters of . We denote the resulting cohomology groups by (resp. ), which are evidently functorial in . We say that an element of is -integral (resp. -rational) to mean that it lies in the image of (resp. in the subspace ).
Example 2.8*.*
If then and . Thus is -rational since each is rational.
Since the coefficient systems on appearing in Definition 2.7 are finite free modules over , we may dualize to define homology groups as well.
Lemma 2.9**.**
There are canonical isomorphisms
[TABLE]
and a canonical short exact sequence
[TABLE]
Proof.
The complex is degreewise free, and the complexes , , and are obtained from it by applying the functors , , and , respectively. ∎
Proposition 2.10** (-rationality of ).**
Let be a simplicial complex of groups of order . The inertial Chern character of any vector bundle is -rational. In fact, there exists a positive integer such that is -integral.
It is worth remarking that the proof we give does not provide an explicit expression for other than for .
Proof.
We first show that the inertial Chern character of any line bundle is -rational. We have for endomorphisms of , so the inertial Chern character of a line bundle splits as the product
[TABLE]
where is the fiberwise character of and is the Chern character in degree one. It thus suffices to show that is rational. We have , so it suffices to show that is integral for some positive integer . Since all isotropy groups of have order , the tensor power descends to a line bundle on the coarse space. Since is an ordinary line bundle over a space (rather than an orbispace), its first Chern class is integral. We have thus shown that is -rational (in fact, we have shown that is -integral).
To treat the case of vector bundles of dimension greater than one, we will use the splitting principle. Given a vector bundle with hermitian metric, let denote the fibration whose fiber over is the space of decompositions of into ordered orthogonal one-dimensional subspaces. In other words, where is the principal -bundle associated to with its chosen metric. We claim that the pullback map
[TABLE]
is injective. The fiber of over a given point is the space of -invariant ordered decompositions of into one-dimensional subspaces. Since the group generated by is abelian, its irreducible representations are one-dimensional, and hence there are plenty of such decompositions which are -invariant, namely when each one-dimensional subspace is contained in some -isotypic piece of . Thus is a disjoint union of iterated projective space bundles, from which the desired injectivity statement follows (for any projective space bundle over an orbispace , the pullback map is split by the map where denotes the tautological line bundle and denotes fiberwise integration).
Since is injective and is exact, we conclude from Lemma 2.9 that , and hence that the domain is torsion. From this and Lemma 2.9 again, it follows that a class in is -rational iff its pullback to is -rational. Now consider the pullback of to . This pullback splits as a direct sum of line bundles, so since the inertial Chern character is additive under direct sum, we conclude that the pullback of to is -rational, and hence that is itself -rational.
It remains to produce an integer such that is -integral for . We claim that there exists a finite orbi-complex carrying a principal -bundle
[TABLE]
with the property that every principal -bundle over a simplicial complex of groups of dimension and isotropy is a pullback of . Since the inertial Chern character of is -rational and is finite, there exists an integer such that times this inertial Chern character is -integral in cohomological degree . By pullback, the same integer works for any vector bundle of rank over a simplicial complex of groups of order . Note that this argument gives no explicit bound on the integer .
It remains to construct . We seek -spaces such that
[TABLE]
where ranges over all finite subgroups of of order . 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 exists given , argue as follows. We have by Hurewicz, and the latter group is finitely generated since is made up of finitely many cells. We define by attaching cells along a choice of finitely many generators of , for each of the finitely many subgroups on our fixed set of representatives. Note that , so any map for can be homotoped to land inside . It follows that the homotopy groups of satisfy the desired vanishing property.
Let us show that any principal -bundle is pulled back from when with isotropy groups of order . That is, we should construct an equivariant map (note that since the target is a space, this is the same as an equivariant map from the coarse space ). By induction on the cells of , it suffices to solve the extension problem for equivariant maps from to for and . This is equivalent to the extension problem for maps , whose positive solution for is one of the defining properties of . ∎
Example 2.11*.*
A previous version of this text claimed a version of Proposition 2.10 with independent of . This stronger result is false; here is a counterexample. Begin with with a line bundle given by multiplication by on . Glue on a disk using an attaching map to a generator of . The line bundle extends since all complex line bundles on a circle are trivial. The inertial Chern character of this line bundle has denominator .
Lemma 2.12** (Bott).**
The image of the Chern character map
[TABLE]
is precisely .
Proof.
This is an immediate corollary of Bott periodicity; for completeness, we include the proof from [25]. Let denote the line bundle with , and let denote the trivial line bundle. According to Bott periodicity, multiplication with defines an isomorphism . In particular, is a free generator. Multiplicativity of the Chern character gives . ∎
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 be a -dimensional simplicial complex of groups of order . There exists a complex vector bundle of rank over , whose fiber over is isomorphic to a direct sum of copies of the regular representation of .
Proof.
The condition on the fibers of amounts to the assertion that where denotes “the characteristic function of the identity” as a function . We will construct satisfying
[TABLE]
(meaning all higher inertial Chern characters vanish). Certainly such a vector bundle exists over the [math]-skeleton of , with rank , since all isotropy groups have order . It therefore suffices to show that any such vector bundle on extends to .
To extend as a vector bundle from to amounts to doing an extension from to for each -simplex. The obstruction to doing this lies in , where . The map is an isomorphism on provided , which is guaranteed by taking say . The Chern character detects by Lemma 2.12, so vanishing of the inertial Chern character and linear independence of characters means that these obstructions all vanish. Thus extends to .
It remains to show that the extension of to can be taken to have trivial inertial Chern character. The space of extensions over a given -simplex is a torsor for , which is again once we take . We will use this freedom to ensure that the inertial Chern character of on vanishes. Consider the long exact sequence
[TABLE]
Since the inertial Chern character in maps to zero in , it is in the image of . In particular, it may be nonzero only in degree . Let us now replace with (which multiplies its inertial Chern character by ) for an appropriate positive integer so that by Proposition 2.10 its inertial Chern character is -integral. Modifying our bundle by an element of (the product over the -simplices of) shifts its inertial Chern character by anything in by Lemma 2.12. We are done since 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 , and since 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 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 depends on the additivity of the Chern character (note that is not suitable for this argument since is a space, hence every bundle pulled back from has trivial isotropy representations). It may prove interesting to intepret this argument within Schwede’s framework of global homotopy theory [31].
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 [30], Gepner–Henriques [18], Behrend [6], Metzler [28], Behrend–Noohi [7], Heinloth [17], or Laumon–Moret-Bailly [23].
Let denote the category of topological spaces and continuous maps, and let denote the 2-category of (essentially) small groupoids. A stack is a functor which satisfies descent, i.e. such that for every topological space and every open cover , the natural functor
[TABLE]
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 . 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 will usually omit the prefix ‘2-’).
The Yoneda lemma implies that the Yoneda functor embeds the category of topological spaces fully faithfully into the 2-category of stacks, and moreover that the natural map from to the groupoid of maps of stacks is an equivalence. The category of topological spaces is complete, and the Yoneda embedding is continuous (commutes with limits). Hence we will make no distinction between a topological space and the associated stack of maps to , nor between objects of and maps (which we will simply write as ).
Every stack has a coarse space (a topological space) which is initial in the category of maps from to topological spaces. Concretely, the points of are the isomorphism classes of maps , and a subset is open iff for every map from a topological space , the inverse image of is an open subset of .
A stack is called representable iff it is in the essential image of the Yoneda embedding (i.e. it is isomorphic to a topological space). A morphism of stacks is called representable iff for every map from a topological space , the fiber product is representable.
For any property of morphisms of topological spaces which is preserved under pullback, a representable morphism of stacks is said to have property iff the pullback has for every map from a topological space . The following are examples of properties of morphisms which are preserved under pullback:
is injective.
is surjective.
