group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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 $\mathbf{H}$
and then afterwards indicate what this amounts to in someimportant special cases of choices of $\mathbf{H}$
In the simplest situation the group action on the coefficients is trivial and one is dealing with cohomology of spaces $X$ that are equipped with a $G$-action (G-spaces). Here a cocycle in equivariant cohomology is an ordinary cocycle $c \in \mathbf{H}(X,A)$ on $X$, together with an equivalence $c \simeq g^\ast c$ coherently for each generalized element $g$ of $G$, hence is a cocycle which is $G$- invariant , but only up to coherent choices of equivalences. Diagrammatically this means that where a non-equivariant cocycle on $X$ with coefficients in $A$ is just a map $c \colon X \to A$ (see at cohomology) an equivariant cocycle is a natural system of diagrams of the form
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 $X//G$. Since a standard model for homotopy quotients is the Borel construction, this kind of equivariant cohomology with trivial $G$-action on the coefficients is also called Borel equivariant cohomology.
In general the group $G$ also acts on the coefficients $A$, and then an equivariant cocycle is a map $c \;\colon\; X \to A$ which is invariant, up to equivalence, under the joint action of $G$ on base space and coefficients. Diagrammatically this is a natural system of diagrams of the form
More concisely this means that an equivariant cocycle is a homotopy fixed point of the non-equivariant cocycle ∞-groupoid $\mathbf{H}(X,A)$:
By the discussion at ∞-action one may phrase this abstractly as follows: spaces and coefficients with $G$-∞-action are objects in the slice (∞,1)-topos of the ambient (∞,1)-topos $\mathbf{H}$
and $G$-equivariant cohomology is the dependent product base change along
of internal homs in the slice over $\mathbf{B}G$:
(This formally recovers the above special case of Borel-equivariant cohomology by the dual incarnation of the projection formula (theone denoted $\overline{\gamma}$ at Wirthmüller context – The comparison maps), according to which $\prod_{\mathbf{B}G}[\rho_X,A]\simeq [\sum_{\mathbf{B}G} \rho_X,A] \simeq [X//G,A]$.)
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 $A$ are the stages in a spectrum object in $\mathbf{H}_{/\mathbf{B}G}$, 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, see e.g. Greenlees-May, p. 16)).
Among the simplest non-trivial example of this $G$-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 $\mathbb{Z}_2$-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 $\mathbb{Z}_2$-equivariant cohomology.
Similarly, equivariant K-theory is topological K-theory not just over spaces with $G$-action, but of vector bundles whose fibers are $G$-representations, and such that the $G$-action on the base intertwines that on the fibers.
On the other extreme, when the $G$-action on the domain space happens to be trivial and only the coefficients have nontrivial $G$-action, then a cocycle in equivariant cohomology is a system of the form
and hence is equivalently a map
to the homotopy fixed points $A^G$ of the coefficients (formed in $\mathbf{H}$! See below for different incarnations ).
Hence we have in summary:
cohomology in the presence of ∞-group $G$ ∞-action:
Borel equivariant cohomology | $\leftarrow$ | general (Bredon) equivariant cohomology | $\rightarrow$ | non-equivariant cohomology with homotopy fixed point coefficients |
---|---|---|---|---|
$\mathbf{H}(X_G, A)$ | trivial action on coefficients $A$ | $[X,A]^G$ | trivial action on domain space $X$ | $\mathbf{H}(X, A^G)$ |
Exactly what the above comes down to depends on the choice of ambient (∞,1)-topos $\mathbf{H}$ and of the way that $G$ is regarded as an ∞-group object of $\mathbf{H}$. Some important choices are the following:
“coarse” equivariance. For $\mathbf{H} =$ ∞Grpd $\simeq L_{whe}$ Top, and $G$ a discrete group, regarded via its delooping groupoid/classifying space $\mathbf{B}G \in \mathbf{H}$, then $\mathbf{H}_{/\mathbf{B}G}$ is presented by the Borel model structure on the category of simplicial sets equipped with $G$-action. (This is also called the coarse equivariant homotopy theory, in view of the next examples). This theory only knows homotopy quotients and homotopy fixed points of $G$ (in particular cofibrant replacement in the Borel model structure is indeed given by the Borel construction and so Borel equivariant cohomology theory appears here whenever the coefficients have trivial $G$-action). In the case tha the domain itself is the points with trivial $G$-action then the equivariant cohomology here is precisely the group cohomology of $G$.
