nLab
equivariant cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Representation theory

Contents

Idea

Equivariant cohomology is cohomology in the presence of and taking into account group-actions (and generally ∞-group ∞-actions) both on the domain space and on the coefficients.

Exactly what this comes down to depends on the choice of ambient (∞,1)-topos H\mathbf{H} and of the way that GG is regarded as an ∞-group object of H\mathbf{H}. Some important choices are the following:

Presentations

under construction

(…) Elmendorf theorem (…) Borel model structure (…)

Borel equivariant cohomology

We first state the general abstract definition of Borel equivariant cohomology and then derive from it the more concrete formulations that are traditionally given in the literature.

For standard cohomology in the (∞,1)-topos H=\mathbf{H} = Top these action groupoids of a group GG acting on a topological space XX are traditionally known as the Borel construction G× GX\mathcal{E}G \times_G X.

Recall from the discussion at cohomology that in full generality we have a notion of cohomology of an object XX with coefficients in an object AA whenever XX and AA are objects of some (∞,1)-topos H\mathbf{H}. The cohomology set H(X,A)H(X,A) is the set of connected components in the hom-object ∞-groupoid of maps from XX to AA: H(X,A)=π 0H(X,A)H(X,A) = \pi_0 \mathbf{H}(X,A).

Recall moreover from the discussion at space and quantity that objects of an (∞,1)-topos of (∞,1)-sheaves have the interpretation of ∞-groupoids with extra structure. For instance for (,1)(\infty,1)-sheaves on a site of smooth test spaces such as Diff these objects have the interpretation of Lie ∞-groupoids.

In this case, for XX some such ∞-groupoid with structure, let X 0XX_0 \hookrightarrow X be its 0-truncation, which is the space of objects of XX, the categorically discrete groupoid underlying XX. We think of the morphisms in XX as determining which points of X 0X_0 are related under some kind of action on X 0X_0, the 2-morphisms as relating these relations on some higher action, and so on. Equivariance means, roughly: functorial transformation behaviour of objects on X 0X_0 with respect to this “action” encoded in the morphisms in XX. This is the intuition that is made precise in the following

In the simple special case that one should keep in mind, XX is for instance the action groupoid X=X 0//GX = X_0//G of the action, in the ordinary sense, of a group GG on X 0X_0: its morphisms xg(x)x \to g(x) connect those objects of X 0X_0 that are related by the action by some group element gGg \in G.

It is natural to consider the relative cohomology of the inclusion X 0XX_0 \hookrightarrow X. Equivariant cohomology is essentially just another term for relative cohomology with respect to an inclusion of a space into a (\infty-)groupoid.

Definition (equivariant cohomology)

In some (∞,1)-topos H\mathbf{H} the equivariant cohomology with coefficient in an object AA of a 0-truncated object X 0X_0 with respect to an action encoded in an inclusion X 0XX_0 \hookrightarrow X is simply the AA-valued cohomology H(X,A)H(X,A) of XX.

More specifically, an equivariant structure on an AA-cocycle c:X 0Ac : X_0 \to A on X 0X_0 is a choice of extension c^\hat c

X 0 A c^ X. \array{ X_0 &\to& A \\ \downarrow & \nearrow_{\hat c} \\ X } \,.

i.e. a lift of cc through the projection H(X,A)H(X 0,A)\mathbf{H}(X,A) \to \mathbf{H}(X_0,A).

Examples

Group cohomology

By comparing the definition of equivariant cohomology with that of group cohomology one sees that group cohomology can be equivalently thought of as being equivariant cohomology of the point.

Equivariant cohomotopy

flavours of
Cohomotopy
cohomology theory
cohomology
(full or rational)
equivariant cohomology
(full or rational)
non-abelian cohomologyCohomotopy
(full or rational)
equivariant Cohomotopy
twisted cohomology
(full or rational)
twisted Cohomotopytwisted equivariant Cohomotopy
stable cohomology
(full or rational)
stable Cohomotopyequivariant stable Cohomotopy

Equivariant bundles

For GG some group let GBundG Bund be the stack of GG-principal bundles. Let KK be some finite group (just for the sake of simplicity of the example) and let KAut(X 0)K \to Aut(X_0) be an action of KK on a space X 0X_0. Let X=X 0//KX = X_0 // K be the corresponding action groupoid.

Then a cocycle in the KK-equivariant cohomology H(X 0//K,GBund)H(X_0//K, G Bund) is

  • a GG-principal bundle PXP \to X on XX;

  • for each kKk \in K an isomorphism of GG-principal bundles λ k:Pk *P\lambda_k : P \to k^* P

  • such that for all k 1,k 2Kk_1, k_2 \in K we have λ k 2λ k 1=λ k 2k 1\lambda_{k_2}\circ \lambda_{k_1} = \lambda_{k_2\cdot k_1}.

