nLab
G-CW approximation

Contents

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Representation theory

Contents

Idea

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.

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)

  • Jay Shah, Theorem 2.9 in: Equivariant algebraic topology, 2010 (pdf, pdf)

Last revised on March 19, 2021 at 07:06:57. See the history of this page for a list of all contributions to it.