nLab
orbifold cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher geometry

Contents

Idea

Since the crucial extra structures carried by an orbifold are

  1. geometric structure (e.g. topological, algebro-geometric, differential-geometric, super-geometric, etc.)

  2. singularities

the cohomology of orbifolds should be such as to provide invariants which are sensitive not just to the underlying plain homotopy type of an orbifold (its shape) but also to this extra structure. This means that orbifold cohomology should, respectively, unify

  1. geometric cohomology (e.g. sheaf hypercohomology, differential cohomology, etc.)

  2. equivariant cohomology in its fine form of Bredon cohomology

in the sense that geometric cohomology is recovered away from the orbifold singularities and equivariant cohomology is recovered right at the singularities, while globally orbifold cohomology provides a unification of these two aspects.

Since any concept of cohomology (as discussed there) is effectively equivalent to the choice of ambient (∞,1)-topos, the question of defining orbifold cohomology is closely related to the question of how exactly to define the (∞,1)-category of orbifolds (usually a (2,1)-category) in the first place. This question is notoriously more subtle than the simple intuitive idea of orbifolds might suggest, as witnessed by the convoluted history of the concept (see e.g. Lerman 08, Introduction).


Deficiency of orbifolds as geometric groupoids

A proposal popular among Lie theorists (Moerdijk-Pronk 97) is to regard an orbifold with local charts U iG iActionsU_i \in G_i Actions (actions of some group on some local model space) as the geometric stack obtained by gluing the corresponding homotopy quotients/quotient stacks U iG iU_i \!\sslash \!G_i. If H\mathbf{H} is the ambient cohesive (∞,1)-topos in which this takes place (for instance H=\mathbf{H} = Smooth∞Groupoids, SuperFormalSmooth∞Groupoids, etc.) and if GGrpDiscGrp(H)G \in Grp \overset{Disc}{\hookrightarrow} Grp(\mathbf{H}) is a discrete group in which all the isotropy groups of the orbifold are contained, this gives an object

(𝒳 faith BG faith)(H /BG) 0H /BG \left( \array{ \mathcal{X} \\ \downarrow^{\mathrlap{faith}} \\ \mathbf{B}G } \phantom{{}^{faith}} \right) \;\in\; \left( \mathbf{H}_{/\mathbf{B}G} \right)_{\leq 0} \hookrightarrow \mathbf{H}_{/\mathbf{B}G}

in the slice (∞,1)-topos over the delooping BG=*G\mathbf{B}G = \ast \sslash G of GG, which is still a 0-truncated object, reflecting that as a functor of groupoids the morphism 𝒳BG\mathcal{X} \to \mathbf{B}G is a faithful functor.

Accordingly, if this is – or were – the correct formalization of the nature of orbifolds 𝒳\mathcal{X}, then the corresponding orbifold cohomology has coefficients given by objects 𝒜H /BG\mathcal{A} \in \mathbf{H}_{/\mathbf{B}G} and cohomology sets being the connected components of the (∞,1)-categorical hom-spaces

H BG(𝒳,𝒜)π 0H /BG(𝒳,𝒜). H_{\mathbf{B}G}\big( \mathcal{X}, \mathcal{A}\big) \;\coloneqq\; \pi_0 \mathbf{H}_{/\mathbf{B}G}\big( \mathcal{X}, \mathcal{A}\big) \,.

This concept of orbifold cohomology does fully reflect the geometric nature of orbifolds. It also reflects some equivariance aspect. For example if 𝒳=*G\mathcal{X} = \ast \sslash G is the one-point orbifold with singularity given by a finite group GG, and if VGRepresentationsV \in G Representations is a linear representation, with K(V,n)GGroupoids /BGDiscH /BGK(V,n)\sslash G \in \infty Groupoids_{/\mathbf{B}G} \overset{Disc}{\hookrightarrow} \mathbf{H}_{/\mathbf{B}G} its Eilenberg-MacLane space, then

H BG(BG,K(V,n)G)H grp n(G,V) H_{\mathbf{B}G}\big( \mathbf{B}G, K(V,n)\sslash G \big) \;\simeq\; H^n_{grp}(G,V)

is the group cohomology of GG with coefficients in VV.

