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equivariant cohomology

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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. This is particularly interesting, and traditionally considered, for some choice of “geometric” cohomology, hence cohomology inside an (∞,1)-topos possibly richer than that of geometrically discrete ∞-groupoids.

We now first describe the idea of forming equivariant cohomology as such in an ambient (∞,1)-topos H\mathbf{H}

and then afterwards indicate what this amounts to in someimportant special cases of choices of H\mathbf{H}

Equivariance

In the simplest situation the group action on the coefficients is trivial and one is dealing with cohomology of spaces XX that are equipped with a GG-action (G-spaces). Here a cocycle in equivariant cohomology is an ordinary cocycle cH(X,A)c \in \mathbf{H}(X,A) on XX, together with an equivalence cg *cc \simeq g^\ast c coherently for each generalized element gg of GG, hence is a cocycle which is GG- invariant , but only up to coherent choices of equivalences. Diagrammatically this means that where a non-equivariant cocycle on XX with coefficients in AA is just a map c:XAc \colon X \to A (see at cohomology) an equivariant cocycle is a natural system of diagrams of the form

X c A ρ X(g) = X c A \array{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{\mathllap{\rho_X(g)}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{=}} \\ X &\underset{c}{\longrightarrow}& A }

Standard examples of this kind of equivariant cocycles are traditional equivariant bundles or cocycles in equivariant de Rham cohomology. This kind of equivariant cocycle is the same as just a single cocycle on the homotopy quotient X//GX//G. Since a standard model for homotopy quotients is the Borel construction, this kind of equivariant cohomology with trivial GG-action on the coefficients is also called Borel equivariant cohomology.

In general the group GG also acts on the coefficients AA, and then an equivariant cocycle is a map c:XAc \;\colon\; X \to A which is invariant, up to equivalence, under the joint action of GG on base space and coefficients. Diagrammatically this is a natural system of diagrams of the form

X c A ρ X(g) ρ A(g) X c A. \array{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{\mathllap{\rho_X(g)}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\rho_A(g)}} \\ X &\underset{c}{\longrightarrow}& A } \,.

More concisely this means that an equivariant cocycle is a homotopy fixed point of the non-equivariant cocycle ∞-groupoid H(X,A)\mathbf{H}(X,A):

H G(X,A)π 0(H(X,A) G). H^G(X,A) \simeq \pi_0(\mathbf{H}(X,A)^G) \,.

By the discussion at ∞-action one may phrase this abstractly as follows: spaces and coefficients with GG-∞-action are objects in the slice (∞,1)-topos of the ambient (∞,1)-topos H\mathbf{H}

GAct (H)H /BG, G Act_\infty(\mathbf{H})\simeq \mathbf{H}_{/\mathbf{B}G} \,,

and GG-equivariant cohomology is the dependent product base change along

BG:H /BGH \underset{\mathbf{B}G}{\prod} \;\colon\; \mathbf{H}_{/\mathbf{B}G} \longrightarrow \mathbf{H}

of internal homs in the slice over BG\mathbf{B}G:

H G(X,A)π 0Γ(BG[X,A]). H^G(X,A) \simeq \pi_0 \Gamma \left( \underset{\mathbf{B}G}{\prod} [X,A] \right) \,.

(This formally recovers the above special case of Borel-equivariant cohomology by the dual incarnation of the projection formula (theone denoted γ¯\overline{\gamma} at Wirthmüller context – The comparison maps), according to which BG[ρ X,A][ BGρ X,A][X//G,A]\prod_{\mathbf{B}G}[\rho_X,A]\simeq [\sum_{\mathbf{B}G} \rho_X,A] \simeq [X//G,A].)

Hence equivariant cohomology is a natural generalization of group cohomology, to which it reduces when the base space is a point.

If here the cohomology is to be \mathbb{Z}-graded this means that the coefficients AA are the stages in a spectrum object in H /BG\mathbf{H}_{/\mathbf{B}G}, which is a spectrum with G-action. These are hence the coefficients for equivariant generalized (Eilenberg-Steenrod) cohomology. (More generally one considers genuine G-spectra in equivariant stable homotopy theory, but as coefficients in cohomology they appear only through their underlying spectra with G-action as above (see below and see e.g. Greenlees-May, p. 16)).

Among the simplest non-trivial example of this GG-equivariance with joint action on domain and coefficients is real oriented generalized cohomology theory such as notably KR-theory, which is equivariance with respect to a 2\mathbb{Z}_2-action. This appears notably in type II string theory on orientifold backgrounds, where the extra group action on the coefficients is exhibited by what is called the worldsheet parity operator. The word “orientifold” is modeled on that of “orbifold” to reflect precisely this extra action (on coefficients) of non-Borel 2\mathbb{Z}_2-equivariant cohomology.

Similarly, equivariant K-theory is topological K-theory not just over spaces with GG-action, but of vector bundles whose fibers are GG-representations, and such that the GG-action on the base intertwines that on the fibers.

On the other extreme, when the GG-action on the domain space happens to be trivial and only the coefficients have nontrivial GG-action, then a cocycle in equivariant cohomology is a system of the form

X c A = ρ A(g) X c A \array{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{=}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\rho_A(g)}} \\ X &\underset{c}{\longrightarrow}& A }

and hence is equivalently a map

cXA G c \;\coloneqq\; X \longrightarrow A^G

to the homotopy fixed points A GA^G of the coefficients (formed in H\mathbf{H}! See below for different incarnations ).

Hence we have in summary:

cohomology in the presence of ∞-group GG ∞-action:

Borel equivariant cohomology\leftarrowgeneral (Bredon) equivariant cohomology\rightarrownon-equivariant cohomology with homotopy fixed point coefficients
H(X G,A)\mathbf{H}(X_G, A)trivial action on coefficients AA[X,A] G[X,A]^Gtrivial action on domain space XXH(X,A G)\mathbf{H}(X, A^G)

Geometricity

Exactly what the above 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:

In general one may (and should) consider equivariant cohomology for any ambient (∞,1)-topos H\mathbf{H} and any ∞-group object GGrp(H)G \in Grp(\mathbf{H}). But traditional literature on equivariant homotopy theory/equivariant cohomology considers specifically only the choice H=PSh (Orb)\mathbf{H} = PSh_\infty(Orb) (and only somewhat implicitly,in fact traditional literature explicitly considers \infty-presheaves on the GG-orbit category Orb GOrb_G. This relates to the above via the standard equivalence PSh (Orb) /BGPSh (Orb /BG)PSh (Orb G)PSh_\infty(Orb)_{/\mathbf{B}G} \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb_G).

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 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)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 theorie? 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

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type∞-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}H
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

A quick introduction is in

More details are in

  • 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)

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

Parts of the above material is taken from a blog discussion between Mike and Urs here.

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.

In differential geometry

Equivariant degree-2 U(1)U(1)-Lie group cohomology is discussed in

Revised on April 16, 2014 02:29:35 by Urs Schreiber (92.68.97.89)