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fixed point spectrum

Contents

Context

Stable Homotopy theory

Representation theory

Contents

Idea

Given a GG-equivariant spectrum, its fixed point spectrum F GXF^G X is the plain spectrum which is the value of the derived functor of the naive GG-fixed point functor on XX. This constitutes a suitably derived functor

F G:GSpectraSpectra F^G \;\colon\; G Spectra \longrightarrow Spectra

(e.g. Schwede 15, def. 7.1, Mandell-May 02, arojnd thm. 3.3)

More generally, for NGN \subset G a normal subgroup, there is the NN-fixed point spectrum regarded as G/NG/N-spectrum, equivariant under the remaining quotient group G/NG/N:

F N:GSpectraG/NSpectra F^N \;\colon\; G Spectra \longrightarrow G/N Spectra

(Mandell-May 02, def. 3.8, Greenlees-Shipley 11, Prop. 3.3)

These are derived right adjoint to the operation of regarding a G/NG/N-spectrum as a GG-spectrum, via the projection GG/NG \to G/N

(Mandell-May 02, theorem 3.12, Greenlees-Shipley 11, Prop. 3.3)

Properties

Relation to equivariant homotopy groups

The plain homotopy groups of the GG-fixed point spectrum of XX are the equivariant homotopy groups of the GG-spectrum XX:

π k(F GX)π k G(X) \pi_k(F^G X) \simeq \pi_k^G(X)

(e.g. Schwede 15, prop. 7.2)

Examples

For equivariant suspension spectra

For Σ G X\Sigma^\infty_G X an equivariant suspension spectrum the fixed point spectrum is given by the tom Dieck splitting formula

F G(Σ G X)[HG]Σ (EW(X) + WHX H). F^G(\Sigma^\infty_G X) \simeq \underset{[H\subset G]}{\oplus} \Sigma^\infty( E W(X)_+ \wedge_{W H} X^H ) \,.

(e.g. Schwede 15, example 7.7)

References

Last revised on December 30, 2018 at 13:27:21. See the history of this page for a list of all contributions to it.