Local systems – flat connections

For X 0X_0 a space and X:=P n(X 0)X := P_n(X_0) a version of its path n-groupoid we have a canonical inclusion X 0P n(X 0)X_0 \hookrightarrow P_n(X_0) of X 0X_0 as the collection of constant paths in X 0X_0.

Consider for definiteness Π(X 0):=Π (X 0)\Pi(X_0) := \Pi_\infty(X_0), the path ∞-groupoid of X 0X_0. (All other cases are in principle obtaind from this by truncation and/or strictification).

Then for AA some coefficient \infty-groupoid, a morphism g:X 0Ag : X_0 \to A can be thought of as classifying a AA-principal ∞-bundle on the space X 0X_0.

On the other hand, a morphism out of P n(X 0)P_n(X_0) is something like a flat connection (see there for more details) on this principal \infty-bundle, also called an AA-local system. (More details on this are at differential cohomology).

Accordingly, an extension of g:X 0Ag : X_0 \to A through the inclusion X 0Π(X)X_0 \hookrightarrow \Pi(X) is the process of equipping a principal \infty-bundle with a flat connection.

Comparing with the above definition of eqivariant cohomology, we see that flat connections on bundles may be regarded as path-equivariant structures on these bundles.

This is therefore an example of equivariance which is not with respect to a global group action, but genuinely a groupoidal one.

Equivariant de Rham cohomology

Remarks

When pairing equivariant cohomology with other variants of cohomology such as twisted cohomology or differential cohomology one has to exercise a bit of care as to what it really is that one wants to consider. A discussion of this is (beginning to appear) at differential equivariant cohomology.

Bredon equivariant cohomology

See also

Preliminary remarks

According to the nPOV on cohomology, if XX and AA are objects in an (∞,1)-topos, the 0th cohomology H 0(X;A)H^0(X;A) is π 0(Map(X,A))\pi_0(Map(X,A)), while if AA is a group object, then H 1(X;A)=π 0(Map(X,BA))H^1(X;A)= \pi_0(Map(X,B A)). More generally, if AA is nn times deloopable, then H n(X;A)=π 0(Map(X,B nA)H^n(X;A) = \pi_0(Map(X, B^n A). In Top, this gives you the usual notions if AA is a (discrete) group, and in general, H 1(X;A)H^1(X;A) classifies principal ∞-bundles in whatever (∞,1)-topos.

Now consider the (,1)(\infty,1)-topos GTopG Top of GG-equivariant spaces, which can also be described as the (∞,1)-presheaves on the orbit category of GG. For any other group Π\Pi there is a notion of a principal (G,Π)(G,\Pi)-bundle (where GG is the group of equivariance, and Π\Pi is the structure group of the bundle), and these are classified by maps into a classifying GG-space B GΠB_G \Pi. So the principal (G,Π)(G,\Pi)-bundles over XX can be called H 0(X;B GΠ)H^0(X;B_G \Pi). If we had something of which B GΠB_G \Pi was a delooping, we could call the principal (G,Π)(G,\Pi)-bundles “H 1(X;?)H^1(X;?)”, but there does not seem to be such a thing. It seems that B GΠB_G \Pi is not connected, in the sense that *B GΠ{*}\to B_G \Pi is not an effective epimorphism and thus B GΠB_G \Pi is not the quotient of a group object in GTopG Top.

GG-equivariant spectra

If we have an object AA of our (,1)(\infty,1)-topos that can be delooped infinitely many times, then we can define H n(X;A)H^n(X;A) for any integer nn by looking at all the spaces Ω nA=B nA\Omega^{-n} A = B^n A. These integer-graded cohomology groups are closely connected to each other, e.g. they often have cup products or Steenrod squares or Poincare duality, so it makes sense to consider them all together as a cohomology theory . We then are motivated to put together all of the objects {B nA}\{B^n A\} into a spectrum object, a single object which encodes all of the cohomology groups of the theory. A general spectrum is a sequence of objects {E n}\{E_n\} such that E nΩE n+1E_n \simeq \Omega E_{n+1}; the stronger requirement that E n+1BE nE_{n+1} \simeq B E_n restricts us to “connective” spectra, those that can be produced by successively delooping a single object of the (,1)(\infty,1)-topos. In Top, the most “basic” spectra are the Eilenberg-MacLane spectra produced from the input of an ordinary abelian group.