“fine” Bredon equivariance. In order to bring in more geometric information one may equip G-spaces with information about the actual $G$-fixed points, not just their homotopy fixed points. By general lore of topos theory this means to have all spaces be probe-able by fixed points, hence to have them be (∞,1)-presheaves on the global equivariant indexing category $Glob$, or if desired just on the global orbit category $Orb$, hence to set $(\mathbf{H} \to \mathbf{B}) = (PSh_\infty(Glob) \to PSh_\infty(Orb))$, where the base (∞,1)-topos is that of orbispaces and $\mathbf{H}$ sitting cohesively over it is the “global equivariant homotopy theory” proper (see there).
Now we have $\mathbf{B}G \in \mathbf{H}$ naturally via the (∞,1)-Yoneda embedding and the slice (∞,1)-topos $\mathbf{B}_{/\mathbf{B}G} \simeq L_{fpwe} G Top$ is the traditional equivariant homotopy theory presented by the “fine” model structure on G-spaces whose weak equivalences are the $H$-fixed point wise weak homotopy equivalences for all suitble subgroups $H \hookrightarrow G$. The spectrum objects here are what are called spectra with G-action or “naive G-spectra”. See at Elmendorf's theorem for details. By the discussion there every object in the fine model structure if fibrant and cofibrant replacement here is given by passage to G-CW complexes, so that the derived hom spaces computing cohomology are the ordinary $G$-fixed points of the mapping spectra from such as G-CW complex into the coefficient spectrum (this is traditionally motivated via detour through genuine G-spectra, see e.f. Greenlees-May, equation (3.7)).
Cohomology with Eilenberg-MacLane object-coefficients in $PSh_\infty(Orb)_{/\mathbf{B}G}$ is what Glen Bredon originally considered as what is now called Bredon cohomology.
fully geometric equivariance. More complete geometric information is retained if one takes $\mathbf{H} =$ ETop∞Grpd or $=$Smooth∞Grpd, which by the discussion at canonical topology means to not only test on moduli stacks $\mathbf{B}G$ of compact Lie groups (as in the global equivariant indexing category) but on all topological/smooth ∞-stacks. Then again $G$ itself embeds canonically, and now its equivariant cohomology now is refined Segal-Brylinski-Lie group cohomology (see the discussion there).
In general one may (and should) consider equivariant cohomology for any ambient (∞,1)-topos $\mathbf{H}$ and any ∞-group object $G \in Grp(\mathbf{H})$. But traditional literature on equivariant homotopy theory/equivariant cohomology considers specifically only the choice $\mathbf{H} = PSh_\infty(Orb)$ (and only somewhat implicitly,in fact traditional literature explicitly considers $\infty$-presheaves on the $G$-orbit category $Orb_G$. This relates to the above via the standard equivalence $PSh_\infty(Orb)_{/\mathbf{B}G} \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb_G)$.
under construction
(…) Elmendorf theorem (…) Borel model structure (…)
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.
Borel equivariant cohomology is the cohomology of action groupoids (homotopy quotients/Borel constructions).
For standard cohomology in the (∞,1)-topos $\mathbf{H} =$ Top these action groupoids of a group $G$ acting on a topological space $X$ are traditionally known as the Borel construction $\mathcal{E}G \times_G X$.
Recall from the discussion at cohomology that in full generality we have a notion of cohomology of an object $X$ with coefficients in an object $A$ whenever $X$ and $A$ are objects of some (∞,1)-topos $\mathbf{H}$. The cohomology set $H(X,A)$ is the set of connected components in the hom-object ∞-groupoid of maps from $X$ to $A$: $H(X,A) = \pi_0 \mathbf{H}(X,A)$.
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 $(\infty,1)$-sheaves on a site of smooth test spaces such as Diff these objects have the interpretation of Lie ∞-groupoids.
