group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic 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
Be?linson-Bernstein localization?
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 $\mathbf{H}$ 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.
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.
flavours of Cohomotopy cohomology theory | cohomology (full or rational) | equivariant cohomology (full or rational) |
---|---|---|
non-abelian cohomology | Cohomotopy (full or rational) | equivariant Cohomotopy |
twisted cohomology (full or rational) | twisted Cohomotopy | twisted equivariant Cohomotopy |
stable cohomology (full or rational) | stable Cohomotopy | equivariant stable Cohomotopy |
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) cohomology | representing spectrum | equivariant cohomology of the point $\ast$ | cohomology of classifying space $B G$ |
---|---|---|---|
(equivariant) ordinary cohomology | HZ | Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$ | |
(equivariant) complex K-theory | KU | representation ring $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$ | Atiyah-Segal completion theorem $R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$ |
(equivariant) complex cobordism cohomology | MU | $MU_G(\ast)$ | completion theorem for complex cobordism cohomology $MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$ |
(equivariant) algebraic K-theory | $K \mathbb{F}_p$ | representation ring $(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$ | Rector completion theorem $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$ |
(equivariant) stable cohomotopy | $K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S | Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$ | Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$ |
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 on $\mathbf{B}G$ | $G$-∞-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 $\mathbf{B}G \to \mathbf{B}H$ | restricted representation |
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) |
Introduction to Borel equivariant cohomology:
Loring Tu, What is… Equivariant Cohomology?, Notices of the AMS, Volume 85, Number 3, March 2011 (pdf, pdf)
Loring Tu, Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies 204, AMS 2020 (ISBN:9780691191744)
Introduction to Bredon equivariant cohomology:
Andrew Blumberg, section 1.4 of Equivariant homotopy theory, 2017 (pdf, GitHub)
John Greenlees, Peter May, section 3 of Equivariant stable homotopy theory (pdf)
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 et al., Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics Volume: 91; 1996 (ISBN:978-0-8218-0319-6, pdf, 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
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)
Discussion of equivariant versions of differential cohomology is in
See also at equivariant de Rham cohomology.
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, Equivariant 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.
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 $X$, this is the ordinary cohomology of (the bare homotopy type of) the Borel construction $X \!\sslash\! G \;\simeq\; X \times_G E G$, 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 $G_x$ are, by definition, finite groups, their classifying spaces $\ast \!\sslash\! G \simeq B G$ 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 $X/G$.
In this form, as an invariant of just $X/G$, 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 $\mathcal{X}$ turns out to be just Borel cohomology with rational coefficients, hence is just Satake’s coarse cohomology – but applied to the inertia orbifold of $\mathcal{X}$. A review that makes this nicely explicit is (see p. 4 and 7):
Hence Chen-Ruan cohomology of a global quotient orbifold is equivalently the rational cohomology (real cohomology, complex cohomology) for the topological quotient space $AutMor(X\!\sslash\!G)/G$ of the space of automorphisms in the action groupoid by the $G$-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 $G$-equivariant Bredon cohomology of $X$ – for one specific choice of equivariant coefficient system (abelian sheaf on the orbit category of $G$), namely for $G/H \mapsto ClassFunctions(H)$.
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 $X/G$ with coefficients in the abelian sheaf whose stalk at $[x] \in X/G$ is the ring of class functions of the isotropy group at $x$; and then appeals to Theorem 5.5 in
for the followup statement that the abelian sheaf cohomology of $X/G$ with coefficient sheaf $\underline{A}$ being “locally constant except for dependence on isotropy groups” is equivalently Bredon cohomology of $X$ with coefficients in $G/H \mapsto \underline{A}_x$ for $Isotr_x = H$. This identification of the coefficient systems is Prop. 6.5 b) in:
See also Section 4.3 of
In summary:
Traditional orbifold cohomology theory is Borel cohomology of underlying Borel construction-spaces, and reduces rationally further to the rational cohomology of underlying naive quotient spaces.
Chen-Ruan cohomology is just the latter rational cohomology but applied after passage to the inertia orbifold. This is equivalent to the Bredon cohomology of the original orbifold, for one specific equivariant coefficient-system.
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 $\mathcal{X}$ which are equivalent to a global quotient orbifold $\mathcal{X} \simeq \prec(X \!\sslash\! G)$ to be the $G$-equivariant K-theory of $X$: $K^\bullet(\mathcal{X}) \;\coloneqq\; K_G^\bullet(X) \,.$
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 October 16, 2021 at 13:31:44. See the history of this page for a list of all contributions to it.