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
and then afterwards indicate what this amounts to in someimportant special cases of choices of
In the simplest situation the group action on the coefficients is trivial and one is dealing with cohomology of spaces that are equipped with a -action (G-spaces). Here a cocycle in equivariant cohomology is an ordinary cocycle on , together with an equivalence coherently for each generalized element of , hence is a cocycle which is - invariant , but only up to coherent choices of equivalences. Diagrammatically this means that where a non-equivariant cocycle on with coefficients in is just a map (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 . Since a standard model for homotopy quotients is the Borel construction, this kind of equivariant cohomology with trivial -action on the coefficients is also called Borel equivariant cohomology.
In general the group also acts on the coefficients , and then an equivariant cocycle is a map which is invariant, up to equivalence, under the joint action of on base space and coefficients. Diagrammatically this is a natural system of diagrams of the form
of internal homs in the slice over :
(This formally recovers the above special case of Borel-equivariant cohomology by the dual incarnation of the projection formula (theone denoted at Wirthmüller context – The comparison maps), according to which .)
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 -graded this means that the coefficients are the stages in a spectrum object in , 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 -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 -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 -equivariant cohomology.
Similarly, equivariant K-theory is topological K-theory not just over spaces with -action, but of vector bundles whose fibers are -representations, and such that the -action on the base intertwines that on the fibers.
On the other extreme, when the -action on the domain space happens to be trivial and only the coefficients have nontrivial -action, then a cocycle in equivariant cohomology is a system of the form
and hence is equivalently a map
Hence we have in summary:
|Borel equivariant cohomology||general (Bredon) equivariant cohomology||non-equivariant cohomology with homotopy fixed point coefficients|
|trivial action on coefficients||trivial action on domain space|
“coarse” equivariance. For ∞Grpd Top, and a discrete group, regarded via its delooping groupoid/classifying space , then is presented by the Borel model structure on the category of simplicial sets equipped with -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 (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 -action). In the case tha the domain itself is the points with trivial -action then the equivariant cohomology here is precisely the group cohomology of .
“fine” Bredon equivariance. In order to bring in more geometric information one may equip G-spaces with information about the actual -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 , or if desired just on the global orbit category , hence to set , where the base (∞,1)-topos is that of orbispaces and sitting cohesively over it is the “global equivariant homotopy theory” proper (see there).
Now we have naturally via the (∞,1)-Yoneda embedding and the slice (∞,1)-topos is the traditional equivariant homotopy theory presented by the “fine” model structure on G-spaces whose weak equivalences are the -fixed point wise weak homotopy equivalences for all suitble subgroups . 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 -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)).
fully geometric equivariance. More complete geometric information is retained if one takes ETop∞Grpd or Smooth∞Grpd, which by the discussion at canonical topology means to not only test on moduli stacks of compact Lie groups (as in the global equivariant indexing category) but on all topological/smooth ∞-stacks. Then again 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 and any ∞-group object . But traditional literature on equivariant homotopy theory/equivariant cohomology considers specifically only the choice (and only somewhat implicitly,in fact traditional literature explicitly considers -presheaves on the -orbit category . This relates to the above via the standard equivalence .
Recall from the discussion at cohomology that in full generality we have a notion of cohomology of an object with coefficients in an object whenever and are objects of some (∞,1)-topos . The cohomology set is the set of connected components in the hom-object ∞-groupoid of maps from to : .
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 -sheaves on a site of smooth test spaces such as Diff these objects have the interpretation of Lie ∞-groupoids.
In this case, for some such ∞-groupoid with structure, let be its 0-truncation, which is the space of objects of , the categorically discrete groupoid underlying . We think of the morphisms in as determining which points of are related under some kind of action on , the 2-morphisms as relating these relations on some higher action, and so on. Equivariance means, roughly: functorial transformation behaviour of objects on with respect to this “action” encoded in the morphisms in . This is the intuition that is made precise in the following
In the simple special case that one should keep in mind, is for instance the action groupoid of the action, in the ordinary sense, of a group on : its morphisms connect those objects of that are related by the action by some group element .
