nLab
G-CW complex

Contents

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Representation theory

Contents

Idea

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

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

Definition

(…)

Remark

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

  • a plain CW-complex-structure

  • on which the GG-action is cellular (sends nn-cells onto nn-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

GG-Manifolds

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:

Proposition

For GG 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 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.

Properties

Closure properties

Proposition

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

Proof

Since for finite GG, a GG-CW complex is the same as a plain CW-complex equipped with a cellular action by GG (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 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.

References

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)

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