Contents

# Contents

## Idea

The concept of $G$-CW complex is to that of CW-complexes as topological G-spaces are to topological spaces: for $G$ a compact topological group, the notion of $G$-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 $D^n$ (“cells”) for a $G$-CW-complex one glues products of disks with $G$-orbits $G/H$ (coset spaces) for compact subgroups $H$.

These are cofibrant spaces used in $G$-equivariant homotopy theory.

## Examples

### $G$-Manifolds

The equivariant triangulation theorem says that if a compact Lie group $G$ acts on a compact smooth manifold $X$, 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:

###### Proposition

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

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

## Properties

### Equivariant CW-approximation

See at G-CW approximation.

### Elmendorf’s theorem

See at Elmendorf's theorem.

## References

The notion of G-CW complexes is, for the case of finite groups $G$, 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 $G$-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 $G$-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 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.