nLab G-CW complex

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

Let the equivariance group $G$ be topological group (typically required to be a compact Lie group, see there).

As $H \underset{clsd}{\subset} G$ ranges over its closed subgroups, consider the coset spaces $G/H$ as G-spaces with respect to the residual left multiplication action by $G$, and regard its product spaces $S^n \times G/H$ and $D^n \times G/H$ with the n-sphere and the n-disk, respectively, for the latter equipped with the trivial action.

This way, the canonical boundary inclusions

$$\iota_n \,\colon\, S^{n-1} \xhookrightarrow{\phantom{--}} D^n$$

considered for ordinary CW-complexes induces $G$-equivariant maps

(1)$$S^{n-1} \times G/H \xhookrightarrow{ \phantom{-} \iota_n \times id_{G/H} \phantom{-} } D^n \times G/H \,,\;\;\;\;\;\; n \in \mathbb{N} ,\, H \underset{clsd}{\subset} G \,.$$

Definition

A $G$-CW-complex $X$ is a G-space isomorphic to the result of a sequence, monotone in cell dimension $n$, of cell attachments via the $G$-equivariant attaching maps (1).

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.

$G$-Surfaces

Simple examples of G-manifolds (above) are surfaces with $G$-action.

Example

Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with the $\mathbb{Z}_{/2}$-action which reflects one of the two coordinate axes:

Properties

Closure properties

Proposition

For $G$ (at least) a finite group , 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. That is this the case is this Proposition.

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 $G$, due to

• Glen Bredon, Section I.1. of Equivariant cohomology theories, Springer Lecture Notes in Mathematics Vol. 34. 1967 (doi:10.1007/BFb0082690)

announced in

• Glen Bredon, Equivariant cohomology theories, Bull. Amer. Math. Soc. Volume 73, Number 2 (1967), 266-268. (educlid:1183528794)

In the broader generality of general topological groups and specifically of compact Lie groups, the notion 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 (1971) 363-374 [irdb:00926/0001786419, 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:

• Tammo tom Dieck, Sections I.1 and I.2 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

• Wolfgang Lück, Sections I.1, I.2 of: Transformation Groups and Algebraic K-Theory, Lecture Notes in Mathematics 1408 (Springer 1989) (doi:10.1007/BFb0083681)

• Peter May et al., Section I.3 of: Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996 (ISBN: 978-0-8218-0319-6 pdf, pdf)

Lecture notes:

• Andrew Blumberg, Equivariant homotopy theory, 2017 (pdf, GitHub)