In this case, for $X$ some such ∞-groupoid with structure, let $X_0 \hookrightarrow X$ be its 0-truncation, which is the space of objects of $X$, the categorically discrete groupoid underlying $X$. We think of the morphisms in $X$ as determining which points of $X_0$ are related under some kind of action on $X_0$, the 2-morphisms as relating these relations on some higher action, and so on. Equivariance means, roughly: functorial transformation behaviour of objects on $X_0$ with respect to this “action” encoded in the morphisms in $X$. This is the intuition that is made precise in the following
In the simple special case that one should keep in mind, $X$ is for instance the action groupoid $X = X_0//G$ of the action, in the ordinary sense, of a group $G$ on $X_0$: its morphisms $x \to g(x)$ connect those objects of $X_0$ that are related by the action by some group element $g \in G$.
It is natural to consider the relative cohomology of the inclusion $X_0 \hookrightarrow X$. Equivariant cohomology is essentially just another term for relative cohomology with respect to an inclusion of a space into a ($\infty$-)groupoid.
In some (∞,1)-topos $\mathbf{H}$ the equivariant cohomology with coefficient in an object $A$ of a 0-truncated object $X_0$ with respect to an action encoded in an inclusion $X_0 \hookrightarrow X$ is simply the $A$-valued cohomology $H(X,A)$ of $X$.
More specifically, an equivariant structure on an $A$-cocycle $c : X_0 \to A$ on $X_0$ is a choice of extension $\hat c$
i.e. a lift of $c$ through the projection $\mathbf{H}(X,A) \to \mathbf{H}(X_0,A)$.
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.
For $G$ some group let $G Bund$ be the stack of $G$-principal bundles. Let $K$ be some finite group (just for the sake of simplicity of the example) and let $K \to Aut(X_0)$ be an action of $K$ on a space $X_0$. Let $X = X_0 // K$ be the corresponding action groupoid.
Then a cocycle in the $K$-equivariant cohomology $H(X_0//K, G Bund)$ is
a $G$-principal bundle $P \to X$ on $X$;
for each $k \in K$ an isomorphism of $G$-principal bundles $\lambda_k : P \to k^* P$
such that for all $k_1, k_2 \in K$ we have $\lambda_{k_2}\circ \lambda_{k_1} = \lambda_{k_2\cdot k_1}$.
For $X_0$ a space and $X := P_n(X_0)$ a version of its path n-groupoid we have a canonical inclusion $X_0 \hookrightarrow P_n(X_0)$ of $X_0$ as the collection of constant paths in $X_0$.
Consider for definiteness $\Pi(X_0) := \Pi_\infty(X_0)$, the path ∞-groupoid of $X_0$. (All other cases are in principle obtaind from this by truncation and/or strictification).
Then for $A$ some coefficient $\infty$-groupoid, a morphism $g : X_0 \to A$ can be thought of as classifying a $A$-principal ∞-bundle on the space $X_0$.
On the other hand, a morphism out of $P_n(X_0)$ is something like a flat connection (see there for more details) on this principal $\infty$-bundle, also called an $A$-local system. (More details on this are at differential cohomology).
Accordingly, an extension of $g : X_0 \to A$ through the inclusion $X_0 \hookrightarrow \Pi(X)$ 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.
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.
See also
According to the nPOV on cohomology, if $X$ and $A$ are objects in an (∞,1)-topos, the 0th cohomology $H^0(X;A)$ is $\pi_0(Map(X,A))$, while if $A$ is a group object, then $H^1(X;A)= \pi_0(Map(X,B A))$. More generally, if $A$ is $n$ times deloopable, then $H^n(X;A) = \pi_0(Map(X, B^n A)$. In Top, this gives you the usual notions if $A$ is a (discrete) group, and in general, $H^1(X;A)$ classifies principal ∞-bundles in whatever (∞,1)-topos.
Now consider the $(\infty,1)$-topos $G Top$ of $G$-equivariant spaces, which can also be described as the (∞,1)-presheaves on the orbit category of $G$. For any other group $\Pi$ there is a notion of a principal $(G,\Pi)$-bundle (where $G$ is the group of equivariance, and $\Pi$ is the structure group of the bundle), and these are classified by maps into a classifying $G$-space $B_G \Pi$. So the principal $(G,\Pi)$-bundles over $X$ can be called $H^0(X;B_G \Pi)$. If we had something of which $B_G \Pi$ was a delooping, we could call the principal $(G,\Pi)$-bundles “$H^1(X;?)$”, but there does not seem to be such a thing. It seems that $B_G \Pi$ is not connected, in the sense that ${*}\to B_G \Pi$ is not an effective epimorphism and thus $B_G \Pi$ is not the quotient of a group object in $G Top$.
