equivariant Whitehead theorem



Stable Homotopy theory

Representation theory



The equivariant Whitehead theorem is the generalization of the Whitehead theorem from (stable) homotopy to (stable) equivariant homotopy theory.

GG-Homotopy equivalences f:XYf \colon X \longrightarrow Y between G-CW complexes are equivalent to maps that induce weak homotopy equivalences f H:X HY Hf^H \colon X^H \longrightarrow Y^H on all fixed point spaces for all closed subgroups HGH \hookrightarrow G (Matumoto 71, Waner 80, theorem 3.4, for review see Blumberg 17, corollary 1.2.14).

For maps F:EFF \colon E \longrightarrow F between genuine G-spectra, they are weak equivalences (isomorphisms in the equivariant stable homotopy category) if they induce isomorphisms on all equivariant homotopy group Mackey functors π n(f):π n(E)π n(F)\pi_n(f)\colon \pi_n(E) \longrightarrow \pi_n(F) (e. g. Greenlees-May 95, theorem 2.4, Bohmann, theorem 3.2).


The original proof seems to be due to

  • T. Matumoto, Equivariant K-theory and Fredholm operators, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 (jairo)

streamlined in

  • Stefan Waner, Equivariant Homotopy Theory and Milnor’s Theorem, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 (JSTOR)

and reviewed in

For the stable case:

Last revised on April 13, 2018 at 09:09:48. See the history of this page for a list of all contributions to it.