nLab RO(G)-grading

Idea

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.

Motivation

Suspension by representation spheres

Since a cohomology theory is, in particular, a functor

$$E^\bullet \colon BasedSpaces^{op} \longrightarrow GradedAbelianGroups$$

satisfying some axioms, then naively a $G$-equivariant cohomology theory could be defined to be a functor

$$E_G^\bullet\colon Based G Spaces^{op} \longrightarrow GradedAbelianGroups \,.$$

This implies in particular the suspension isomorphism: for $n\in \mathbb{N}$ there is, for each $G$-space $X$, an identification

$$E_G^\bullet(X) \simeq E_G^{\bullet + n}(S^n \wedge X) \,,$$

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

$$E_G^\bullet(X) \simeq E_G^{\bullet + V}(S^V \wedge X) \,,$$

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).

Representation by $G$-spectra

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.

Induced transfer and Mackey functors

(…)

Definition

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

$$X(V) \coloneqq \mathbf{L}(\mathbb{R}^n, V)_+ \wedge_{O(n)} X_n \,,$$

where

• $\mathbf{L}(\mathbb{R}^n, V)$ is the linear isometries $\mathbb{R}^n \to V$ (orthogonal bases);

(e.g. Schwede 15 (2.2))

The $RO(G)$-graded equivariant homotopy group of $X$ is (in the notation used there)

$$\pi_V^G(X) \coloneqq \underset{\longrightarrow_{\mathrlap{n}}}{\lim} [S^{V + n \rho_G}, X(n \rho_G)]_G \,.$$

(e.g. Schwede 15, p. 40)

Properties

Relation to intrinsic twisting

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)

Related concepts

• G-universe

References

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 (arXiv:2008.05551, pdf)

• Justin Noel, Equivariant cohomology of representation spheres and $Pic(S_G)$-graded homotopy groups, 2013 (pdf)