nLab equivariant Whitehead theorem

Idea

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 $G/H$ are themselves G-CW-complexes, which follows for compact Lie groups, e.g. by the equivariant triangulation theorem.)

Then: $G$-homotopy equivalences $f \colon X \longrightarrow Y$ between G-CW complexes are equivalent to equivariant weak homotopy equivalences, hence to maps that induce weak homotopy equivalences $f^H \colon X^H \longrightarrow Y^H$ on all fixed point spaces for all closed subgroups $H \hookrightarrow G$.

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 $F \colon E \longrightarrow F$ 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).

Related concepts

• G-CW approximation theorem, equivariant cellular approximation theorem

• Elmendorf's theorem

• tom Dieck splitting

References

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

• Takao Matumoto, Theorem 5.3 in: On $G$-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:

• Andrew Blumberg, Section 1.2 in: Equivariant homotopy theory, 2017 (pdf, GitHub)

A proof for $G$-ANRs is due to:

• Ioan James, Graeme Segal, Theorem 1.1 in: On equivariant homotopy type, Topology Volume 17, Issue 3, 1978, Pages 267-272 (doi:10.1016/0040-9383(78)90030-7)

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

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

Textbook account for $G$-ANRs:

• Tammo tom Dieck, Prop. 8.2.4, 8.2.6 in: Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766, Springer 1979 (doi:10.1007/BFb0085965)

For the case of stable equivariant homotopy theory:

• Anna Marie Bohmann, Basic notions of equivariant stable homotopy theory (pdf)