If we have an object $A$ of our $(\infty,1)$-topos that can be delooped infinitely many times, then we can define $H^n(X;A)$ for any integer $n$ by looking at all the spaces $\Omega^{-n} A = B^n A$. 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 $\{B^n A\}$ 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 $\{E_n\}$ such that $E_n \simeq \Omega E_{n+1}$; the stronger requirement that $E_{n+1} \simeq B E_n$ restricts us to “connective” spectra, those that can be produced by successively delooping a single object of the $(\infty,1)$-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 $G Top$, and the resulting notion of spectrum is called a naive G-spectrum: a sequence of $G$-spaces $\{E_n\}$ with $E_n \simeq \Omega E_{n+1}$. Any naive $G$-spectrum represents a cohomology theory on $G$-spaces. The most “basic” of these are “Eilenberg-Mac Lane $G$-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 $G$-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 $G$-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 $\Omega^n = Map(S^n, -)$, we look at $\Omega^V = Map(S^V,-)$, where $V$ is a finite-dimensional representation of $G$ and $S^V$ is its one-point compactification. Now if $A$ is a $G$-space that can be delooped “$V$ times,” we can define $H^V(X;A) = \pi_0(Map(X,\Omega^{-V} A)$. If $A$ can be delooped $V$ times for all representations $V$, then our integer-graded cohomology theory can be expanded to an RO(G)-graded cohomology theory, with cohomology groups $H^\alpha(X;A)$ for all formal differences of representations $\alpha = V - W$. The corresponding notion of spectrum is a genuine G-spectrum, which consists of spaces $E_V$ for all representations $V$ such that $E_V \simeq \Omega^{W-V} E_W$. A naive Eilenberg-Mac Lane $G$-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 $RO(G)$-graded Bredon cohomology theory .
$RO(G)$-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 $G$-manifolds: if $M$ is a $G$-manifold, then we can embed it in a representation $V$ (generally not a trivial one!) and by Thom space arguments, obtain a Poincare duality theorem involving a dimension shift of $\alpha$, where $\alpha$ is generally not an integer (and, apparently, not even uniquely determined by $M$!). Unfortunately, however, $RO(G)$-graded Bredon cohomology is kind of hard to compute.
For more see at equivariant stable homotopy theory and global equivariant stable homotopy theory.
$\mathbb{Z}_2$-equivariant cohomology theories: KR-theory, MR-theory
modular group-equivariance: modular equivariant elliptic 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
equivariant stable homotopy theory, global equivariant stable homotopy theory
equivariant rational homotopy theory, rational equivariant stable homotopy theory
equivariant K-theory, equivariant operator K-theory, equivariant KK-theory
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):
homotopy type theory | representation theory |
---|---|
pointed connected context $\mathbf{B}G$ | ∞-group $G$ |
dependent type | ∞-action/∞-representation |
dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |
context extension along $\mathbf{B}G \to \ast$ | trivial representation |
dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |
dependent product of internal hom along $\mathbf{B}G \to \ast$ | equivariant cohomology |
dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |
context extension along $\mathbf{B}G \to \mathbf{B}H$ | |
dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |
spectrum object in context $\mathbf{B}G$ | spectrum with G-action (naive G-spectrum) |
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
Michael Hill, Michael Hopkins, Douglas Ravenel, The Arf-Kervaire problem in algebraic topology: Sketch of the proof (pdf)
(with an eye towards application to the Arf-Kervaire invariant problem)
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
Tammo tom Dieck, Bordism of $G$-manifolds and integrability theorems, Topology 9 (1970) 345-358
William Abram, Equivarisnt complex cobordism, 2013 (web, pdf)
William Abram, Igor Kriz, The equivariant complex cobordism ring of a finite abelian group (pdf)
The following articles discuss equivariant formal group laws:
John Greenlees, Equivariant formal group laws and complex oriented cohomology theories, Homology Homotopy Appl. Volume 3, Number 2 (2001), ii-451 (EUCLID)
William Abram, On the equivariant formal group law of the equivariant complex cobordism ring, (arXiv:1309.0722)
(also Abrams 13a, section III).
See also the references at equivariant elliptic cohomology.
Equivariant degree-2 $U(1)$-Lie group cohomology is discussed in