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equivariant Whitehead theorem
Contents
Context
Stable Homotopy theory
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
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.
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