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RO(G)-grading

Contents

Context

Stable Homotopy theory

Representation theory

Contents

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 GG. This is often called RO(G)RO(G)-grading.

Motivation

Suspension by representation spheres

Since a cohomology theory is, in particular, a functor

E :BasedSpaces opGradedAbelianGroups E^\bullet \colon BasedSpaces^{op} \longrightarrow GradedAbelianGroups

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

E G :BasedGSpaces opGradedAbelianGroups. E_G^\bullet\colon Based G Spaces^{op} \longrightarrow GradedAbelianGroups \,.

This implies in particular the suspension isomorphism: for nn\in \mathbb{N} there is, for each GG-space XX, an identification

E G (X)E G +n(S nX), E_G^\bullet(X) \simeq E_G^{\bullet + n}(S^n \wedge X) \,,

where in the smash product S nXS^n \wedge X the group GG is to be taken to act trivially on the sphere S nS^n. From this perspective it is desirable to have an analogous relation also for smashing with spheres on which the group GG acts nontrivially. Since we may think of S nS^n as being the representation sphere of the trivial action of GG on n\mathbb{R}^n, this leads one to demand that E E^\bullet is graded not just by the integers, but by any linear representation VV, so that one has equivariant suspension isomorphisms

E G (X)E G +V(S VX), E_G^\bullet(X) \simeq E_G^{\bullet + V}(S^V \wedge X) \,,

where now S VS^V denotes the representation sphere of VV. 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 GG-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 GG-equivariant cohomology theories also become representables. This is the equivariant stable homotopy theory of genuine G-spectra.

Induced transfer and Mackey functors

(…)

Definition

For XX a GG-equivariant spectrum modeled as an orthogonal spectrum equipped with a GG-action, and for VV a linear GG-representation on a real vector space of dimension nn, then the value of XX in RO(G)RO(G)-degree VV is

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

where

(e.g. Schwede 15 (2.2))

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

π V G(X)lim n[S V+nρ G,X(nρ G)] G. \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)RO(G) is a great convenience, but it is not the thing most intrinsic to the mathematics. Ignore the multiplication on RO(G)RO(G), which is irrelevant to its use for grading theories, and think of it just as an abelian group. Send a representation VV to the isomorphism class of the suspension GG-spectrum of the one-point compactification S VS^V. This induces a homomorphism from RO(G)RO(G) into the Picard group Pic(HoG𝒮)Pic(Ho G\mathcal{S}) of the stable homotopy category of GG-spectra, namely the abelian group of equivalence classes of GG-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(HoG𝒮)Pic(Ho G\mathcal{S}), but that group is much less convenient than RO(G)RO(G). (P. May, comment on MO, Jul 2014)

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. G. 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)Pic(S_G)-graded homotopy groups, 2013 (pdf)

Last revised on March 1, 2019 at 06:58:39. See the history of this page for a list of all contributions to it.