nLab fixed point spectrum

Idea

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

$$F^G \;\colon\; G Spectra \longrightarrow Spectra$$

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

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

$$F^N \;\colon\; G Spectra \longrightarrow G/N Spectra$$

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

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

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

Properties

Relation to equivariant homotopy groups

The plain homotopy groups of the $G$-fixed point spectrum of $X$ are the equivariant homotopy groups of the $G$-spectrum $X$:

$$\pi_k(F^G X) \simeq \pi_k^G(X)$$

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

Examples

For equivariant suspension spectra

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

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

(e.g. Schwede 15, example 7.7)

Related concepts

• fixed point space

• homotopy fixed points

• geometric fixed point spectrum

• categorical fixed point spectrum?

• tom Dieck splitting

References

• L. Gaunce Lewis, Jr., Splitting theorems for certain equivariant spectra, Memoirs of the AMS, number 686, March 2000, Volume 144 (pdf)

• Michael Mandell, Peter May, Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 2003i:55012 (pdf, K-theory archive)

• John Greenlees, Brooke Shipley, p. 18 of An algebraic model for rational torus-equivariant spectra (arXiv:1101.2511)

• John Greenlees, Brooke Shipley, Fixed point adjunctions for equivariant module spectra, Algebr. Geom. Topol. 14 (2014) 1779-1799 (arXiv:1301.5869)

• Stefan Schwede, Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)