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.
is an embedding, meaning that it is a homeomorphism onto its image.
is a closed embedding.
is étale, meaning that for every there exists an open neighborhood such that is an open embedding.
is separated, meaning that for every distinct pair with , there exist open neighborhoods which are disjoint . (This is equivalent to the relative diagonal being a closed embedding.)
is universally closed, meaning that is closed for every . This is equivalent to the assertion that for every and every collection of open sets covering , there exists a finite subcollection which covers for some open neighborhood . (One proof of this equivalence goes via yet a third equivalent condition, namely that every net with has a subnet converging to some .)
is proper, meaning that it is separated and universally closed.
admits local sections, meaning that there is an open cover such that every restriction admits a section.
is a finite covering space, meaning that there is an open cover such that every restriction is isomorphic to for some integer .
Each of these properties is also closed under composition, and thus also under fiber products, meaning that for maps and over , if both and have then also has (indeed, is a pullback of ).
For any stack , open (resp. closed) embeddings are in natural bijection with open (resp. closed) subsets of .
To check that a given map of spaces satisfies one of the properties above, it is often helpful to make use of the fact that these are all local on the target, meaning that for every open cover , if has for every , then so does . This leads to the following generalization for maps of stacks: if is a representable morphism of stacks and is a representable morphism of stacks admitting local sections, then has iff has . In fact, in this statement we need not assume that is representable, just that it admit local sections in the generalized sense that for every map from a topological space , there exists an open cover such that each admits a section. We thus say that “ 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 be a map of stacks, and let be a map of stacks admitting local sections. If is representable, then so is .
Proof.
By replacing and with their pullbacks under for a topological space , we may assume without loss of generality that is representable. Since admits local sections, we may replace with the composition where is an open cover. Now each is representable by assumption, and gluing these spaces together on their common overlaps gives a topological space representing . ∎
A complex vector bundle over a stack is a representable map together with maps and (both over ) such that for every map from a topological space , there exists an open cover and integers such that is isomorphic to 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\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O consists of a pair of topological spaces (‘objects’) and (‘morphisms’), two maps M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O (‘source’ and ‘target’), a map (‘identity’), an involution (‘inverse’), and a map (‘composition’) satisfying the axioms of a groupoid. A topological groupoid M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O presents a stack [M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O] defined as follows. An object of [M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O](X) is an open cover together with maps and satisfying a compatibility condition, and an isomorphism in [M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O](X) consists of maps satisfying a compatibility condition. There is a natural map O\to[M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O] (take the trivial open cover ) and the fiber product O\times_{[M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O]}O is naturally identified with . The morphism O\to[M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O] admits local sections (by definition), so since O\times_{[M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O]}O\to O is representable, it follows by descent that O\to[M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O] is representable. Conversely, a representable map admitting local sections from a topological space to a stack determines a topological groupoid U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U presenting . Indeed, the fiber product is representable (since and are), it admits two maps to (the two projections), an involution (exchanging the two factors), and a composition map (forgetting the middle factor), and one can check using the stack property that the natural map [U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U]\to X is an equivalence. A stack for which there exists a representable map admitting local sections from a space is called topological, and such is called an atlas for .
For a topological stack with atlas , for any property which descends along maps admitting local sections, the map has iff both maps have , and the diagonal has iff the map has . In particular, for any topological stack , the diagonal is representable, and thus every map from a topological space is representable. More generally, for any map of topological stacks , the relative diagonal is representable (by descent from its pullback for an atlas ). If is representable and is topological, then so is (if is an atlas for , then its pullback is an atlas for ), and the relative diagonal has iff the relative diagonal of atlases has (the latter is the pullback of the former under the map , which is representable and admits local sections since it is a pullback of ).
For a topological space , a topological group , and a continuous group action , we may consider the action groupoid G\times V\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}V with source and target maps and . The stack associated to this groupoid is denoted and is called the stack quotient of the action . If is discrete, the two maps are étale, so is étale. If is compact and is Hausdorff, the map is universally closed (factor as which is a closed embedding since is Hausdorff and which is universally closed since is compact), so has universally closed diagonal. If is Hausdorff, then the map is separated, so has separated diagonal. Thus if is compact Hausdorff and is Hausdorff then has proper diagonal.
A groupoid presentation of a topological stack also gives a description of its coarse space as follows. For an atlas , consider the equivalence relation on given by the image of . There is a map (which is tautological once we regard as [U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U]), inducing a map , which is a bijection, essentially by definition. Since an open substack of pulls back to an open subset of invariant under , it follows that is open and is thus a homeomorphism. In particular, it follows that the coarse space of the stack quotient is the usual topological quotient of by .
An action is called locally trivial iff admits a cover by -invariant open sets where acts by left multiplication on and trivially on . If is locally trivial, then the natural map from the stack quotient to the topological quotient is an equivalence. Indeed, this assertion is local on , so it suffices to consider the case of , where it holds by inspection.
Definition 3.2**.**
A (separated) orbispace is a stack for which there exists a representable étale surjection from a topological space (an ‘étale atlas’), and the diagonal is proper.
Proposition 3.3**.**
A stack is an orbispace iff is Hausdorff and there exists an open cover where are finite discrete groups acting on Hausdorff spaces .
(Similar results include [6, Theorem 1.108] and [30, Proposition 14.10].)
Proof.
Let be an orbispace, and let us show that there is an open cover . Fix an étale atlas , and let . The automorphism group is finite and discrete since is proper. Since the two projections U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U are étale, for every there exists an open neighborhood such that each projection restricted to is an open embedding (this gives another proof that has the discrete topology). Since is finite and is separated, we may take these to be disjoint. Now the complement of is closed and disjoint from , so it projects to a closed (by properness) subset of disjoint from . Hence there is an open neighborhood such that . Thus is a disjoint union of pieces indexed by , and each piece maps homeomorphically to under each projection. By further shrinking , we may assume that the map respects composition (this is possible since composition is continuous). It follows that V\times_{X}V\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}V is an action groupoid G\times V\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}V for an action . Since expresses the diagonal of as a composition of proper maps, we conclude that is Hausdorff. Now the map of groupoids (G\times V\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}V)=(V\times_{X}V\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}V)\to(U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U) induces a map of stacks . To see that this is an open embedding, let denote the orbit of under the morphisms U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U. Since the projections U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U are étale, it follows that is open, and hence [V^{+}\times_{X}V^{+}\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}V^{+}]\to[U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U] is an open embedding of stacks; denote by this open substack, so is an étale atlas. Thus the topological groupoid V\times_{X}V=V\times_{Z}V\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}V presents , so is an open embedding as desired.
Let us now show that for any orbispace , its coarse space is Hausdorff. We saw earlier that is the quotient of by the image of (which is an equivalence relation). This equivalence relation is closed since is proper, so is Hausdorff provided the quotient map is open. Now openness of does not depend on which atlas we take: there are étale maps over , which means that is open iff is open. Moreover, openness of can be checked locally on , so we may assume without loss of generality that . Hence it is enough to note that the quotient map (induced by the canonical étale atlas ) is open. Thus is Hausdorff.
Finally, let us show that if is Hausdorff and there is an open cover , then is an orbispace. The maps are representable étale, so is an étale atlas. To show that the diagonal is proper, it is equivalent to show that is proper. This reduces to showing that is proper for any pair . Since is Hausdorff, the map a closed embedding (as it is a pullback of the diagonal of ), so it follows that is proper iff is proper. Now for the purposes of studying the latter map , we may as well shrink , , and so that . Now the diagonal of is proper, hence so is . ∎
Corollary 3.4**.**
Every orbispace has an étale atlas with Hausdorff; equivalently, every étale atlas has locally Hausdorff.
Proof.
