G-CW complex




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Representation theory



The concept of GG-CW complex is to that of CW-complexes as topological G-spaces are to topological spaces: for GG a compact topological group, the notion of GG-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 nD^n (“cells”) for a GG-CW-complex one glues products of disks with GG-orbits G/HG/H (coset spaces) for compact subgroups HH.

These are cofibrant spaces used in GG-equivariant homotopy theory.



The equivariant triangulation theorem says that if a compact Lie group GG acts on a compact smooth manifold XX, 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:


(G-representation spheres are G-CW-complexes)

For GG a compact Lie group (e.g. a finite group) and VRO(G)V \in RO(G) a finite-dimensional orthogonal GG-linear representation, the representation sphere S VS^V admits the structure of a G-CW-complex.


Elmendorf’s theorem

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

See at Elmendorf's theorem.


Good lecture notes are

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)

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