Groupoids and Orbifold Cohomology, Part 2

Transcription

1 Groupoids and Orbifold Cohomology, Part 2 Dorette Pronk (with Laura Scull) Dalhousie University (and Fort Lewis College) Groupoidfest 2011, University of Nevada Reno, January 22, 2012

2 Motivation Orbifolds: Spaces which are locally of the form R n /G for a finite group G. Equivariant homotopy theory: Homotopy theory for G-spaces, for a fixed group G. What can equivariant homotopy theory tell us about orbifold homotopy theory?

3 Outline Orbifolds and Groupoids - An Overview Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Twisted/Local Coefficients Orbifolds

4 What is an orbifold? An orbifold is a paracompact Hausdorff space with an equivalence class of orbifold atlases. Local charts are of the form U = Ũ/G for some finite group acting on an open set Ũ Rn via diffeomorphisms, ρ G : G Diffeo (Ũ), U V G and they are locally compatible. We denote an orbifold chart by (Ũ, G, ρ G, ϕ).

5 Examples 1 Manifolds (no non-trivial isotropy groups) 2 A global quotient of a properly discontinuous group G acting on a manifold M Eg M = S 1 with G = Z/2 action The orbifold consists of the orbit space M/G together with the data about the isotropy groups.

6 Examples 3 (Thurston) The teardrop orbifold: Z/n e 4 When a compact Lie group L acts on a manifold X with finite isotropy groups, the orbit space X/L is an orbifold. Such orbifolds are called representable. 5 The teardrop orbifold can be obtained by S 1 acting on S 3 via λ[z 1, z 2 ] = [λ n z 1, z 2 ]

7 Describing Manifolds with Groupoids A manifold is a smooth topological equivalence relation identifications objects M

8 Describing Orbifolds with Groupoids Introduce symmetries into this picture: The notion of groupoid generalizes both the notion of a group and the notion of an equivalence relation,

9 Smooth Groupoids A Lie groupoid G is a groupoid in the category of smooth manifolds π 1 i G mor s,gobj,t G mor m G mor G mor π 2 s u t G obj such that the source and target maps are submersions, and all the usual equations are satisfied.

10 Lie Groupoid Examples Manifolds 1: G obj = G mor = M with all structure maps identities. Manifolds 2: G obj is the disjoint union of charts and G mor is the disjoint union of all the intersections of pairs of charts (with source and target maps the appropriate embeddings). Lie groups: G obj = { } and G mor = L, a Lie group. : for a Lie group L acting on a manifold M, there is a translation groupoid L M, L L M µ 1 M L M (ι,a) L M a M π 2

11 Orbifolds and Smooth Groupoids Example 1: a Global Quotient Orbifold Take X = S 1 with the Z/2-action by reflection. reflection id morphisms objects

12 Example 2: a Single Chart Orbifold We model an order 3 cone with 2/3 1/3 morphisms id objects

13 Example 3: an orbifold atlas with several charts For the teardrop orbifold we obtain: 2/3 1/3 morphisms id X id objects

14 Equivalent atlases give rise to Morita equivalent groupoids. Morita equivalence of smooth groupoids can be described in terms of smooth essential equivalences between groupoids.

15 Essential Equivalences, I A (smooth) essential equivalence φ: G H satisfies the following two properties: 1 (Essentially surjective) is a surjective submersion, G obj Hobj H mor H obj G H obj obj φ may not be onto the objects of H, but every object in H is isomorphic to an object in the image of G.

16 Essential Equivalences, II 2 (Fully faithful) G mor (s,t) G obj G obj φ H mor (s,t) φ φ H obj H obj is a pullback, G H The local isotropy structure is preserved.

17 Morita Equivalent Groupoids Two Lie groupoids G and H are called Morita equivalent if there exists a third Lie groupoid K with essential equivalences ϕ ψ G K H. This is an equivalence relation on groupoids, because essential equivalences of Lie groupoids are stable under weak pullbacks (iso-comma-squares).

18 Morita Equivalent Presentations, I We can describe the same orbifold with different groupoids (corresponding to equivalent atlases): A line segment can be presented as morphisms objects

19 Morita Equivalent Presentations, II It can also be presented as morphisms objects

20 Morita Equivalent Presentations, III Or as: morphisms objects

21 Morita Equivalent Presentations, IV Here is our order 3 cone again. 2/3 1/3 morphisms id objects

22 Morita Equivalent Presentations, IV And here is another presentation 2/3 2/3 2/3 2/3 1/3 1/3 1/3 1/3 id id morphisms id id objects

23 Orbifold Groupoids An orbifold groupoid is a groupoid which is Morita equivalent to a proper étale groupoid. A groupoid G is étale when both maps s, t : G mor G obj are étale; A Lie groupoid G is Morita equivalent to an étale groupoid if and only if all its isotropy groups are discrete; A groupoid G is proper when the map (s, t): G mor G obj G obj is proper. An orbifold is representable precisely when its atlas groupoid is Morita equivalent to a translation groupoid L M.