However, this definition does not reflect Bredon-equivariant cohomology around the orbifold singularities. Instead, it really given (geometric/stacky refinement) of cohomology with local coefficients.

\,

Lift of orbifolds to singular-cohesive homotopy theory

Hence the proposal of Moerdijk-Pronk 97, that an orbifold should be regarded as a certain geometric stack, is missing something. It was briefly suggested in Schwede 17, Introduction, Schwede 18, p. ix-x that the missing aspect is provided by global equivariant homotopy theory, but details seem to have been left open.

Here we discuss how to define the required orbifold cohomology in detail and in general. We combine the differential cohesion for the geometric aspect with the cohesion of global equivariant homotopy theory that was observed and highlighted in Rezk 14.

The following may serve as intuition for the issue with the nature of orbifolds:

Envision the picture of an orbifold singularity and hold a mathemagical magnifying glass over the singular point. Under this magnification you can see resolved the singular point as a fuzzy fattened point, to be called 𝔹G\mathbb{B}G.

Removing the magnifying glass, what one sees with the bare eye depends on how one squints:

  • The physicist says that what he sees is a singular point, but a point after all. This is the plain quotient *=*/G\ast = \ast / G.

  • The Lie geometer says that what she sees is a point transforming under the GG-action that fixes it, hence the homotopy quotient groupoid BG=*G\mathbf{B}G =\ast \sslash G.

These are two opposite extreme aspects of the orbifold singularity 𝔹G\mathbb{B}G, but the orbifold singularity itself is more than both of these aspects. The real nature of an orbifold singularity is in fact a point, not a big classifying space BG\mathbf{B} G (recall that already B 2=P \mathbf{B}\mathbb{Z}_2 = \mathbb{R}P^\infty), but it is a point that also remembers the group action, for that characterizes how the singularity is being singular:

orbifold singularity𝔹G < Aopposite extremeaspects of orbifold singularity plain quotient*=*/G homotopy quotientBG=*G \array{ && { \text{orbifold singularity} \atop {\mathbb{B}G} } \\ & {}^{\mathllap{\boxed{\lt}}}\swarrow & {{\phantom{A}} \atop { \text{opposite extreme} \atop \text{aspects of orbifold singularity} }} & \searrow^{\mathrlap{ \boxed{\subset} }} \\ { \text{plain quotient} \atop {\ast = \ast/G} } && && { \text{homotopy quotient} \atop { \mathbf{B}G = \ast \sslash G } } }

This “unity of opposites” may be captured by the modalities on the singular-cohesive \infty -topos of singular-smooth \infty -groupoids. Its intrinsic cohomology accomodates a good notion of orbifold cohomology (SaSc 2020).

History

According to Abramovich 05, p. 42:

On December 7, 1995 Maxim Kontsevich delivered a history-making lecture at Orsay, titled String Cohomology. This is what is now know, after Chen-Ruan 00, as orbifold cohomology, Kontsevich’s lecture notes described the orbifold and quantum cohomology of a global quotient orbifold. Twisted sectors, the age grading, and a version of orbifold stable maps for global quotients are all there.

The same lecture also introduced motivic integration.

References

Traditional orbifold cohomology

Traditionally, the cohomology of orbifolds has, by and large, been taken to be simply the ordinary cohomology of (the plain homotopy type of) the geometric realization of the topological/Lie groupoid corresponding to the orbifold.

For the global quotient orbifold of a G-space XX, this is the ordinary cohomology of (the bare homotopy type of) the Borel construction XGX× GEGX \!\sslash\! G \;\simeq\; X \times_G E G , hence is Borel cohomology (as opposed to finer versions of equivariant cohomology such as Bredon cohomology).

A dedicated account of this Borel cohomology of orbifolds, in the generality of twisted cohomology (i.e. with local coefficients) is in:

Moreover, since the orbifold’s isotropy groups G xG_x are, by definition, finite groups, their classifying spaces *GBG\ast \!\sslash\! G \simeq B G have purely torsion integral cohomology in positive degrees, and hence become indistinguishable from the point in rational cohomology (and more generally whenever their order is invertible in the coefficient ring).

