Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

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

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

For maps $F \colon E \longrightarrow F$ 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 $\pi_n(f)\colon \pi_n(E) \longrightarrow \pi_n(F)$ (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.