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Translation groupoids and orbifold cohomology. (English) Zbl 1197.57026

Can. J. Math. 62, No. 3, 614-645 (2010); erratum ibid. 69, No. 4, 851-853 (2017).
This paper demonstrates how results from equivariant homotopy theory can be applied to the study of orbifolds. In particular, the authors consider representable orbifolds, orbifolds which can be expressed as the quotient of a manifold by a compact Lie group acting with finite stablizers. It was conjectured by [A. Adem, J. Leida, and Y. Ruan, Orbifolds and stringy topology. Cambridge Tracts in Mathematics 171. Cambridge: Cambridge University Press, (2007; Zbl 1157.57001)] that all orbifolds are representable; all effective orbifolds are representable, and results regarding noneffective orbifolds have been obtained by A. Henriques and D. S. Metzler [Trans. Am. Math. Soc. 356, 2481–2499 (2004; Zbl 1060.58013)]. Note that a representation of an orbifold as a \(G\)-space is in general not unique.
The main result of this paper is that the inclusion functor of the \(2\)-category of smooth translation groupoids and equivariant maps into the \(2\)-category of translation groupoids as a full subcategory of that of Lie groupoids induces an equivalence of the associated bicategories of fractions. In particular, restricting to only those morphisms between translation groupoids that are determined by equivariant maps does not lose important information. It follows that representations of orbifolds as \(G\)-spaces can be used to define invariants for orbifolds.
As applications, the authors show that if a \(G\)-space \(X\) is a representation of an orbifold, the equivariant \(K\)-theory \(K_G(X)\) is an invariant of the orbifold, and hence is independent of the representation. Similarly, the authors develop a definition of Bredon cohomology for representable orbifolds in terms of \(G\)-spaces and show that this yields an orbifold invariant.

MSC:

57R18 Topology and geometry of orbifolds
57R19 Algebraic topology on manifolds and differential topology
57R91 Equivariant algebraic topology of manifolds
57S15 Compact Lie groups of differentiable transformations
55N91 Equivariant homology and cohomology in algebraic topology
19L47 Equivariant \(K\)-theory
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D35 Structured objects in a category (MSC2010)
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