By Proposition 3.3 there is an open cover with Hausdorff, so is an étale atlas with Hausdorff. Now for any two étale atlases , consideration of the surjective étale maps shows that is locally Hausdorff iff is locally Hausdorff. Given any étale atlas with locally Hausdorff and any open cover with Hausdorff, the disjoint union is an étale atlas with Hausdorff. ∎
Corollary 3.5**.**
For any étale atlas on an orbispace, there exists an open cover as in Proposition 3.3 such that each map factors through an open embedding .
Proof.
Let be given and fix any open cover as in Proposition 3.3. Since is étale, it admits local sections, and hence by replacing each with an open cover of itself, we may assume that each projection admits a section (which is thus an open embedding). Now the resulting maps are étale by the factorization , so by again replacing each with an open cover, we may assume they are open embeddings.
Alternatively, we could note that the open cover produced by the proof of Proposition 3.3 is in fact of the desired form. ∎
Corollary 3.6**.**
A map of orbispaces is representable iff it is injective on isotropy groups. In particular, an orbispace is a space iff 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 which is injective on isotropy groups is representable.
Since representability descends under maps admitting local sections, it suffices to show that the fiber product is representable for some étale atlas . Note that has trivial isotropy since has trivial isotropy and is injective on isotropy groups (being a pullback of ).
We claim that is an orbispace, provided we take to be Hausdorff (which we can by Corollary 3.4). The pullback of an étale atlas is an étale atlas (since is representable, being a pullback of ). The diagonal of is the composition . The second map is proper as it is a pullback of . To analyze the first map , note that it pulls back to under the map which admits local sections (being a pullback of ). Since is separated, its pullback is also separated, which implies each projection Y^{\prime}\times_{Y}Y^{\prime}\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}Y^{\prime} is separated since is Hausdorff, which implies is separated. Hence its pullback is separated, so is a closed embedding, hence proper. Thus is proper, and we conclude that is an orbispace.
We are thus reduced to showing that an orbispace with trivial isotropy is a space. By Proposition 3.3, we know that is given locally by for Hausdorff and finite discrete acting freely (since has trivial isotropy). Free actions with Hausdorff and finite are locally trivial, so we conclude that the map is an equivalence. Alternatively, we could note that the chart near a given constructed in the proof of Proposition 3.3 in fact satisfies 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 is a subset consisting of open sets such that implies . A covering sieve on is a sieve such that . An open cover is said to generate the covering sieve on consisting of those open sets which are contained in some .
Definition 4.1**.**
A connection sieve on a map of spaces is a covering sieve on such that (1) for , the composition is an open embedding, and (2) for with , either or .
Note that for sieves satisfying condition (1), condition (2) is equivalent to condition (2*′) for either or . If is an inclusion of covering sieves and is a connection sieve on , then so is . In particular, if and are connection sieves on then so is . To check that an open cover generates a connection sieve, it is enough to check axioms (1) and (2′*) for the open sets .
Definition 4.2**.**
A map is called strongly étale iff 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 below). 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 for a composition is given by those elements of a fixed connection sieve for whose image lies in a fixed connection sieve for ). A disjoint union of strongly étale maps is strongly étale (take disjoint union of connection sieves), and the projection is strongly étale for any discrete space . 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 with the subspace topology. Every double cover of or is trivial. A given double cover of has, however, many distinct extensions to , indexed by functions modulo those functions which extend continuously to zero.
Now suppose given a double cover of together with a connection sieve on it. The restriction of this data to remembers the extension of the double cover to (use the elements of the connection sieve projecting to open sets of the form ).
Let denote the double cover obtained by gluing two copies of over via some map . If is strongly étale, then we can take any connection sieve on it, restrict to the common , and deduce that the two extensions of the double cover of to coincide, in other words that extends continuously to zero. Hence the double cover is strongly étale iff extends continuously to zero.
Recall that a topological space is called paracompact iff every open cover admits a locally finite refinement [13]. If is paracompact and Hausdorff, then there exists a partition of unity subordinate to any given locally finite open cover , namely functions with such that (recall that the support of a function is by definition the complement of the largest open set over which vanishes identically).
Lemma 4.4**.**
If is paracompact Hausdorff, then a map is strongly étale iff there exists an open cover such that each map is strongly étale.
Proof.
Fix an open cover and connection sieves on . Since is paracompact, we may assume that our open cover is locally finite. Using a partition of unity subordinate to this open cover, we may find another open cover indexed by non-empty finite subsets of the original index set, such that and unless or (explicitly, we may take to be the locus where ). We may now define a connection sieve on as the union over of . ∎
An orbispace will be called paracompact (resp. coarsely finite-dimensional, -dimensional) iff its coarse space is.
Proposition 4.5**.**
Every paracompact orbispace has an étale atlas for which the projections U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U are strongly étale. In fact, there exists such of the form for an open cover as in Proposition 3.3.
Proof.
Fix an open cover as in Proposition 3.3. Since is paracompact, we may shrink the spaces (-equivariantly) so as to ensure that the associated open cover of coarse spaces is locally finite. Choose a partition of unity subordinate to the open cover . Let denote the open subset where . We will show that the étale atlas has the desired property.
It suffices to show that for any pair of open embeddings and pair of functions supported inside and , respectively, the projection is strongly étale. We begin by considering the map , which is a finite covering space over its open image (by pullback from , which is a finite covering space by descent from its pullback ). Thus every point of has a neighborhood over which is strongly étale. Even better, since is Hausdorff and is finite, each -orbit inside has a neighborhood over which is strongly étale. In other words, each point of has a neighborhood over which is strongly étale.
Now since is paracompact, there exists a locally finite open cover of by together with open subsets over which is strongly étale. Fix a partition of unity supported inside and supported inside , that is . Now the patching procedure for connection sieves from the proof of Lemma 4.4 shows that is strongly étale over the complement of . In particular, it follows by restriction that is strongly étale. ∎
A simplicial complex is a pair consisting of a set (“vertices”) and a set (“simplices”) of finite subsets of such that contains all singletons and implies . The star of a simplex in a simplicial complex is the subcomplex consisting of all simplices with .
A map of simplicial complexes is a map of vertex sets which maps simplices to simplices (the image of an element of is an element of ). A map of simplicial complexes is called injective iff the map on vertex sets (hence also the map on simplices) is injective. A map of simplicial complexes is called étale (resp. locally injective) iff the induced maps on stars are isomorphisms (resp. injective). We will call a map of simplicial complexes sufficiently étale iff every simplex (equivalently, every vertex) is the image of a simplex at which 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 of a simplicial complex is the set of tuples with such that , topologized by declaring that the realization of the complete simplex on vertices has the usual topology and that a realization is given the strongest topology for which (the realization of) every map from a complete simplex to 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\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O consists of simplicial complexes and together with structure maps satisfying the axioms of a groupoid, where both maps M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O are locally injective. Local injectivity of the two maps M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O implies that the natural map is a homeomorphism, and thus the geometric realization \left\|M\right\|\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}\left\|O\right\| is a topological groupoid. If denotes the subcomplex consisting of those simplices for which it is not the case that the first projection is étale at every simplex mapped to under the second projection, then the natural map \left\|O\right\|\setminus\left\|\partial O\right\|\to[\left\|M\right\|\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}\left\|O\right\|] is étale. A locally injective simplicial complex groupoid M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O is called sufficiently étale iff this map is surjective (equivalently, every vertex of is -isomorphic to one not in ).
The abstract simplex category has objects finite totally ordered sets and has morphisms weakly order preserving maps; every object of is isomorphic to for a unique integer . A simplicial object in a category is a functor , and the category of simplicial objects in is denoted . If is complete (resp. cocomplete) then so is , and limits (resp. colimits) are calculated pointwise.
