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 (product topological spaces) of disks with $G$-orbits $G/H$ (coset spaces) for compact subgroups $H$.

These are cofibrant objects in the $G$-fine model structure on topological G-spaces.

Definition

(…)

Remark

For finite equivariance groups $G$, a $G$-CW-complex structure may be identified with

• a plain CW-complex-structure

• on which the $G$-action is cellular (sends $n$-cells onto $n$-cells, respecting their boundaries)

• which on cells that are sent to themselves actually restricts to the identity (i.e. cells that are fixed under the action are actually fixed point-wise).

This special case was the original definition on Bredon 1967b, I.1.

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:

Proposition

For $G$ a compact Lie group, every closed smooth G-manifold admits the structure of a G-CW complex.

This is due to Matumoto 72, Prop. 4.4, Illman 72a, Thm. 3.1, Cor. 4.1, Illman 72b, Thm. 2.6, Illman 73, Thm. 2.1, Illman 83, Thm. 7.1, Cor. 7.2 – review in ALR 07, theorem 3.2, see also Waner 80, p. 6.

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

Closure properties

Proposition

For $G$ a finite group (at least), the product of two $G$-CW-complexes in compactly generated weak Hausdorff spaces is itself a $G$-CW-complex.

Proof

Since for finite $G$, a $G$-CW complex is the same as a plain CW-complex equipped with a cellular action by $G$ (Rem. ) it is clear that for this structure to be preserved by the product operation it is sufficient that the products of underlying cells constitute a CW-complex, hence that products preserve CW-complexes in compactly generated Hausdorff spaces. This is this case by this Prop..

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)

and, independently, due to:

• Sören Illman, Chapter I of: Equivariant algebraic topology, Princeton University 1972 (pdf)

• Sören Illman, Section 2 of: 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)

• Sören Illman, Section 2 of: Equivariant algebraic topology, Annales de l’Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 (doi:10.5802/aif.458)

(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 September 13, 2021 at 07:42:42. See the history of this page for a list of all contributions to it.