nLab equivariant Whitehead theorem



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Representation theory



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

Assume that the equivariance group be a compact Lie group (Matumoto 71, Illman 72, James-Segal 78, Waner 80). (This assumption is used, e.g. in Waner 80, Rem. 7.4, to guarantee that Cartesian products of coset spaces G/HG/H are themselves G-CW-complexes, which follows for compact Lie groups, e.g. by the equivariant triangulation theorem.)

Then: GG-homotopy equivalences f:XYf \colon X \longrightarrow Y between G-CW complexes are equivalent to equivariant weak homotopy equivalences, hence 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.

This is due to Matumoto 71, Thm. 5.3, Illman 72, Prop. 2.5, Waner 80, Theorem 3.4, see also James-Segal 78, Thm. 1.1 with Kwasik 81, review in Shah 10, Blumberg 17, Cor. 1.2.14.

An analogous statement holds in stable equivariant homotopy theory:

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).


Proofs for general G-CW-complexes (for GG a compact Lie group) are due to

  • Takao Matumoto, Theorem 5.3 in: On GG-CW complexes and a theorem of JHC Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA 18, 363-374, 1971 (PDF)

following a partial result in

  • Takao Matumoto, Lemma 4.3 in: Equivariant K-theory and Fredholm operators, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 (pdf, pdf)

and, independently, due to

  • Sören Illman, Prop. 3.3 in: Equivariant Algebraic Topology, Princeton University 1972 (pdf)

  • Sören Illman, Prop. 2.5 in: Equivariant singular homology and cohomology for actions of compact lie groups (doi:10.1007/BFb0070055) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), Proceedings of the Second Conference on Compact Transformation Groups Lecture Notes in Mathematics, vol 298. Springer 1972 (doi:10.1007/BFb0070029)

See also:

  • Stefan Waner, Theorem 3.4 in: Equivariant Homotopy Theory and Milnor’s Theorem, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 (jstor:1998061)

Discussion in equivariant stable homotopy theory:

  • John Greenlees, Peter May, Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

Review and Lecture notes:

A proof for GG-ANRs is due to:

Proof that these GG-ANRs have the equivariant homotopy type of G-CW-complexes (for GG a compact Lie group):

  • Slawomir Kwasik, On the Equivariant Homotopy Type of GG-ANR’s, Proceedings of the American Mathematical Society Vol. 83, No. 1 (Sep., 1981), pp. 193-194 (2 pages) (jstor:2043921)

Textbook account for GG-ANRs:

For the case of stable equivariant homotopy theory:

Last revised on October 31, 2021 at 05:36:57. See the history of this page for a list of all contributions to it.