We will consider only simplicial sets (objects of the category ) and simplicial groupoids (objects of the category ). A simplicial set (or groupoid) will be denoted , where is its set (or groupoid) of -simplices. We denote by the standard -simplex, given by . The Yoneda lemma implies that .
A map of simplicial sets is called injective iff it is so levelwise (i.e. every is injective). A map is called étale (resp. locally injective) iff for every map in , the induced map is a bijection (resp. injective) (it is equivalent to impose this condition only for ). A map is called étale (resp. locally injective) at a given -simplex of iff is a bijection (resp. injective) (this condition at a given implies the same at any preimage of under any structure map of ). A map is called sufficiently étale iff the -simplices of at which the map is étale surject onto the -simplices of (it is equivalent to impose this condition only for ). These notions generalize to maps of simplicial groupoids by replacing ‘injectivity’ and ‘surjectivity’ for maps of sets with ‘full faithfulness’ and ‘essential surjectivity’ for functors of groupoids. These properties are all preserved under pullback and closed under composition.
The geometric realization of a simplicial set is the colimit of over all maps . Geometric realization is cocontinuous, and the natural map is bijective, however it need not be a homeomorphism even for finite limits (for example, the case of binary products is discussed in [27, Theorem 14.3, Remark 14.4] and [38, 29, 9]). The map is a homeomorphism if at least one of the maps and is locally injective. Thus a locally injective simplicial set groupoid M_{\bullet}\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O_{\bullet} determines a topological groupoid \left\|M_{\bullet}\right\|\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}\left\|O_{\bullet}\right\|.
Let us now introduce the geometric realization of any simplicial groupoid which is étale, meaning that it admits a locally injective sufficiently étale map from a simplicial set . For any simplicial groupoid and any locally injective map , the pair of simplicial sets U_{\bullet}\times_{X_{\bullet}}U_{\bullet}\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U_{\bullet} forms a locally injective simplicial set groupoid, whose geometric realization \left\|U_{\bullet}\times_{X_{\bullet}}U_{\bullet}\right\|\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}\left\|U_{\bullet}\right\| thus defines a topological stack. The geometric realization of is defined as this topological stack [\left\|U_{\bullet}\times_{X_{\bullet}}U_{\bullet}\right\|\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}\left\|U_{\bullet}\right\|] associated to any locally injective sufficiently étale map .
Lemma 4.6**.**
The geometric realization of an étale simplicial groupoid is well defined.
Proof.
Let be locally injective and sufficiently étale. Let , and consider the map of simplicial set groupoids (U^{\prime\prime}_{\bullet}\times_{X_{\bullet}}U^{\prime\prime}_{\bullet}\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U^{\prime\prime}_{\bullet})\to(U_{\bullet}\times_{X_{\bullet}}U_{\bullet}\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U_{\bullet}). Since is locally injective and sufficiently étale, it follows that this map induces an isomorphism of topological stacks, and the same applies to in place of . ∎
Lemma 4.7**.**
The geometric realization of an étale simplicial groupoid with finite isotropy is an orbispace.
Proof.
All geometric realizations are Hausdorff, so is separated, hence has separated diagonal. Since has finite isotropy, the map has finite fibers, which combined with local injectivity of U_{\bullet}\times_{X_{\bullet}}U_{\bullet}\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U_{\bullet} implies that is universally closed, hence has universally closed diagonal. We have thus shown that has proper diagonal.
To construct an étale atlas for , let denote the simplicial subset consisting of those simplices of at which is not étale. Then is étale. To see that it is surjective, it suffices to show that is surjective (note that surjectivity descends under surjective maps such as ), and this follows since is sufficiently étale. ∎
A simplicial complex gives rise to a simplicial set (its barycentric subdivision) whose -simplices are chains of simplices (in other words, is the nerve of ). Barycentric subdivision preserves injectivity, local injectivity, étale, and sufficiently étale. There is a natural identification of geometric realizations . Moreover, for a locally injective sufficiently étale simplicial complex groupoid M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O, there is a natural identification [\left\|M\right\|\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}\left\|O\right\|]=\left\|[b_{\bullet}M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}b_{\bullet}O]\right\| (the simplicial groupoid [b_{\bullet}M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}b_{\bullet}O] is étale since M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O is sufficiently étale).
A simplicial complex of groups also admits a barycentric subdivision which is a simplicial groupoid. In fact, we will define for any simplicial complex equipped with a functor from the face poset to groupoids (a simplicial complex of groups determines such a functor which sends to ). The groupoid of -simplices in the barycentric subdivision is now defined as the groupoid of functors from the category to the category whose objects are pairs and and whose morphisms are inclusions covered by maps . The barycentric subdivision of a simplicial complex of groups is étale, as can be seen as follows. For any and , we consider the functor given by . Applying the nerve construction from just above to this functor, we obtain a simplicial set mapping to , which is the required locally injective map which is étale over . An essentially equivalent discussion (albeit without barycentrically subdividing) appears in [8, 12.24–12.25]. Since is étale, it has a geometric realization which we also write as . When are finite groups (or, more generally, groupoids with finite isotropy), then the geometric realization is an orbispace by Lemma 4.7, and this is what we have been calling the orbispace presented by the simplicial complex of finite groups .
Lemma 4.8**.**
Let M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O be a locally injective simplicial complex groupoid with the following properties:
The vertices of every simplex of are pairwise non-isomorphic via .
If simplices and of have vertex sets which are isomorphic via , then and are themselves isomorphic via .
Then there is a simplicial complex of groups giving rise to the same simplicial groupoid as M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O.
Proof.
The hypotheses imply that there is a simplicial complex whose vertices are the isomorphism classes in the vertex groupoid V(M)\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O(M), and whose simplices are the -isomorphism classes of simplices of . Now M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O defines a functor , and there is a natural isomorphism between and the simplicial groupoid [b_{\bullet}M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}b_{\bullet}O]. By definition, all the groupoids have a single isomorphism class, and all the functors are faithful (this follows from local injectivity of M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O). Choosing (independently) a base object of each shows comes from a simplicial complex of groups. ∎
The nerve of a collection of open sets is the simplicial complex whose vertices are the indices with , and in which a collection of indices spans a simplex (i.e. ) iff . A partition of unity subordinate to a locally finite open cover defines a map from to the geometric realization of the nerve of the open cover.
Proposition 4.9**.**
For any paracompact orbispace , there exists a simplicial complex of groups and a map which is injective on isotropy groups. Moreover, we may take to be locally finite, and we may take the groups for vertices to be isotropy groups of points of . If is coarsely finite-dimensional (resp. -dimensional), then may be taken to be finite-dimensional (resp. -dimensional).
Proof.
We begin with an open cover with the properties guaranteed by Proposition 4.5, and we set . Since is paracompact, by -equivariantly shrinking the spaces , we may assume that the associated open cover of coarse spaces is locally finite. We fix a covering sieve on which is invariant under the ‘exchange’ (i.e. ‘inverse’) involution of and is a connection sieve for both projections U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U. We also fix a partition of unity subordinate to the open cover
Denote by and the fibers over , so ; similarly define and with . These sets are finite since the are finite and is locally finite. Furthermore, they have the discrete topology, since is Hausdorff and is Hausdorff (since is Hausdorff and is separated).
The Hausdorff property implies that the inclusions and admit retractions defined in some open neighborhood. Now the inverse images of small open neighborhoods (i.e. open substacks ) form a basis of neighborhoods of and , so for sufficiently small , these inverse images are naturally disjoint unions and . Note that this applies only inside and for which and are non-empty: the full inverse image of inside may intersect other nontrivially. By shrinking further, we may ensure that the retraction (U_{x}\times_{X}U_{x}\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U_{x})\to(m(x)\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}o(x)) is a map of groupoids. For later purposes, let us also take small enough so that:
If then .
