nLab G-CW approximation



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Representation theory



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

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

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


  • 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)

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