geometric representation theory
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In equivariant stable homotopy theory a genuine G-spectrum has homotopy groups graded not just by the integers, but by the (additive group underlying) the representation ring of $G$. This is often called $RO(G)$-grading.
Since a cohomology theory is, in particular, a functor
satisfying some axioms, then naively a $G$-equivariant cohomology theory could be defined to be a functor
This implies in particular the suspension isomorphism: for $n\in \mathbb{N}$ there is, for each $G$-space $X$, an identification
where in the smash product $S^n \wedge X$ the group $G$ is to be taken to act trivially on the sphere $S^n$. From this perspective it is desirable to have an analogous relation also for smashing with spheres on which the group $G$ acts nontrivially. Since we may think of $S^n$ as being the representation sphere of the trivial action of $G$ on $\mathbb{R}^n$, this leads one to demand that $E^\bullet$ is graded not just by the integers, but by any linear representation $V$, so that one has equivariant suspension isomorphisms
where now $S^V$ denotes the representation sphere of $V$. This more general grading is, for historical reasons, called “RO(G)-grading”, and an equivariant cohomology theory equipped with this extra structure is called genuine (as opposed to the “naive” case with grading just over the integers).
Since by the Brown representability theorem, in the absence of a group action a cohomology theory is represented by a spectrum, it is natural to construct a category in which genuine $G$-equivariant cohomology theories also become representables. This is the equivariant stable homotopy theory of genuine G-spectra.
(…)
For $G$ a finite group, $X$ a $G$-equivariant spectrum modeled as an orthogonal spectrum equipped with a $G$-action, and for $V$ a linear $G$-representation on a real vector space of dimension $n$, the value of $X$ in $RO(G)$-degree $V$ is
where
(e.g. Schwede 15 (2.2))
The $RO(G)$-graded equivariant homotopy group of $X$ is (in the notation used there)
(e.g. Schwede 15, p. 40)
In equivariant cohomology theory, the use of $RO(G)$ is a great convenience, but it is not the thing most intrinsic to the mathematics. Ignore the multiplication on $RO(G)$, which is irrelevant to its use for grading theories, and think of it just as an abelian group. Send a representation $V$ to the isomorphism class of the suspension $G$-spectrum of the one-point compactification $S^V$. This induces a homomorphism from $RO(G)$ into the Picard group $Pic(Ho G\mathcal{S})$ of the stable homotopy category of $G$-spectra, namely the abelian group of equivalence classes of $G$-spectra that are invertible under the smash product. That homomorphism is neither a monomorphism nor an epimorphism. See (Fausk-Lewis-May 01) for a discussion of that Picard group. Logically, equivariant cohomology theories really should be graded on $Pic(Ho G\mathcal{S})$, but that group is much less convenient than $RO(G)$. (P. May, comment on MO, Jul 2014)
A standard reference is
Peter May, chapters IX.5, X and XIII of 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. (pdf)
Stefan Schwede, section 4 (from page 40 on) in Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)
Halvard Fausk, L. Gaunce Lewis, Peter May, The Picard group of equivariant stable homotopy theory, Advances in Mathematics Volume 163, Issue 1, 15 October 2001, Pages 17–33 (pdf)
Justin Noel, Equivariant cohomology of representation spheres and $Pic(S_G)$-graded homotopy groups, 2013 (pdf)
Last revised on November 23, 2020 at 06:37:02. See the history of this page for a list of all contributions to it.