nLab
equivariant stable homotopy theory

under construction

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 ${ℝ}^{\mathrm{\infty }}$).

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 ${\Omega }^V:=[S^V,X]$ 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

  $$EV\stackrel{\simeq }{\to }{\Omega }^{W-V}E.$$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…

• Mackey functor

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

• Anna Marie Bohmann, Basic notions of equivariant stable homotopy theory (pdf)

• John Greenlees, Peter May, Equivariant stable homotopy theory (pdf)

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

• Stefan Schwede, Brooke Shipley, Classification of stable model categories (pdf)

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)