24 of Two translation groupoids are Morita equivalent if and only if they can be connected by a span of equivariant essential equivalences. Every equivariant essential equivalence G X H Y can be written as a composition of a quotient essential equivalence, G X G/K X/K for K G which acts freely on X and an inclusion essential equivalence, L Z H (H L Z ) for L H.

25 What have we learned so far? When we restrict ourselves to representable orbifolds, we may restrict ourselves to translation groupoids. Equivariant homotopy invariants are orbifold invariants iff they are invariant under the equivariant Morita equivalences G X G/K X/K for K G which acts freely on X; L Z H (H L Z ) for L H.

26 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Theories Equivariant cohomology comes in two flavours: Borel cohomology and Bredon cohomology. Let X be a G-space. Borel Cohomology: Let EG be a free contractible G-space. Then the equivariant cohomology of X (with coefficients in an abelian group A) is considered to be the ordinary cohomology of the orbit space EG G X (the Borel space). This is equal to the sheaf cohomology for G-sheaves on X with constant coefficients, i.e., the cohomology of the classifying space B(G X). This cohomology is Morita invariant, so there is a version of this type of equivariant cohomology for orbifolds.

27 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds The Problem with Borel (Johann Leida) However, this homotopy theory does not tell the whole story: EG G X does not capture the G-homotopy type of X. Let D be the disk with a smooth fixed-point free action of I, the icosahedral group. The map D {pt} is an equivariant map into the point orbifold with isotropy group I. This map is a non-equivariant homotopy equivalence, which gives rise to a homotopy equivalence EI I D BI However, it is clear that we do not want to consider I D and I {pt} as the same orbifold (even up to homotopy).

28 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Bredon Cohomology of G-Spaces Let G be a Lie group or a topological group, and X a G-space. Idea Study the homotopy of X in terms of its diagram of fixed point subspaces X H := {x; h x = x for all h H} for all closed subgroups H G, with arrows given by natural inclusions and the action of G. This diagram is indexed by the orbit category O G of homogeneous G-spaces G/H.

29 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Properties of Bredon Cohomology, I For Bredon cohomology, the coefficients are given by a functor O op G Ab. Bredon cohomology is more general than Borel cohomology: Bredon cohomology agrees with Borel cohomology when the coefficient functor is constant on the objects of O G.

30 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Properties of Bredon Cohomology, II Bredon cohomology has a number of useful properties: it is related to K-theory and can be used to prove a Riemann-Roch theorem; it is more closely related to Chen-Ruan cohomology than Borel cohomology; it is the right cohomology for equivariant obstruction theory; it gives rise to an equivariant Serre spectral sequence for equivariant fibrations.

31 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds The Orbit Category Definition The orbit 2-category O G has Objects: G-sets G/H, for H G; Arrows: G-maps G/H G/K. Note: for a G-space X, a G-map ϕ: G/H X is determined by x = ϕ(eh); moreover, x X H. O G (G/H, G/K ) = (G/K ) H is a topological space. 2-Cells: homotopy classes of paths. Definition The homotopy orbit category ho G is the category of orbit types G/H, with homotopy classes (i.e., connected components) of G-equivariant morphisms as arrows.

32 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Equivariant Coefficient Systems A G-space X gives rise to a functor Φ X : O op G Spaces, Φ X (G/H) = Map G (G/H, X) = X H. An equivariant coefficient system (with constant coefficients) is a functor ho op G Ab.

33 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Bredon Cohomology There is an equivariant chain complex: C (X)(G/H) = C (X H /W 0 H) where W 0 H is the identity component of the Weyl group WH = NH/H. (This quotient is taken, because we want to consider singular simplices up to G-homotopies.) For each n, this gives a coefficient system C n (X). For any equivariant coefficient system A, there is a cochain complex: C n G (X; A) = Hom G(C n (X), A). The Bredon cohomology of X is the cohomology of this cochain complex: HBr (X; A) = H G (X; A) = H (CG (X; A))

34 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Example Let G = Z/2 act on S 1 by reflection in the line connecting the north and south pole. The orbit category O Z/2 : Coefficient systems: τ B ι A, such that τ 2 = 1 B and τι = ι. Examples: 1. B = A = Z and all structure maps are identities; σ 2. B = Z Z, A = Z, τ is interchange and ι is the diagonal; 3. B = 0 and A = Z. The resulting cohomology groups are: 1. the cohomology of the orbitspace; 2. the cohomology of S 1 ; 3. the cohomology of the fixed point set. G 0.