Therefore, in the special case of rational/real/complex coefficients, the traditional orbifold Borel cohomology reduces further to an invariant of just (the homotopy type of) the naive quotient underlying an orbifold. For global quotient orbifolds this is the topological quotient space X/GX/G.

In this form, as an invariant of just X/GX/G, the real/complex/de Rham cohomology of orbifolds was originally introduced in

following analogous constructions in

Since this traditional rational cohomology of orbifolds does, hence, not actually reflect the specific nature of orbifolds, a proposal for a finer notion of orbifold cohomology was famously introduced (motivated from orbifolds as target spaces in string theory, hence from orbifolding of 2d CFTs) in

However, Chen-Ruan cohomology of an orbifold 𝒳\mathcal{X} turns out to be just Borel cohomology with rational coefficients, hence is just Satake’s coarse cohomology – but applied to the inertia orbifold of 𝒳\mathcal{X}. A review that makes this nicely explicit is (see p. 4 and 7):

  • Emily Clader, Orbifolds and orbifold cohomology, 2014 (pdf)

Hence Chen-Ruan cohomology of a global quotient orbifold is equivalently the rational cohomology (real cohomology, complex cohomology) for the topological quotient space AutMor(XG)/GAutMor(X\!\sslash\!G)/G of the space of automorphisms in the action groupoid by the GG-conjugation action.

On the other hand, it was observed in (see p. 18)

that for global quotient orbifolds Chen-Ruan cohomology indeed is equivalent to a GG-equivariant Bredon cohomology of XX – for one specific choice of equivariant coefficient system (abelian sheaf on the orbit category of GG), namely for G/HClassFunctions(H)G/H \mapsto ClassFunctions(H).

Or rather, Moerdijk 02, p. 18 observes that the Chen-Ruan cohomology of a global quotient orbifold is equivalently the abelian sheaf cohomology of the naive quotient space X/GX/G with coefficients in the abelian sheaf whose stalk at [x]X/G[x] \in X/G is the ring of class functions of the isotropy group at xx; and then appeals to Theorem 5.5 in

  • Hannu Honkasalo, Equivariant Alexander-Spanier cohomology for actions of compact Lie groups, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 (jstor:24492569)

for the followup statement that the abelian sheaf cohomology of X/GX/G with coefficient sheaf A̲\underline{A} being “locally constant except for dependence on isotropy groups” is equivalently Bredon cohomology of XX with coefficients in G/HA̲ xG/H \mapsto \underline{A}_x for Isotr x=HIsotr_x = H. This identification of the coefficient systems is Prop. 6.5 b) in:

See also Section 4.3 of

In summary:

This suggests, of course, that more of proper equivariant cohomology should be brought to bear on a theory of orbifold cohomology. A partial way to achieve this is to prove for a given equivariant cohomology-theory that it descends from an invariant of topological G-spaces to one of the associated global quotient orbifolds.

For topological equivariant K-theory this is the case, by

Therefore it makes sense to define orbifold K-theory for orbifolds 𝒳\mathcal{X} which are equivalent to a global quotient orbifold 𝒳(XG) \mathcal{X} \simeq \prec(X \!\sslash\! G) to be the GG-equivariant K-theory of XX: K (𝒳)K G (X). K^\bullet(\mathcal{X}) \;\coloneqq\; K_G^\bullet(X) \,.

This is the approach taken in

Exposition and review of traditional orbifold cohomology, with an emphasis on Chen-Ruan cohomology and orbifold K-theory, is in:

Proper orbifold cohomology

Discussion of orbifold cohomology in the context of Bredon cohomology:

The suggestion to regard orbifold cohomology in global equivariant homotopy theory:

Spelling out this suggestion of Schwede 17, Intro, Schwede 1, p. ix-x8:

Based on results of or related to:

The observation of the cohesion of global- over G-equivariant homotopy theory is due to:

A formulation of orbifold cohomology in the singular-cohesive \infty -topos of singular-smooth \infty -groupoids is then discussed in:

Last revised on October 26, 2021 at 06:49:34. See the history of this page for a list of all contributions to it.