nLab G-CW approximation

Idea

The generalization of CW-approximation from plain homotopy theory to $G$-equivariant homotopy theory is called $G$-CW approximation:

For suitable equivariance groups $G$, every topological G-space $X$ receives a $G$-equivariant function $Q X \overset{f}{\longrightarrow} X$ from a G-CW complex $Q X$, such that this restricts to a weak homotopy equivalence $f^H \;\colon (Q X)^H \to X^H$ on $H$-fixed loci, for all suitable subgroups $H \subset G$.

This should hold for $G$ a compact Lie group (such as a finite group) and $H$ ranging over its closed subgroups.

Related concepts

• equivariant Whitehead theorem

• Elmendorf's theorem

• tom Dieck splitting

References

• Peter May et al., Thm 3.6 in: 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)