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equivariant Whitehead theorem

Contents

Context

Stable Homotopy theory

Representation theory

Contents

Idea

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).

References

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.