If then .
If then over all of .
Each of these conditions can be ensured on its own, and since the open cover is locally finite, we can ensure all at once.
We also shrink so as to ensure that each (our chosen connection sieve), which has the following implication: given with , the relation is a partial bijection between lifts of and . Furthermore, the domain of this bijection is as large as possible: for (so and for some ) with , we get a full bijection between the inverse images of and inside and (this follows since the projection is a finite covering space of degree over ).
We now consider the nerves and . Note that for simplices in these nerves, namely subsets or with or , the maps or are injective. The natural maps on index sets U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U determine maps of nerves
[TABLE]
These maps are locally injective; indeed, local injectivity means that for every with and every projecting to , there is at most one lift of with , and this is a direct consequence of our assumption that every . Let us now argue that there is a natural composition map
[TABLE]
More precisely, we claim that for non-empty finite subsets with and with projecting to (under the first and second projections, respectively), the subset defined by applying the composition map to and also satisfies . This claim follows from the property that every (indeed, this property implies that ). We have thus defined a locally injective simplicial complex groupoid
[TABLE]
We now show that this locally injective simplicial complex groupoid is sufficiently étale. Every isomorphism class of vertex (equivalently, every with ) has a representative with . To show that these representatives are étale, let be a morphism with source and target , and let . We must show that there is a bijection between morphisms and . The morphisms in question all have source inside , so we really can consider just for the present purpose. Now the bulleted conditions on from above imply that since and , we have . Thus over the map is a finite covering space of degree . Thus all points have the same number of lifts, so it follows that the connection sieve property gives us a bijection between lifts.
We have already seen above that our sufficiently étale locally injective simplicial complex groupoid satisfies the first hypothesis of Lemma 4.8, and the second hypothesis follows from the partial bijection property derived above from the connection sieve. Thus by Lemma 4.8, there is a simplicial complex of groups giving rise to the same simplicial groupoid b_{\bullet}(Z,G)=[b_{\bullet}M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}b_{\bullet}O] and thus (since M\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}O is sufficiently étale) to the same geometric realization \left\|(Z,G)\right\|=[\left\|M\right\|\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}\left\|O\right\|]. The groups associated to vertices are by definition isotropy groups of the vertex groupoid V(M)\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}V(O), which are by definition isotropy groups of points of .
To conclude, it remains to define a map which is injective on isotropy groups. To define this map, we shrink the so that the open cover is locally finite, and choose a partition of unity subordinate to the open cover . These maps lift to maps supported inside and supported inside . The collection of these lifts defines a map of topological groupoids
[TABLE]
where is the disjoint union of the open loci where (the maps and do not define a map over all of U\times_{X}U\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}U due to the fact that and may not be the full inverse images of inside and ). It remains to check that this map is injective on isotropy groups; in other words, for we must show that the map is injective. Distinct elements of are, in particular, distinct lifts of , which therefore cannot lie in any common since is injective, so we see that the map is indeed injective on isotropy groups. ∎
Proof of Theorem 1.1.
Apply Proposition 4.9 to find a simplicial complex of finite groups and a map which is injective on isotropy groups. Now Theorem 2.13 applies to to give a vector bundle , and the pullback of this bundle to satisfies the desired property since is injective on isotropy groups. ∎
5 From vector bundles to principal bundles
We derive Corollaries 1.2 and 1.3 from Theorem 1.1.
A hermitian inner product on a complex vector bundle is a map satisfying , for , and for (meaning, these conditions are imposed on the fiber over each map ).
Lemma 5.1**.**
A complex vector bundle over a paracompact orbispace admits a hermitian inner product.
Proof.
Begin with an étale atlas such that the pullback of to every is trivial. By Corollary 3.5, we may refine this cover further so that each map is the composition of with an open embedding . The pullback of to each is trivial, hence admits a hermitian inner product; by averaging, we may make it -invariant thus giving a hermitian inner product on the restriction of to each . Since is paracompact, there is a partition of unity subordinate to this open cover of , and hence is the desired hermitian inner product on . ∎
Corollary 5.2**.**
Every short exact sequence of vector bundles over a paracompact orbispace splits. ∎
Given a rank complex vector bundle with hermitian inner product, the associated frame bundle is defined by declaring that a map ( a topological space) is a map together with an isomorphism under which the pullback of is the standard hermitian inner product on . There is an action of on (by precomposition with automorphisms of respecting its hermitian inner product), giving the structure of a principal bundle over , meaning that for every map from a topological space, there is an open cover such that is isomorphic to with acting by left multiplication on the first factor.
Proof of Corollary 1.2.
Let be the rank complex vector bundle produced by Theorem 1.1. By Lemma 5.1, there exists a hermitian inner product on . The total space of the associated principal bundle has trivial isotropy since the isotropy groups of act faithfully on the fibers of .
We claim that is a Hausdorff topological space. By Corollary 3.6, it is enough to show that is an orbispace. Since is representable, the pullback of an étale atlas for is an étale atlas for . The diagonal of may be expressed as the composition . The second map is proper, being a pullback of the diagonal of . To check that the first map is proper, it suffices by descent to show that its pullback under an étale atlas is proper. This pullback is proper since is a principal bundle (of topological spaces). We have thus shown that is a topological space.
The map is representable and admits local sections (by definition), so the topological groupoid P\times_{X}P\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}P presents the stack . It can be seen by inspection that P\times_{X}P\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}P is the action groupoid of the action on . ∎
Let denote the category of topological spaces equipped with a maximal atlas of charts from open sets of with smooth transition maps (a smooth manifold an object of whose underlying topological space is Hausdorff). A smooth structure on a stack is a substack of the pullback of under the forgetful functor (meaning that is a full subcategory of for ); maps lying in (the essential image of) are then called smooth. Given a map of stacks and a smooth structure on , we may consider the pullback smooth structure on , defined as those maps whose composition with is smooth. A map of stacks equipped with smooth structures is called smooth iff the composition of any smooth map with is a smooth map (equivalently, is contained inside the pullback of ). Stacks with smooth structures form a 2-category just like stacks. This category is complete, and limits are calculated by taking the limit of the underlying stacks and declaring a map to be smooth iff every induced map is smooth. The category embeds fully faithfully into the category of stacks with smooth structures, by sending to the Yoneda functor of its underlying topological space, equipped with the smooth structure consisting of those maps which are morphisms in .
Definition 5.3**.**
A smooth orbifold is an orbispace equipped with a smooth structure such that for every (equivalently, some) étale atlas , we have when is equipped with the pullback of the smooth structure on .
By Proposition 3.3, a stack with a smooth structure is a smooth orbifold iff is Hausdorff and there is an open cover of by for smooth manifolds equipped with smooth actions of finite groups .
Proof of Corollary 1.3.
Let be the complex vector bundle produced by Theorem 1.1. It suffices to define a smooth structure on , because then we may follow the arguments of Lemma 5.1 and Corollary 1.2 in the smooth category.
We begin by arguing that a complex vector bundle over a smooth orbifold has a smooth structure in a neighborhood of any given point of . We may thus assume that for a smooth manifold acted on smoothly by a finite group , and that the pullback of to is trivial. By shrinking this chart further (and adjusting ), we may assume that our given point of corresponds to a point fixed by . Choose any trivialization where denotes the fiber over . By averaging, we may make this map equivariant, and it remains an isomorphism in a neighborhood of . We thus obtain a trivialization of near in which the action of is constant (independent of the coordinate). The standard smooth structure in this trivialization is thus, in particular, invariant under the action of , giving the desired smooth structure on near our given point of .
