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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{equivariant cohomology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory} [[!include representation theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{IdeaEquivariance}{Equivariance}\dotfill \pageref*{IdeaEquivariance} \linebreak \noindent\hyperlink{IdeaGeometricity}{Geometricity}\dotfill \pageref*{IdeaGeometricity} \linebreak \noindent\hyperlink{presentations}{Presentations}\dotfill \pageref*{presentations} \linebreak \noindent\hyperlink{Borel}{Borel equivariant cohomology}\dotfill \pageref*{Borel} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{group_cohomology}{Group cohomology}\dotfill \pageref*{group_cohomology} \linebreak \noindent\hyperlink{equivariant_cohomotopy}{Equivariant cohomotopy}\dotfill \pageref*{equivariant_cohomotopy} \linebreak \noindent\hyperlink{equivariant_bundles}{Equivariant bundles}\dotfill \pageref*{equivariant_bundles} \linebreak \noindent\hyperlink{local_systems__flat_connections}{Local systems -- flat connections}\dotfill \pageref*{local_systems__flat_connections} \linebreak \noindent\hyperlink{equivariant_de_rham_cohomology}{Equivariant de Rham cohomology}\dotfill \pageref*{equivariant_de_rham_cohomology} \linebreak \noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak \noindent\hyperlink{Bredon}{Bredon equivariant cohomology}\dotfill \pageref*{Bredon} \linebreak \noindent\hyperlink{preliminary_remarks}{Preliminary remarks}\dotfill \pageref*{preliminary_remarks} \linebreak \noindent\hyperlink{equivariant_spectra}{$G$-equivariant spectra}\dotfill \pageref*{equivariant_spectra} \linebreak \noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak \noindent\hyperlink{multiplicative_equivariant_cohomology}{Multiplicative equivariant cohomology}\dotfill \pageref*{multiplicative_equivariant_cohomology} \linebreak \noindent\hyperlink{examples_3}{Examples}\dotfill \pageref*{examples_3} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{InComplexOrientedGeneralizedCohomologyTheory}{In complex oriented generalized cohomology theory}\dotfill \pageref*{InComplexOrientedGeneralizedCohomologyTheory} \linebreak \noindent\hyperlink{in_differential_geometry}{In differential geometry}\dotfill \pageref*{in_differential_geometry} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} \emph{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]] $\mathbf{H}$ \begin{itemize}% \item \emph{\hyperlink{IdeaEquivariance}{Equivariance}} \end{itemize} and then afterwards indicate what this amounts to in someimportant special cases of choices of $\mathbf{H}$ \begin{itemize}% \item \emph{\hyperlink{IdeaGeometricity}{Geometricity}} \end{itemize} \hypertarget{IdeaEquivariance}{}\subsubsection*{{Equivariance}}\label{IdeaEquivariance} 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]] $c \simeq g^\ast c$ [[coherence|coherently]] for each [[generalized element]] $g$ of $G$, hence is a cocycle which is $G$- \emph{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 \emph{[[cohomology]]}) an equivariant cocycle is a natural system of [[diagrams]] of the form \begin{displaymath} \itexarray{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{\mathllap{\rho_X(g)}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{=}} \\ X &\underset{c}{\longrightarrow}& A } \end{displaymath} 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 \textbf{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 \emph{joint} action of $G$ on base space and coefficients. Diagrammatically this is a natural system of diagrams of the form \begin{displaymath} \itexarray{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{\mathllap{\rho_X(g)}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\rho_A(g)}} \\ X &\underset{c}{\longrightarrow}& A } \,. \end{displaymath} More concisely this means that an equivariant cocycle is a [[homotopy fixed point]] of the non-equivariant [[cocycle]] [[∞-groupoid]] $\mathbf{H}(X,A)$: \begin{displaymath} H^G(X,A) \simeq \pi_0(\mathbf{H}(X,A)^G) \,. \end{displaymath} By the discussion at \emph{[[∞-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]] $\mathbf{H}$ \begin{displaymath} G Act_\infty(\mathbf{H})\simeq \mathbf{H}_{/\mathbf{B}G} \,, \end{displaymath} and $G$-equivariant cohomology is the [[dependent product]] [[base change]] along \begin{displaymath} \underset{\mathbf{B}G}{\prod} \;\colon\; \mathbf{H}_{/\mathbf{B}G} \longrightarrow \mathbf{H} \end{displaymath} of [[internal homs]] in the slice over $\mathbf{B}G$: \begin{displaymath} H^G(X,A) \simeq \pi_0 \Gamma \left( \underset{\mathbf{B}G}{\prod} [X,A] \right) \,. \end{displaymath} (This formally recovers the above special case of Borel-equivariant cohomology by the dual incarnation of the [[projection formula]] (the one denoted $\overline{\gamma}$ at \emph{\href{Wirthmüller+context#TheComparisonMaps}{Wirthm\"u{}ller 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. \hyperlink{GreenleesMay}{Greenlees-May, p. 16})). Among the simplest non-trivial example of this $G$-equivariance with joint action on domain and coefficients is [[real-oriented cohomology|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 \begin{displaymath} \itexarray{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{=}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\rho_A(g)}} \\ X &\underset{c}{\longrightarrow}& A } \end{displaymath} and hence is equivalently a map \begin{displaymath} c \;\coloneqq\; X \longrightarrow A^G \end{displaymath} to the [[homotopy fixed points]] $A^G$ of the coefficients (formed in $\mathbf{H}$! See \hyperlink{IdeaGeometricity}{below} for different incarnations ). Hence we have in summary: [[!include equivariant cohomology -- table]] \hypertarget{IdeaGeometricity}{}\subsubsection*{{Geometricity}}\label{IdeaGeometricity} 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: \begin{itemize}% \item \textbf{``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 [[presentable (∞,1)-category|presented]] by the [[Borel model structure]] on the category of [[simplicial sets]] equipped with $G$-action. (This is also called the \emph{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 $G$. \item \textbf{``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 [[cohesion|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 \emph{[[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. \hyperlink{GreenleesMay}{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 \emph{[[Bredon cohomology]]}. \item \textbf{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 \emph{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). \end{itemize} 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 \href{over-%28infinity%2C1%29-topos#SheavesOnBigSite}{standard} equivalence $PSh_\infty(Orb)_{/\mathbf{B}G} \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb_G)$. \hypertarget{presentations}{}\subsection*{{Presentations}}\label{presentations} \begin{quote}% under construction \end{quote} (\ldots{}) [[Elmendorf theorem]] (\ldots{}) [[Borel model structure]] (\ldots{}) \hypertarget{Borel}{}\subsection*{{Borel equivariant cohomology}}\label{Borel} We first state the [[category theory|general abstract]] definition of \emph{[[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 groupoid]]s 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 [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-topos of (∞,1)-sheaves]] have the interpretation of [[∞-groupoid]]s 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 ∞-groupoid]]s. In this case, for $X$ some such [[∞-groupoid]] with structure, let $X_0 \hookrightarrow X$ be its 0-truncation, which is the [[space]] of [[object]]s of $X$, the [[discrete groupoid|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. \textbf{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. \begin{udefn} In some [[(∞,1)-topos]] $\mathbf{H}$ the \textbf{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 \textbf{equivariant structure} on an $A$-cocycle $c : X_0 \to A$ on $X_0$ is a choice of extension $\hat c$ \begin{displaymath} \itexarray{ X_0 &\to& A \\ \downarrow & \nearrow_{\hat c} \\ X } \,. \end{displaymath} i.e. a lift of $c$ through the projection $\mathbf{H}(X,A) \to \mathbf{H}(X_0,A)$. \end{udefn} \hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples} \hypertarget{group_cohomology}{}\paragraph*{{Group cohomology}}\label{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 \textbf{equivariant cohomology of the point}. \hypertarget{equivariant_cohomotopy}{}\paragraph*{{Equivariant cohomotopy}}\label{equivariant_cohomotopy} [[!include flavours of cohomotopy -- table]] \hypertarget{equivariant_bundles}{}\paragraph*{{Equivariant bundles}}\label{equivariant_bundles} For $G$ some [[group]] let $G Bund$ be the [[stack]] of $G$-[[principal bundle]]s. 