Now we can do all of this in GTopG Top, and the resulting notion of spectrum is called a naive G-spectrum: a sequence of GG-spaces {E n}\{E_n\} with E nΩE n+1E_n \simeq \Omega E_{n+1}. Any naive GG-spectrum represents a cohomology theory on GG-spaces. The most “basic” of these are “Eilenberg-Mac Lane GG-spectra” produced from coefficient systems, i.e. abelian-group-valued presheaves on the orbit category. The cohomology theory represented by such an Eilenberg-Mac Lane GG-spectrum is called an (integer-graded) Bredon cohomology theory.

It turns out, though, that the cohomology theories arising in this way are kind of weird. For instance, when one calculates with them, one sees torsion popping up in odd places where one wouldn’t expect it. It would also be nice to have a Poincare duality theorem for GG-manifolds, but that fails with these theories. The solution people have come up with is to widen the notion of “looping” and “delooping” and thereby the grading:

instead of just looking at Ω n=Map(S n,)\Omega^n = Map(S^n, -), we look at Ω V=Map(S V,)\Omega^V = Map(S^V,-), where VV is a finite-dimensional representation of GG and S VS^V is its one-point compactification. Now if AA is a GG-space that can be deloopedVV times,” we can define H V(X;A)=π 0(Map(X,Ω VA)H^V(X;A) = \pi_0(Map(X,\Omega^{-V} A). If AA can be delooped VV times for all representations VV, then our integer-graded cohomology theory can be expanded to an RO(G)-graded cohomology theory, with cohomology groups H α(X;A)H^\alpha(X;A) for all formal differences of representations α=VW\alpha = V - W. The corresponding notion of spectrum is a genuine G-spectrum, which consists of spaces E VE_V for all representations VV such that E VΩ WVE WE_V \simeq \Omega^{W-V} E_W. A naive Eilenberg-Mac Lane GG-spectrum can be extended to a genuine one precisely when the coefficient system it came from can be extended to a Mackey functor, and in this case we get an RO(G)RO(G)-graded Bredon cohomology theory .

RO(G)RO(G)-graded Bredon cohomology has lots of formal advantages over the integer-graded theory. For instance, the torsion that popped up in odd places before can now be seen as arising by “shifting” of something in the cohomology of a point in an “off-integer dimension,” which was invisible to the integer-graded theory. Also there is a Poincare duality for GG-manifolds: if MM is a GG-manifold, then we can embed it in a representation VV (generally not a trivial one!) and by Thom space arguments, obtain a Poincare duality theorem involving a dimension shift of α\alpha, where α\alpha is generally not an integer (and, apparently, not even uniquely determined by MM!). Unfortunately, however, RO(G)RO(G)-graded Bredon cohomology is kind of hard to compute.

For more see at equivariant stable homotopy theory and global equivariant stable homotopy theory.

Examples

Multiplicative equivariant cohomology

For multiplicative cohomology theories there is a further refinement of equivariance where the equivariant cohomology groups are built from global sections on a sheaf over cerain systems of moduli spaces. For more on this see at

Examples

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point *\ast
cohomology
of classifying space BGB G
(equivariant)
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
(equivariant)
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)KU G(*)compl.KU G(*)^KU(BG)R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
(equivariant)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
(equivariant)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
(equivariant)
stable cohomotopy
K𝔽 1Segal 74K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}Hrestricted representation
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

References

General

Introduction to Borel equivariant cohomology:

  • Loring Tu, What is… Equivariant Cohomology?, Notices of the AMS, Volume 85, Number 3, March 2011 (pdf, pdf)

Introduction to Bredon equivariant cohomology:

Textbooks and lecture notes include

  • Tammo tom Dieck, section 7 of Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766, Springer 1979

  • Peter May, Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

  • Matvei Libine, Lecture Notes on Equivariant Cohomology (arXiv)

  • Sébastien Racanière, Lecture on Equivariant Cohomology, 2004 (pdf)

For a brief modern survey see also the first three sections of

Discussion of equivariant versions of differential cohomology is in

  • Andreas Kübel, Andreas Thom, Equivariant Differential Cohomology, Transactions of the American Mathematical Society (2018) (arXiv:1510.06392)

See also at equivariant de Rham cohomology.

In complex oriented generalized cohomology theory

Equivariant complex oriented cohomology theory is discussed in the following articles.

  • Michael Hopkins, Nicholas Kuhn, Douglas Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), 553-594 (publisher, pdf)

    (This deals with “naive” Borel-equivariant complex oriented cohomology, but discusses general character expressions and explicit formulas for equivariant K(n)-cohomology.)

Specifically equivariant complex cobordism cohomology is discussed in

The following articles discuss equivariant formal group laws:

See also the references at equivariant elliptic cohomology.

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:

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:

Last revised on September 11, 2020 at 03:41:49. See the history of this page for a list of all contributions to it.