Having shown the existence of smooth structures locally, we now show how to patch them together. To show that there exists a smooth structure on over all of , it suffices to show that for open subsets and smooth structures on and , there exists a smooth structure on restricting to the given smooth structure on (indeed, this allows us to patch together smooth structures over arbitrary unions of open sets, by choosing a well ordering and adding one open set at a time). To prove this pairwise patching statement, it suffices to show that an isomorphism of smooth vector bundles may be approximated by a smooth isomorphism (then apply this to the identity map on equipped with the restrictions of the two given smooth structures). Since is locally the quotient of Euclidean space by a finite group action, it is locally metrizable; since it is paracompact and Hausdorff, it is thus metrizable. Thus every open subset of is metrizable, hence paracompact. We may now conclude with a smooth partition of unity argument. ∎
6 -theory of orbispaces
We derive Corollary 1.4 from Theorem 1.1. We then derive from Corollary 1.4 some basic properties of the -theory of finite rank vector bundles on orbispaces (some of this derivation is borrowed, with various technical differences, from [24, §3] and [33]). In what follows, ‘vector bundle’ can be taken to mean either real vector bundle or complex vector bundle (always of finite rank).
Proof of Corollary 1.4.
Denote our given map by , and let be the given vector bundle on . Let be any vector bundle on satisfying the conclusion of Theorem 1.1.
We first consider the local situation on . Fix , and let be an open embedding sending fixed by to and inducing an isomorphism (such an open embedding exists by the proof of Proposition 3.3). By shrinking , we may ensure that the pullbacks of and to are trivial. Choose any map on from the pullback of to the pullback of (some integer ) which is injective and -equivariant at , and average it to make it equivariant everywhere. This produces over a map from to which is injective at . Now since is paracompact Hausdorff, hence normal, there exists a continuous function supported inside with . Multiplying by this function yields a map defined on all of which is injective at . Note that the integer may be taken independent of since the rank of is bounded.
We now combine the above maps to give an everywhere injective map as follows. Fix an open cover and maps defined over all of which are injective over . Since is coarsely finite-dimensional, we may refine this open cover so that it is locally finite and has nerve of dimension for some integer . As in the proof of Lemma 4.4, there is yet another open cover of by open sets indexed by the non-empty subsets of the index set of the , such that and unless or . Note that these conditions imply that unless and that if and . Now let be a partition of unity subordinate to the open cover . We may now define our desired everywhere injective map (so ) by the formula , where is any choice of index in the set , and is the inclusion of the th direct summand.
We have thus constructed an injection over . ∎
Definition 6.1**.**
For any stack , denote by the set of isomorphism classes of vector bundles on of bounded rank.
Direct sum of vector bundles equips with the structure of an abelian monoid. A map of stacks induces a pullback map .
The reason we restrict attention to vector bundles of bounded rank is so that we may appeal to Corollary 1.4. A counterexample of Lück–Oliver [24, Example 3.11] shows that the -theory of vector bundles of unbounded rank fails to be a cohomology theory on (possibly infinite) simplicial complexes of groups, whereas we shall see below that the -theory of vector bundles of bounded rank is a cohomology theory for such.
For an abelian monoid , its group completion is the quotient of the free abelian group on the underlying set of by the subgroup generated by for . The map is universal (initial) among maps from to an abelian group.
Definition 6.2**.**
is the group completion of for any stack .
A map of stacks induces a pullback map . The functor is finitely additive, in the sense that and the natural map is an isomorphism. In the present context of vector bundles of bounded rank, additivity does not hold for general infinite disjoint unions.
Eventually, we will restrict attention to the -theory of orbispaces satisfying the hypothesis of Theorem 1.1. However, we will impose such hypotheses gradually, as they become relevant.
To show that -theory is homotopy invariant, the following is the key assertion.
Lemma 6.3**.**
For any paracompact orbispace , every vector bundle on is pulled back from .
Proof.
Let a vector bundle over be given.
We first discuss the local structure around a given point . Fix an open embedding and a lift of fixed by . We consider the pullback bundle on , which is a -equivariant vector bundle. For each , there exist open neighborhoods and such that is trivial. By compactness of , we may assume is independent of . Replacing with , we may assume that for each there exists an open neighborhood such that is trivial. A trivialization induces a map which is the identity over . Averaging makes this map -equivariant, so it descends to a map which is still the identity over . It is thus an isomorphism over some ; another compactness argument and shrinking of ensures that we have a collection of isomorphisms which are the identity over . Patching these together via a partition of unity on and further shrinking produces an isomorphism
[TABLE]
which is the identity over . Any vector bundle over is pulled back from , and hence we conclude that is pulled back from , for some neighborhood of .
We now globalize. Begin with an open cover and over each an isomorphism which is the identity over . Note that for any and any , the map determines an isomorphism
[TABLE]
which is the specialization of an isomorphism between the pullbacks of under the two projections . Using this observation, we may now patch together the into an isomorphism as follows. Fix a partition of unity subordinate to the open cover , and fix an arbitrary total ordering of the index set of the open cover. For , let denote the indices of the open sets containing , ordered as in the fixed total order, and consider the composition of isomorphisms
[TABLE]
This fiberwise description now translates into the desired isomorphism of vector bundles . ∎
Two maps will be called homotopic iff there exists a map whose restrictions to and are the two given maps. A map is called a homotopy equivalence iff there exists a map such that the compositions and are homotopic to the respective identity maps.
Corollary 6.4** (Homotopy Invariance).**
If is a paracompact orbispace, then homotopic maps induce the same map . ∎
For any map of abelian monoids , the quotient is, as a set, the quotient of by the equivalence relation iff there exist with , equipped with the descent of the monoid operation from . The map is initial among maps from to an abelian monoid sending to zero. If (the image inside of) is an abelian group, then the kernel of the quotient map is precisely . The group completion is the quotient of by the diagonal submonoid .
Definition 6.5**.**
For any map of stacks , the relative -theory is defined as follows. We consider the set of isomorphism classes of triples where are vector bundles on of bounded rank and is an isomorphism between their pullbacks to . Now is an abelian monoid under direct sum, and it contains as the submonoid of triples of the form . The relative -theory is the quotient
[TABLE]
Note that a priori is merely an abelian monoid, not an abelian group.
A map (i.e. a commutative square) induces a map . There is a natural map given by sending to the formal difference , and the composition is zero. The map is an isomorphism (since is simply ).
A pair shall mean that is a closed substack of . A map of pairs means a map sending inside . Note that paracompactness, coarse finite-dimensionality, being an orbispace, and the hypothesis of Theorem 1.1 all pass to closed substacks.
Remark 6.6*.*
Even for very nice orbispace pairs , we cannot in general form the quotient in a reasonable way. For example, for an orbispace , if we could reasonably define as an orbispace, it would follow that is (globally) the quotient of a topological space by a finite group action (a very special property).
For any pair , let denote the pair . Here denotes the union of closed substacks of (recall that closed substacks of a stack are in bijective correspondence with closed subsets of , so by union of closed substacks we mean union of subsets of ). There is a natural map of pairs . A map of pairs induces a map .
Lemma 6.7**.**
For any pair where is a paracompact orbispace, is an abelian group.
Proof.
Let be a triple representing an arbitrary element of . We claim that the triple is an inverse to it. It suffices to show that
[TABLE]
are isomorphic. In other words, it suffices to show that is the restriction to of an automorphism of . By Lemma 6.3, there exist isomorphisms . In such coordinates, the desired automorphism of may be given by
[TABLE]
on and the identity on (note that to specify a map of vector bundles over a stack, it suffices to specify its restriction to each member of a finite cover by closed substacks, subject to the requirement that these restrictions agree on their pairwise overlaps). ∎
Lemma 6.8**.**
For any pair where is a paracompact orbispace, the map is surjective.
Proof.