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 \begin{itemize}% \item a $G$-[[principal bundle]] $P \to X$ on $X$; \item for each $k \in K$ an [[isomorphism]] of $G$-principal bundles $\lambda_k : P \to k^* P$ \item such that for all $k_1, k_2 \in K$ we have $\lambda_{k_2}\circ \lambda_{k_1} = \lambda_{k_2\cdot k_1}$. \end{itemize} \hypertarget{local_systems__flat_connections}{}\paragraph*{{Local systems -- flat connections}}\label{local_systems__flat_connections} 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 on a bundle|connection]] (see there for more details) on this principal $\infty$-bundle, also called an $A$-[[local system]]. (More details on this are at [[schreiber: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 \textbf{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 [[groupoid]]al one. \hypertarget{equivariant_de_rham_cohomology}{}\paragraph*{{Equivariant de Rham cohomology}}\label{equivariant_de_rham_cohomology} \begin{itemize}% \item [[equivariant de Rham cohomology]] \end{itemize} \hypertarget{remarks}{}\subsubsection*{{Remarks}}\label{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 [[schreiber:differential equivariant cohomology]]. \hypertarget{Bredon}{}\subsection*{{Bredon equivariant cohomology}}\label{Bredon} See also \begin{itemize}% \item [[Bredon cohomology]] \end{itemize} \hypertarget{preliminary_remarks}{}\subsubsection*{{Preliminary remarks}}\label{preliminary_remarks} 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 [[groupoid object in an (∞,1)-category|group object]], then $H^1(X;A)= \pi_0(Map(X,B A))$. More generally, if $A$ is $n$ times [[delooping|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 ∞-bundle]]s in whatever [[(∞,1)-topos]]. Now consider the $(\infty,1)$-topos $G Top$ of $G$-equivariant spaces, which can also be described as the [[(∞,1)-presheaf|(∞,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 [[groupoid object in an (∞,1)-category|group object]] in $G Top$. \hypertarget{equivariant_spectra}{}\subsubsection*{{$G$-equivariant spectra}}\label{equivariant_spectra} 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 group]]s are closely connected to each other, e.g. they often have [[cup product]]s or [[Steenrod square]]s or [[Poincare duality]], so it makes sense to consider them all together as a \emph{[[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 spectrum|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 \textbf{[[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 \textbf{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 ``[[loop space object|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 [[delooping|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 \textbf{[[RO(G)-grading|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 \textbf{[[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 \textbf{$RO(G)$-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 \emph{[[equivariant stable homotopy theory]]} and \emph{[[global equivariant stable homotopy theory]]}. \hypertarget{examples_2}{}\subsubsection*{{Examples}}\label{examples_2} \begin{itemize}% \item $\mathbb{Z}_2$-equivariant cohomology theories: [[KR-theory]], [[MR-theory]] \item [[modular group]]-equivariance: [[modular equivariant elliptic cohomology]] \end{itemize} \hypertarget{multiplicative_equivariant_cohomology}{}\subsection*{{Multiplicative equivariant cohomology}}\label{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 \begin{itemize}% \item \href{http://ncatlab.org/nlab/show/A+Survey+of+Elliptic+Cohomology+-+A-equivariant+cohomology#equivariant}{Equivariant multiplicative cohomology} \item [[equivariant elliptic cohomology]] \end{itemize} \hypertarget{examples_3}{}\subsection*{{Examples}}\label{examples_3} [[!include Segal completion -- table]] \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[equivariance]], [[equivariant structure]] \item [[equivariant bundle]], [[equivariant connection]] \item [[equivariant differential topology]] \item [[equivariant stable homotopy theory]], [[global equivariant stable homotopy theory]] \item [[equivariant rational homotopy theory]], [[rational equivariant stable homotopy theory]] \item [[Segal conjecture]] \item [[equivariant K-theory]], [[equivariant operator K-theory]], [[equivariant KK-theory]] \begin{itemize}% \item [[Baum-Connes conjecture]], [[Green-Julg theorem]], [[Atiyah-Segal completion theorem]] \item [[quantization commutes with reduction]] \end{itemize} \item [[equivariant elliptic cohomology]] \item [[orbifold cohomology]] \end{itemize} [[!include homotopy type representation theory -- table]] \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Introduction to [[Borel equivariant cohomology]]: \begin{itemize}% \item [[Loring Tu]], \emph{What is\ldots{} Equivariant Cohomology?