G-CW complex




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


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topological homotopy theory

Representation theory



The concept of GG-CW complex is to that of CW-complexes as topological G-spaces are to topological spaces: for GG a compact topological group, the notion of GG-CW-complex is much like that of CW-complex, only that where in the latter case one builds a topological space from gluing of disks D nD^n (“cells”) for a GG-CW-complex one glues products of disks with GG-orbits G/HG/H (coset spaces) for compact subgroups HH.

These are cofibrant spaces used in GG-equivariant homotopy theory.



The equivariant triangulation theorem says that if a compact Lie group GG acts on a compact smooth manifold XX, then the manifold admits an equivariant triangulation. In particular it thus has the structure of a G-CW complex.

(Illman 83, theorem 7.1, corollary 7.2) Recalled as (ALR 07, theorem 3.2). See also Waner 80, p. 6 who attributes this to Matumoto 71

Moreover, if the manifold does have a boundary, then its G-CW complex may be chosen such that the boundary is a G-subcomplex. (Illman 83, last sentence above theorem 7.1)

In particular:


(G-representation spheres are G-CW-complexes)

For GG a compact Lie group (e.g. a finite group) and VRO(G)V \in RO(G) a finite-dimensional orthogonal GG-linear representation, the representation sphere S VS^V admits the structure of a G-CW-complex.


Equivariant cellular approximation

See at equivariant cellular approximation theorem.

Equivariant CW-approximation

See at G-CW approximation.

Equivariant Whitehead theorem

See at equivariant Whitehead theorem.

Elmendorf’s theorem

See at Elmendorf's theorem.


The notion of G-CW complexes is, for the case of finite groups GG, due to

announced in

In the broader generality of general topological groups and specifically of compact Lie groups, the nition of G-CW-complexes and their equivariant Whitehead theorem is due to:

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

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

(Which, in hindsight and with Elmendorf's theorem, gives a deeper justification for the parametrization over the orbit category already proposed in Bredon 67a, Bredon 67b.)

  • 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)

Proof that 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 accounts:

Lecture notes:

Last revised on March 19, 2021 at 07:05:52. See the history of this page for a list of all contributions to it.