35 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Orbifold Bredon Cohomology Let ϕ: G X H Y be an essential equivalence of orbifold groupoids. This gives rise to a functor ϕ: ho G ho H. We need to show that for any coefficient system A on ho H there is a coefficient system ϕ A on ho G, such that HBr (X, ϕ A) = HBr (Y, A). We need to show that for any coefficient system B on ho G there is a coefficient system ϕ B on ho H, such that H Br (X, B) = H Br (Y, ϕ B).

36 Group Homomorphisms and Coefficient Systems Any group homomorphism ϕ: G H induces functors ho G Ab hoop G ϕ ϕ ϕ ho H Ab hoop H Proposition (Moerdijk Svensson) If φ: G H is any group homomorphism, then H H (H φ,g X, A) = H G (X, φ A) where H φ,g X = H X/(k, gx) (kφ(g), x). Proposition Let r X A be the restriction of the diagram A to ho G,X. If r X A = r X B then H O G (X, A) = H O G (X, B).

37 Which Cohomology? Bredon Cohomology Bredon Cohomology for Orbifolds Orbifold Bredon Cohomology Proposition For any orbifold Morita equivalence (f, ϕ): (X, G) (Y, H) and any orbifold system of coefficients A on ho G, the system ϕ A is equivalent to A, since r X ϕ ϕ A = r X A. Theorem Let X be an orbifold and G X a translation groupoid representation. For any orbifold system of coefficients A on ho G, HBr (X, [A]) is well-defined; that is, if H Y is another representation for X, there is a corresponding system A on ho H such that HBr (X, [A]) = HBr (Y, [A ]).

38 Twisted/Local Coefficients Orbifolds Local/Twisted Coefficients in Ordinary Cohomology Let (X, x 0 ) be a pointed space with local coefficients M (i.e., M may be viewed as a sheaf on X which is locally constant). Note that M x0 is a π-module, where π = π 1 (X, x 0 ) Let p : X X be the universal covering. Then X is a π-space. (Eilenberg) H n (X; M) = H n π( X; M 0 ). Application 1: classical obstruction theory (where one takes the coefficients to be the higher homotopy groups). Application 2: the Serre spectral sequence for fibrations of topological spaces.

39 Twisted/Local Coefficients Orbifolds Bredon cohomology with twisted coefficients was defined independently by [A. Mukherjee, G. Mukherjee, 1996] and [I. Moerdijk, J.A. Svensson, 1993]. The Mukherjees defined it for arbitrary topological groups, and generalized the method for constant coefficients, by using an equivariant fundamental groupoid as their new domain for coefficients, and creating an appropriate twisted system of singular chains. Moerdijk and Svensson defined it only for discrete groups, but they represented the Bredon cohomology of a G-space X as the cohomology of a category G (X). G. Mukherjee and N. Pandey (2002) showed that for discrete groups, the two cohomology theories are isomorphic.

40 Twisted/Local Coefficients Orbifolds Categories for Equivariant Bredon Cohomology, I Let X be a G-space where G is a topological group. Laura Scull and I have constructed a category p X : G (X) O G with a quotient px d : d G (X) ho G, such that for any A: ho op G Ab, H n (B d G (X), (pd X ) A) = HG n (X, A). Our proof is a straight generalization of the one given by Moerdijk and Svensson. G (X) is created out of all the singular simplices (and simplicial in the fixed point spaces X H with a Grothendieck construction over O G.

41 Orbifolds and Groupoids - An Overview Twisted/Local Coefficients Orbifolds Categories for Equivariant Bredon Cohomology, II The equivariant fundamental groupoid Π G (X) has a discretized version Π d G (X) which fits in a commutative diagram d G (X) vx d Π d G (X) p X q X ho G For twisted coefficients A: Π d G (X) Ab, H (B d G (X), (v X d ) A) = HG (X, A), where the latter is as defined by the Mukherjees.

42 Orbifolds and Groupoids - An Overview Twisted/Local Coefficients Orbifolds Theorem (P Scull) Any essential equivalence of orbifold groupoids (f, ϕ): G X H Y induces weak equivalences of categories d G (X) d H (Y ) and Πd G (X) Π d H (Y ) which fit into a commutative diagram G (X) d Π d G (X) ho G d H (Y ) Π d H (Y ) ho H and consequently give rise to isomorphisms in cohomology.

43 Twisted/Local Coefficients Orbifolds Ongoing/Future Work An orbifold Serre spectral sequence Orbifold obstruction theory Generalize these constructions to orbifold atlas groupoids (and possibly all orbifold groupoids). Connections with Chen Ruan cohomology and orbifold K -theory.