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 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

Elmendorf’s theorem

The collection of $G$-CW-complexes has a full embedding into the (infinity,1)-presheaves on the orbit category $Orb(G)$. This is given by sending a $G$-CW complex, $Y$, to the presheaf sending $G/H$ to $Y^H$, the subspace of $Y$ fixed by $H$.

See at Elmendorf's theorem.

References

Good lecture notes are

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

A standard reference is

• Peter May, sections 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. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

  Section X.2 there discusses the generalization to RO(G)-grading.

See also

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

• 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)

• Jay Shah, Equivariant algebraic topology, pdf

• Sören Illman, The equivariant triangulation theorem for actions of compact Lie groups, Math. Ann. 262 (1983), no. 4, 487–501 (web)

• A. Adem, J. Leida and Y. Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (pdf)

• A. Adem, J. Leida and Y. Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (pdf)