}, Notices of the AMS, Volume 85, Number 3, March 2011 (\href{https://www.ams.org/notices/201103/rtx110300423p.pdf}{pdf}, [[TuEquivariantCohomology.pdf:file]]) \end{itemize} Introduction to [[Bredon equivariant cohomology]]: \begin{itemize}% \item [[Andrew Blumberg]], section 1.4 of \emph{Equivariant homotopy theory}, 2017 (\href{https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf}{pdf}, \href{https://github.com/adebray/equivariant_homotopy_theory}{GitHub}) \item [[John Greenlees]], [[Peter May]], section 3 of \emph{Equivariant stable homotopy theory} (\href{http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf}{pdf}) \end{itemize} Textbooks and lecture notes include \begin{itemize}% \item [[Tammo tom Dieck]], section 7 of \emph{[[Transformation Groups and Representation Theory]]}, Lecture Notes in Mathematics 766, Springer 1979 \item [[Peter May]], \emph{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\textasciitilde{}na, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (\href{http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf}{pdf}) \item Matvei Libine, \emph{Lecture Notes on Equivariant Cohomology} (\href{http://arxiv.org/abs/0709.3615}{arXiv}) \item Sébastien Racanière, \emph{Lecture on Equivariant Cohomology}, 2004 ([[RacaniereEquivariant04.pdf:file]]) \end{itemize} For a brief modern survey see also the first three sections of \begin{itemize}% \item [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], \emph{The Arf-Kervaire problem in algebraic topology: Sketch of the proof} ([[HHRKervaire.pdf:file]]) (with an eye towards application to the [[Arf-Kervaire invariant problem]]) \item blog on \href{http://www.aimath.org/wiki/localization/index.php/Main_Page}{Localization techniques in Equivariant Cohomology} \end{itemize} Discussion of equivariant versions of [[differential cohomology]] is in \begin{itemize}% \item Andreas Kübel, [[Andreas Thom]], \emph{Equivariant Differential Cohomology}, Transactions of the American Mathematical Society (2018) (\href{https://arxiv.org/abs/1510.06392}{arXiv:1510.06392}) \end{itemize} See also at \emph{[[equivariant de Rham cohomology]]}. \hypertarget{InComplexOrientedGeneralizedCohomologyTheory}{}\subsubsection*{{In complex oriented generalized cohomology theory}}\label{InComplexOrientedGeneralizedCohomologyTheory} Equivariant [[complex oriented cohomology theory]] is discussed in the following articles. \begin{itemize}% \item [[Michael Hopkins]], [[Nicholas Kuhn]], [[Douglas Ravenel]], \emph{Generalized group characters and complex oriented cohomology theories}, J. Amer. Math. Soc. 13 (2000), 553-594 (\href{http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00332-5/}{publisher}, \href{http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf}{pdf}) (This deals with ``naive'' Borel-equivariant complex oriented cohomology, but discusses general [[character]] expressions and explicit formulas for equivariant [[Morava K-theory|K(n)]]-cohomology.) \end{itemize} Specifically equivariant [[complex cobordism cohomology]] is discussed in \begin{itemize}% \item [[Tammo tom Dieck]], \emph{Bordism of $G$-manifolds and integrability theorems, Topology 9 (1970) 345-358} \item [[William Abram]], \emph{Equivariant complex cobordism}, 2013 (\href{http://deepblue.lib.umich.edu/handle/2027.42/99796}{web}, \href{http://deepblue.lib.umich.edu/bitstream/handle/2027.42/99796/abramwc_1.pdf?sequence=1}{pdf}) \item [[William Abram]], [[Igor Kriz]], \emph{The equivariant complex cobordism ring of a finite abelian group} (\href{http://www.math.lsa.umich.edu/~ikriz/cobordism13022-1.pdf}{pdf}) \end{itemize} The following articles discuss equivariant [[formal group laws]]: \begin{itemize}% \item [[John Greenlees]], \emph{Equivariant formal group laws and complex oriented cohomology theories}, Homology Homotopy Appl. Volume 3, Number 2 (2001), ii-451 (\href{http://projecteuclid.org/euclid.hha/1139840255}{EUCLID}) \item [[William Abram]], \emph{On the equivariant formal group law of the equivariant complex cobordism ring}, (\href{http://arxiv.org/abs/1309.0722}{arXiv:1309.0722}) (also \hyperlink{Abrams13a}{Abrams 13a, section III}). \end{itemize} See also the references at \emph{[[equivariant elliptic cohomology]]}. \hypertarget{in_differential_geometry}{}\subsubsection*{{In differential geometry}}\label{in_differential_geometry} Equivariant degree-2 $U(1)$-[[Lie group cohomology]] is discussed in \begin{itemize}% \item [[Kai Behrend]], [[Ping Xu]], Bin Zhang, \emph{Equivariant gerbes over compact simple Lie groups} (\href{http://arxiv.org/abs/math/0306183v1}{arXiv}) \end{itemize} [[!redirects Borel cohomology]] [[!redirects equivariant cohomology theory]] [[!redirects equivariant cohomology theories]] \end{document}