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global equivariant stable homotopy theory

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Stable Homotopy theory

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cohomology

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Idea

What is called global equivariant stable homotopy theory is a variant of equivariant stable homotopy theory where spectra are equipped with GG-infinity-actions “for all compact Lie groups GG at once”.

Often this is referred to just as “global stable homotopy theory” or even just “global homotopy theory”. But there is also unstable global equivariant homotopy theory.

More precisely, given an orthogonal spectrum XX, then every representation ρ:GO(n)\rho \colon G \to O(n) of a compact Lie group on the Cartesian space n\mathbb{R}^n by orthogonal group actions induces a GG-equivariant spectrum and hence a notion of GG-equivariant homotopy groups.

One says that a morphism of orthogonal spectra is a global equivariant equivalence if it induces isomorphisms on all GG- equivariant homotopy groups, for all GG, this way. (This definition appears for instance as (Schwede 13, def. 2.9), there referred to just as “global equivalence”. See also at equivariant Whitehead theorem.)

The global equivariant stable homotopy category 𝒢ℋ\mathcal{GH} is the (simplicial) localization of the category of orthogonal spectra at these global equivariant equivalences (this is a stable (infinity,1)-category/triangulated category.)

Since a global equivariant equivalence is in particular an ordinary weak homotopy equivalence of spectra, there is a canonical functor

U:𝒢ℋ𝒮ℋ U \;\colon\; \mathcal{GH} \longrightarrow \mathcal{SH}

from the global equivariant to the ordinary stable homotopy category.

This functor has a (derived) left adjoint and right adjoint, which are both full and faithful functors, hence which exhibit discrete object/codiscrete object structure

𝒢ℋ RUL 𝒮ℋ \array{ \mathcal{GH} &\stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\underset{R}{\leftarrow}}}& \mathcal{SH} }

on stable/triangulated categories (Schwede 13, theorem IV 5.2)

Global Borel-type equivariant cohomology is in the image of the right adjoint RR (Schwede 13, example IV 5.12)

Properties

Relation to plain stable homotopy theory

The forgetful functor from global stable homotopy theory to plain stable homotopy theory exhibits a recollement.

(…)

Examples

References

A comprehensive textbook account is in

Survey includes

Original articles are

  • L. Gaunce Lewis, Jr., Peter May, M. Steinberger, chapter II of Equivariant stable homotopy theory. Lecture Notes in Mathematics, 1213, Springer-Verlag, 1986

  • John Greenlees, Peter May, section 5 of Localization and completion theorems for MUMU-module spectra Ann. of Math. (2) 146 (1997), 509-544.

Discussion specifically in terms of equivariant orthogonal spectra is in

Discussion for collections of finite subgroups includes

Last revised on September 2, 2018 at 08:30:49. See the history of this page for a list of all contributions to it.