nLab equivariant stable homotopy theory

Context

Stable Homotopy theory

stable homotopy theory

Contents

under construction

Contents

Idea

If in the context of stable homotopy theory the topological spaces and spectra are equipped with an action of a topological group $G$ the theory refines to a $G$-equivariant version. This is to equivariant homotopy theory (roughly) as stable homotopy theory is to homotopy theory.

Basic definitions

The definition of $G$-spectrum is typically given in generalization of the definition of coordinate-free spectrum.

A $G$-universe in this context is a infinite dimensional real inner product space equipped with a linear $G$-action that is the direct sum of countably many copies of a given set of (finite dimensional? -DMR) representations of $G$, at least containing the trivial representation on $ℝ$ (so that $U$ contains at least a copy of ${ℝ}^{\infty }$).

Each such subspace of $U$ (representation contained in $U$? -DMR) is called an indexing space . For $V\subset W$ indexing spaces, write $W-V$ for the orthogonal complement of $V$ in $W$. Write ${S}^{V}$ for the one-point compactification of $V$ and for $X$ any (pointed) topological space write ${\Omega }^{V}:=\left[{S}^{V},X\right]$ for the corresponding (based) sphere space.

A $G$-space in the following means a pointed topological space equipped with a continuous action of the topological group $G$ that fixes the base point. A morphism of $G$-spaces is a continuous map that fixes the basepoints and is $G$-equivariant.

A weak equivalence of $G$-spaces is a morphism that induces isomorphism on all $H$-fixed homotopy groups (…)

A $G$-spectrum $E$ (indexed on the chosen universe $U$) is

• for each indexing space $V\subset U$ a $G$-space $EV$;

• for each pair $V\subset W$ of indexing spaces a $G$-equivariant homeomorphism

$EV\stackrel{\simeq }{\to }{\Omega }^{W-V}E\phantom{\rule{thinmathspace}{0ex}}.$E V \stackrel{\simeq}{\to} \Omega^{W-V} E \,.

A morphism $f:E\to E\prime$ of $G$-spectra is for each indexing space $V$ a morphism of $G$-spaces ${f}_{V}:EV\to E\prime V$, such that this makes the obvious diagrams commute.

This becomes a category with weak equivalences by setting:

a morphism $f$ of $G$-spectra is a weak equivalence of $G$-spectra if for every indexing space $V$ the component ${f}_{V}$ is a weak equivalence of $G$-spaces (as defined above).

This may be expressed directly in terms of the notion of homotopy group of a $G$-spectrum: this is …

Topological Mackey-functors

… see the references below, for the moment…

Equivariant cohomology

The notion of cohomology relevant in equivariant stable homotopy theory is the flavor of equivariant cohomology (see there for details) called Bredon cohomology.

References

The characterization of $G$-equivariant functors in terms of topological Mackey-functors is discussed in example 3.4 (i) of

Something on modelling the equivariant stable category using functors on all (nice) $G$-spaces (instead of on just the orbit category) is in

• Andrew Blumberg, Continuous functors as a model for the equivariant stable homotopy category ((arXiv:math.AT/0505512)

The May recognition theorem? for $G$-spaces and genuine $G$-spectra is discussed in

• Costenoble and Warner, Fixed set systems of equivariant infinite loop spaces Transactions of the American mathematical society, volume 326, Number 2 (1991) (JSTOR)
Revised on November 18, 2013 12:52:15 by Urs Schreiber (82.169.114.243)