nLab homotopy fixed point

Idea

The refinement of the notion of fixed point/invariant to homotopy theory. Also homotopy invariant

See at

• Invariants – For infinity-group actions

• Infinity-action – Invariants

Properties

Presentation

Generally, for $G$ an ∞-group and $X$ equipped with an ∞-action, then the homotopy fixed points is the base change along/dependent product over the delooping $\mathbf{B}G$:

$$(-)^{h G} \;\colon\; G Act(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G} \overset {\;\;\;\prod_{\mathbf{B}G}\;\;\;} {\longrightarrow} \mathbf{H} \,.$$

By the general discussion at dependent product this is given by forming sections, which by the discussion at ∞-action here means forming sections of the $X$-associated ∞-bundle to the universal principal ∞-bundle

$$\array{ X &\longrightarrow& X \sslash G \\ && \big\downarrow \\ && \mathbf{B}G \,. }$$

Such sections

are equivalently homotopy-equivariant function out of the point equipped with the trivial $G$-action:

$$X^{h G} \simeq G Act(\ast, X) \,.$$

For geometrically discrete ∞-group ∞-actions, hence for $\mathbf{H} =$ ∞Grpd, it is the Borel model structure which presents the homotopy theory. By the discussion there, the derived hom space is computed as the hom-space out of plain actions of simplicial groups out of the total space of the simplicial universal principal bundle $\mathbf{E}G = W G$. Therefore in this case one finds

$$X^{h G} \simeq R Hom_G(\ast, X) \simeq Hom_G(E G, X) \,.$$

This is the form in which homotopy fixed points are often defined in traditional literature (going back to Thomason 83 and as in the historical discussion of the Sullivan conjecture).

Relation to periodicity theorems

In the context of complex oriented cohomology theory

… (Hill-Hopkins-Ravenel, theorem 8.4)…

Related concepts

• fixed point space

• crossed homomorphism

• Sullivan conjecture

• invariant theory

• fixed point spectrum

• geometric fixed point spectrum

• categorical fixed point spectrum?

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theory	representation theory
pointed connected context $\mathbf{B}G$	∞-group $G$
dependent type on $\mathbf{B}G$	$G$-∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$	coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$	trivial representation
dependent product along $\mathbf{B}G \to \ast$	homotopy invariants/∞-group cohomology
dependent product of internal hom along $\mathbf{B}G \to \ast$	equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$	induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$	restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$	coinduced representation
spectrum object in context $\mathbf{B}G$	spectrum with G-action (naive G-spectrum)

References

The original article:

• Robert Thomason, The homotopy limit problem, pages 407-419 in: Haynes Miller, Stewart Priddy (eds.) Proceedings of the Northwestern Homotopy Theory Conference, Contemporary Mathematics 19, AMS 1983 (doi:10.1090/conm/019, pdf)

Further discussion, for groupoids and stacks:

• R. Virk, On fixed point stacks (arXiv:2112.06871)

In the context of complex oriented cohomology theory:

• Michael Hill, Michael Hopkins, Douglas Ravenel, Theorem 8.4 in: The Arf-Kervaire problem in algebraic topology: Sketch of the proof (pdf)