# 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.

## Properties

### For $G$-manifolds

If a compact Lie group $G$ acts on a compact smooth manifold $X$, then the manifold admit an equivariant triangulation. In particular it has the structure of a G-CW complex.

(Illman 83, theorem 7.1, corollary 7.2) Recalled as (ALR 07, theorem 3.2).

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)

### 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

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

• 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)

Revised on June 23, 2016 05:22:08 by Urs Schreiber (131.220.184.222)