It is natural to consider the relative cohomology of the inclusion . Equivariant cohomology is essentially just another term for relative cohomology with respect to an inclusion of a space into a (-)groupoid.
In some (∞,1)-topos the equivariant cohomology with coefficient in an object of a 0-truncated object with respect to an action encoded in an inclusion is simply the -valued cohomology of .
More specifically, an equivariant structure on an -cocycle on is a choice of extension
i.e. a lift of through the projection .
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 some group let be the stack of -principal bundles. Let be some finite group (just for the sake of simplicity of the example) and let be an action of on a space . Let be the corresponding action groupoid.
Then a cocycle in the -equivariant cohomology is
Consider for definiteness , the path ∞-groupoid of . (All other cases are in principle obtaind from this by truncation and/or strictification).
Then for some coefficient -groupoid, a morphism can be thought of as classifying a -principal ∞-bundle on the space .
On the other hand, a morphism out of is something like a flat connection (see there for more details) on this principal -bundle, also called an -local system. (More details on this are at differential cohomology).
Accordingly, an extension of through the inclusion is the process of equipping a principal -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.
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.
According to the nPOV on cohomology, if and are objects in an (∞,1)-topos, the 0th cohomology is , while if is a group object, then . More generally, if is times deloopable, then . In Top, this gives you the usual notions if is a (discrete) group, and in general, classifies principal ∞-bundles in whatever (∞,1)-topos.
Now consider the -topos of -equivariant spaces, which can also be described as the (∞,1)-presheaves on the orbit category of . For any other group there is a notion of a principal -bundle (where is the group of equivariance, and is the structure group of the bundle), and these are classified by maps into a classifying -space . So the principal -bundles over can be called . If we had something of which was a delooping, we could call the principal -bundles ””, but there does not seem to be such a thing. It seems that is not connected, in the sense that is not an effective epimorphism and thus is not the quotient of a group object in .
If we have an object of our -topos that can be delooped infinitely many times, then we can define for any integer by looking at all the spaces . 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 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 such that ; the stronger requirement that restricts us to “connective” spectra, those that can be produced by successively delooping a single object of the -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 , and the resulting notion of spectrum is called a naive G-spectrum: a sequence of -spaces with . Any naive -spectrum represents a cohomology theory on -spaces. The most “basic” of these are “Eilenberg-Mac Lane -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 -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 -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 , we look at , where is a finite-dimensional representation of and is its one-point compactification. Now if is a -space that can be delooped ” times,” we can define . If can be delooped times for all representations , then our integer-graded cohomology theory can be expanded to an -graded cohomology theory, with cohomology groups for all formal differences of representations . The corresponding notion of spectrum is a genuine G-spectrum, which consists of spaces for all representations such that . A naive Eilenberg-Mac Lane -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 -graded Bredon cohomology theory .
-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 -manifolds: if is a -manifold, then we can embed it in a representation (generally not a trivial one!) and by Thom space arguments, obtain a Poincare duality theorem involving a dimension shift of , where is generally not an integer (and, apparently, not even uniquely determined by !). Unfortunately, however, -graded Bredon cohomology is kind of hard to compute.
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
|homotopy type theory||representation theory|
|pointed connected context||∞-group|
|dependent sum along||coinvariants/homotopy quotient|
|context extension along||trivial representation|
|dependent product along||homotopy invariants/∞-group cohomology|
|dependent product of internal hom along||equivariant cohomology|
|dependent sum along||induced representation|
|context extension along|
|dependent product along||coinduced representation|
|spectrum object in context||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
(with an eye towards application to the Arf-Kervaire invariant problem)
Equivariant complex oriented cohomology theory is discussed in the following articles.
Specifically equivariant complex cobordism cohomology is discussed in
Tammo tom Dieck, Bordism of -manifolds and integrability theorems, Topology 9 (1970) 345-358
The following articles discuss equivariant formal group laws:
(also Abrams 13a, section III).
See also the references at equivariant elliptic cohomology.
Equivariant degree-2 -Lie group cohomology is discussed in