# CW complex and group action

This is a general question and any reference or related result will be extremely helpful.

Suppose $X$ is a Hausdorff topological space. Suppose G (a countable group) acts on it. Let $Y=X/G$ be the quotient space which is a CW complex. Can we say something about the cell structure of $X$ (if exists.)? If not in general is there some results which gives condition on the action (for example proper or cocompact) which gives us cell structures in $X$. In particular I am interested in the following question.

Under what condition on $X$ and on the group action $X$ will be a CW complex?

• A simple condition: when $X\rightarrow X/G$ is a covering space. – Fernando Muro Oct 4 '13 at 8:51
• You might be interested in work of Gerald Schwarz on quotients of semi-algebraic sets by compact Lie groups acting through representations (or maybe not, since you say below that you don't have any geometric structure on $G$). He shows that the orbit spaces are again semi-algebraic, hence triangulable. – Dan Ramras Oct 7 '13 at 1:23
• @Dan Thanks. I will definitely look at it. I am always interested in these type of results, its just that my "problem" doesn't have any geometric structures :). – Cusp Oct 7 '13 at 4:22

This is not an answer but I think points to a line of investigation.

We know for a CW-complex $X$ that $X$ is the colimit of its skeleta $X^n$, and that $X^{n+1}$ is obtained from $X^n$ by attaching cells, which again is a colimit operation.

I now refer to the general theory of fibred exponential laws, which have been investigated over many years by Peter Booth, also with colleagues at Memorial University, and we wrote two papers:

(1) Spaces of partial maps, fibred mapping spaces and the compact-open topology, Gen. Top. Appl. 8 (1978) 181-195.

(2) On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps, Gen. Top. Appl. (1978) 165-179.

both available from my publications list as [26,27], and from the Journal's web page. These papers show that under suitable restrictions, or by working in a convenient category of spaces, see Section 7 of (1), and Section 8 of (2), that if $p:Y \to X$ is a map of spaces then the pullback functor $p^*: Top/X \to Top/Y$ has a right adjoint and so preserves colimits. So if $Y^n= p^{-1}X^n$ you get that $Y$ is the colimit of the $Y^n$.

Now you also have a chance of analysing the construction of $Y^{n+1}$ from $Y^n$. I think this easily gives the case mentioned by Fernando, when $p$ is a covering: that result goes back to JHCW, with a direct proof.

You need to look carefully at the pullbacks of the elements of the construction of $X^{n+1}$. Perhaps you also need to look at generalisations of CW-complexes. See for example Minian, G.; Ottina, M. A geometric decomposition of spaces into cells of different types. J. Homotopy Relat. Struct. 1 (2006) 245–271 and its sequel.

• For those wondering, that "JHCW" in the penultimate paragraph of Ronnie's answer stands for JHC Whitehead. – Vidit Nanda Oct 6 '13 at 12:51
• Thanks Vidit. The result is (N) in section 5 of Combinatorial Homotopy I. That paper is still amazing to me in the way it deals with the essential matters. – Ronnie Brown Oct 6 '13 at 13:42
• @Brown Thanks for the references. I have read the paper of JHCW. Its really amazing but I think its the definition of CW complex which is more surprising than the results. – Cusp Oct 6 '13 at 14:14

you may wish to check the following paper:

Illman, Sören

Math. Ann. 262 (1983), no. 4, 487–501.

Let G be a compact Lie group.

Theorem 7.1: Let M be a smooth $G$-manifold with or without boundary. Then there exists an equivariant triangulation of $M$.

Corollary 7.2: Let M be a smooth $G$-manifold with or without boundary. Then $M$ can be given an equivariant CW complex structure.

• Thanks for the references. The second corollary looks interesting, but the spaces I am working with are far from being manifolds. Also the groups does not have geometric structures. – Cusp Oct 6 '13 at 14:16