We claim that every vector bundle on is pulled back from . Let be a vector bundle on , let denote its restriction to , and let us show that . The identity map is an identification of and over , and by Lemma 6.3 there is an identification of and over which agrees with the identity over . These isomorphisms thus patch together to define the desired isomorphism.
Since every vector bundle on is pulled back from , it follows that is surjective, so we are done. ∎
Lemma 6.9**.**
For every vector bundle over an orbispace , there exists an open cover such that the restriction of to each is pulled back from (equivalently, the pullback of to is -equivariantly trivial).
Proof.
Every point has an open neighborhood of the form by Proposition 3.3. Choose a lift of , and by shrinking and replacing with the stabilizer of , assume that fixes . Choose a map from to the trivial bundle which is the identity at , average this map to make it -equivariant, and shrink so that this map is an isomorphism over all of . ∎
Lemma 6.10**.**
Let be a finite group action on a Hausdorff topological space. If is paracompact, then so is .
Proof.
Let an open covering be given. For every , there exists an open neighborhood of which is -invariant and whose -translates are disjoint and each contained in some (possibly different) . The images of such in form an open covering of . Refining this to a locally finite covering and taking inverse images in gives the desired locally finite refinement of the original covering. ∎
Lemma 6.11**.**
Let be a paracompact orbispace, let be a vector bundle over , and let be a closed substack. Every section of extends to a section of .
Proof.
Fix an open cover such that the pullback of to each is trivialized -equivariantly (Lemma 6.9). Let be a partition of unity subordinate to this covering. Now is a closed subset of a paracompact Hausdorff space, hence paracompact Hausdorff; its inverse image inside is thus paracompact Hausdorff by Lemma 6.10 (recall is Hausdorff). Thus by the Tietze extension theorem, our given section on extends to . We can make it -equivariant by averaging, so our given section on extends to . Now is an open cover of , so pick another partition of unity subordinate to this cover, and use it to patch together the extended sections on each . ∎
Lemma 6.12**.**
For any paracompact orbispace pair , the map is an isomorphism.
Proof.
We know from the proof of Lemma 6.8 that every vector bundle on is pulled back from . Given this fact, it suffices to show that the pullback map is injective. Thus suppose that we are given two triples and whose pullbacks to coincide, meaning that there are isomorphisms on intertwining and over . Now these maps extend to all of by Lemma 6.11, and these extended maps are isomorphisms over open neighborhood of . Since is universally closed (being a pullback of ), this neighborhood contains for some open neighborhood of . Now a paracompact Hausdorff space is normal, so by Urysohn’s lemma there exists a continuous function supported inside which is identically on . Pulling back under the graph of defines isomorphisms whose restrictions to intertwine and , showing that the original triples on are isomorphic. ∎
Proposition 6.13** (Relative Homotopy Invariance).**
For any map of paracompact orbispace pairs whose constituent maps and are individually homotopy equivalences, the map is an isomorphism.
Proof.
We begin by showing that the two maps (pullback under and ) coincide (from which it follows that homotopic maps of pairs induce the same map on relative ). Since vector bundles on are pulled back from by Proposition 6.3, this amounts to showing that triples represent the same element of if and are homotopic. The construction from the proof of Lemma 6.7 shows that the homotopy between and the identity gives rise to an isomorphism between the pullbacks of these triples to , which is enough by Lemma 6.12.
Since homotopic maps of pairs induce the same map on relative , it follows that homotopy equivalences of pairs induce isomorphisms on relative . In light of Lemma 6.12, it thus suffices to show that a map of pairs whose constituent maps and are individually homotopy equivalences induces a homotopy equivalence of pairs . This is well known: given , let and be homotopy inverses to and . To define a map which is on and on , we need a homotopy between the two maps . A distinguished homotopy class of such homotopies is furnished by composing further with the homotopy equivalence and fixing homotopies between and and between and . One then checks that this map is a homotopy inverse to the map . ∎
Lemma 6.14**.**
Let be a paracompact orbispace. Every vector bundle over a closed substack is the restriction of a vector bundle over some open substack containing .
Proof.
Fix a locally finite open cover such that the pullback of to each is trivialized -equivariantly (Lemma 6.9). Such trivializations evidently extend to . We patch together these extensions on as follows.
Choose closed substacks with and whose interiors cover . We thus have a collection of transition functions satisfying and over . We now execute the following operation for every pair inductively according to an arbitrary well-ordering of such pairs. For , do nothing. For , choose an extension of from to using Lemma 6.11. This extension remains an isomorphism in a neighborhood of . Since (the coarse space of) is paracompact Hausdorff, hence normal, we may choose a closed substack whose interior contains and over which the extension of is an isomorphism. Now this new over gives unique extensions of the remaining to such that the cocycle condition is satisfied. We now replace with and go on to the next pair of indices. Note that since our cover is locally finite, the set remains closed even after possibly infinitely many steps.
After processing every pair , our extended transition functions define a vector bundle on a closed substack whose restriction to is , and by construction is contained in the interior of . ∎
We now come to the exactness and excision properties of -theory, where we will finally make use of Corollary 1.4.
For the proof of exactness, we will make use of the following notion. Let us call a map of abelian monoids cofinal iff for every there exist and with . The quotient is an abelian group iff is cofinal. The conclusion of Corollary 1.4 for a map is that the pullback map is cofinal (recalling from Corollary 5.2 that every inclusion of vector bundles over a paracompact orbispace is split).
Proposition 6.15** (Exactness).**
Let be an orbispace satisfying the hypothesis of Theorem 1.1, let be closed substacks, let be their intersection, and let be their union. The following sequence is exact:
[TABLE]
Proof.
The following sequence is exact
[TABLE]
in the sense that the image of the first map coincides with the inverse image of under the second map. Indeed, given a triple for which extends to , we may glue this extension to over to lift from to . Quotienting this sequence by , we conclude that
[TABLE]
is exact. Now is cofinal by Corollary 1.4, so is an abelian group, so the kernel of the map from to its quotient by is precisely . This quotient is simply , so we are done. ∎
For the proof of excision, we will make use of the following alternative description of relative -theory in terms of direct limits. Let denote the category whose objects are vector bundles on of bounded rank and whose morphisms are homotopy classes of injective maps. The category is filtered, meaning that (1) it is non-empty, (2) for every pair of objects , there exist morphisms , and (3) for every pair of morphisms x\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}y there exists a morphism such that the two compositions coincide. Indeed, (1) we have the zero vector bundle, (2) given vector bundles and , they both admit a morphism to , and (3) given two injections , they become homotopic after composing with the inclusion .
For any map of stacks and any vector bundle on , let denote the set of isomorphism classes of pairs consisting of a vector bundle on of bounded rank and an isomorphism . If is a paracompact orbispace, forms a directed system over . Indeed, given an inclusion , there is a natural map since the inclusion has a(n automatically unique up to homotopy) splitting by Corollary 5.2. For such , we claim that may be expressed as the direct limit
[TABLE]
Elements of the direct limit above are, by definition, equivalence classes of triples . Two triples and are equivalent in the above direct limit iff they become isomorphic after pushing both to a common , which is the same as saying and are isomorphic after adding elements of .
Recall that a functor between filtered categories is called cofinal iff (1) for every in the target there exists a morphism , and (2) for every two morphisms x\mathrel{\vbox{{\halign{#\cr\nointerlineskip\cr\rightarrow\rightarrow\crcr}}}}F(y), there exists a morphism such that the compositions coincide. The significance of cofinality is that pulling back under a cofinal functor induces an isomorphism on direct limits:
[TABLE]
is an isomorphism for any cofinal functor between filtered categories and and any directed system of sets over .
The conclusion of Corollary 1.4 for is equivalent to the assertion that the pullback functor is cofinal. Indeed, always satisfies condition (2) since any two inclusions become homotopic upon postcomposing with , and condition (1) is precisely the conclusion of Corollary 1.4.
Proposition 6.16** (Excision).**
Let be a representable map of orbispaces satisfying the hypothesis of Theorem 1.1. Let be arbitrary, and let denote its pullback along . If there is an open cover such that is an isomorphism over and is an isomorphism over , then the natural map is an isomorphism.
Proof.
We may write the map in question in terms of direct limits as
[TABLE]
By Corollary 1.4, we may replace the second direct limit with the corresponding direct limit over . It thus suffices to show that the pullback map is an isomorphism for every vector bundle on of bounded rank with . Since is an isomorphism over and , this coincides with the map , which is a bijection since is an isomorphism over . ∎
Given the significance of the hypothesis of Theorem 1.1, it is essential to show that this property is preserved under various natural operations.
Lemma 6.17**.**
For any topological stack and any locally compact Hausdorff space , the natural map is an isomorphism.
Proof.
For any atlas , the induced map is a topological quotient map, meaning it is surjective and a subset of the target is open iff its inverse image in the source is open. Topological quotient maps are preserved by taking product with a locally compact Hausdorff space [37, Lemma 4], so is also a topological quotient map. Since is also an atlas, the map is also a topological quotient map. Now is a bijection, so we are done. ∎
Lemma 6.18**.**
If a topological space is coarsely finite-dimensional, then so is for any closed subset .
Proof.
We begin with the case . Let a cover be given. For every and , there exists a pair of open neighborhoods and such that is contained in some . Fixing and using compactness of , we see that there are finitely many such covering . Hence there exists an open neighborhood and an such that is contained in some for every . We consider the collection of all such pairs . Since is coarsely finite-dimensional, there is a collection of such pairs such that the nerve of the covering is finite-dimensional. Now cover by for integers .
Next, we consider the case . Let a cover of be given. The intersection of this cover with has a refinement with finite-dimensional nerve by the case . Consider this refinement intersected with , and take union over all integers to obtain a refinement of the original cover of with finite-dimensional nerve.
Finally, by induction we obtain the case , and the general case follows since coarse finite-dimensionality passes to closed subsets. ∎
Corollary 6.19**.**
If satisfies the hypothesis of Theorem 1.1, then so does for any closed subset .
Let us now recall the Puppe sequence, which produces from Propositions 6.15 and 6.16 a long exact sequence. Let be any pair of orbispaces satisfying the hypothesis of Theorem 1.1. Let . We now have maps of pairs
[TABLE]
(note that these pairs satisfy the hypothesis of Theorem 1.1 by Corollary 6.19). We also have maps up to inverting maps inducing isomorphisms on . Namely, these ‘connecting maps’ are given by
[TABLE]
where we note that the domain admits a natural map to (which induces an isomorphism on by Proposition 6.13) and the target a natural map from . To see that this second map also induces an isomorphism on , factor it as
[TABLE]
and observe that the first map is an isomorphism by excision Proposition 6.16 (and some manipulation involving adding to the second space of both pairs, which does not change their homotopy type but makes it possibly to apply Proposition 6.16) and the second map is an isomorphism by Proposition 6.13 (a homotopy inverse to is given by projection to , and the second terms are both homotopy equivalent to ).
Definition 6.20**.**
For any pair , we define222To keep track of signs, one should really write this as etc., however for the purposes of our presentation here, we will not be so precise.
[TABLE]
Given this definition, we can write the Puppe sequence above in a more familiar form. Namely, for an orbispace satisfying the hypothesis of Theorem 1.1, we have a sequence of the form
[TABLE]
functorial in the pair .
Proposition 6.21** (Long Exact Sequence).**
The sequence above is exact.
Proof.
Each of the following three triples satisfies the hypotheses of Proposition 6.15:
[TABLE]
and the last pair is homotopy equivalent to ∎
We now define a multiplicative structure on -theory. There is a natural map sending to . To define the tensor product map on relative -theory, we will consider the following alternative model based on chain complexes.
Definition 6.22**.**
Let be a pair. Consider bounded complexes of vector bundles on which are exact over . Let us call two such complexes homotopic iff they are the restrictions to and of a complex on acyclic over (homotopy is an equivalence relation). Denote the set of homotopy classes of such complexes by , which is an abelian monoid under direct sum, and define
[TABLE]
This quotient is an abelian group: indeed, the sum of a complex and its shift is homotopic to the mapping cone of the identity map.
Proposition 6.23**.**
There is a natural isomorphism for paracompact orbispace pairs .
Proof.
There is a natural map defined as follows. Given a triple , extend the isomorphism to a neighborhood of using Lemma 6.11, and multiply by a cutoff function supported inside this neighborhood to obtain a globally defined map which is an isomorphism over . This complex is well defined up to homotopy, so we have defined a map . This map evidently sends to (the image of) , so it defines a map as desired.
We may also define a natural map as follows. Given a complex , choose a metric on each using Lemma 5.1, and note that the map is an isomorphism wherever is exact. Since all metrics are homotopic and is homotopy invariant by Proposition 6.13, sending to the triple gives a well defined map . The composition is evidently the identity.
To finish, it suffices to show that is surjective. Equivalently, we are to show that every element of is represented by a complex concentrated in degrees . To do this, we use the following “folding” operation. Given a complex concentrated in degrees , the differential is injective over , hence over an open neighborhood of inside . If is in fact everywhere injective, then we may choose a metric on and split it as , showing that the subcomplex is a direct summand, so our complex represents the same class in as one concentrated in degrees . Now consider the general case where is not assumed everywhere injective. By replacing our given complex by its direct sum with the mapping cone of the identity map concentrated in degrees , we may assume that there exists an injection which agrees with over a neighborhood of . Now multiply by a cutoff function supported inside and positive on (this is homotopic to the original differential), and take the convex interpolation between and as a further homotopy. This ensures that the differential is everywhere injective, so our complex represents the same class in as one concentrated in degrees . The dual operation shows that any complex concentrated in degrees represents the same class in as one concentrated in degrees . Combining these two operations, we can put anything in degrees . ∎
Tensor product at the level of evidently defines commutative and associative maps
[TABLE]
for paracompact orbispace pairs and for which is paracompact. These induce, by inspection, associative and graded commutative maps
[TABLE]
under the same hypotheses (note that a paracompact space times a compact space is paracompact).
We conclude with a discussion of Bott periodicity. So far, our discussion has applied equally to complex vector bundles as to real vector bundles, however we now restrict to complex vector bundles. The Bott element is represented by the complex of vector bundles on (which contains as the unit disk).
Proposition 6.24**.**
For any finite simplicial complex of groups with subcomplex , multiplication by the Bott element is an isomorphism for all .
Proof.
By the long exact sequence, excision, the five lemma, and finite additivity, we are reduced to the case of . By the definition of , we are further reduced to the case of , namely to showing that multiplication by the Bott element
[TABLE]
is an isomorphism for all , which is well known [5] [2, Theorem 4.3] [33, §3]. ∎
In fact, the argument above shows more generally that for any complex line bundle over a finite simplicial complex of groups , pullback followed by multiplication with the relative Bott class in (represented by the complex on ) defines an isomorphism (and, yet more generally, that pullback and multiplication with the Koszul complex defines an isomorphism for any complex vector bundle ).
Definition 6.25**.**
The periodic -theory of a pair is the direct limit
[TABLE]
over multiplication by . Whereas is defined only in non-positive degrees, is defined in all degrees.
Since direct limits are exact, is, like , a cohomology theory for orbispace pairs satisfying the hypothesis of Theorem 1.1. When the natural map is an isomorphism, provides a natural extension of from non-positive degrees to all degrees.
Corollary 6.26**.**
For any finite simplicial complex of groups with subcomplex , the natural map is an isomorphism in non-positive degrees. ∎
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