Reference request: if $G$ is homotopic to CW complex, then $EG$ is contractible.

Question

I am looking for a reference that supplies a proof of the following fact.

If $G$ is homotopic to a CW complex, then $EG$ is contractible.

Any help would be well-appreciated.

Answer 1

First of all, I do not like to answer unclear questions and this question is left unclear by both OP and Harambe. Nevertheless, here is one interpretation of what is being asked.

First of all, it is ambiguous what $EG$ is. In one interpretation (which I will be using below), given a topological group $G$, a classifying space $EG$ is a weakly contractible topological space $EG$ equipped with a principal $G$-bundle structure $\xi: EG\to BG$, such that $EG$ is universal in the category of principal $G$-bundles $P$ over CW complexes, meaning that given any principal $G$-bundle $P\to Z$ (with base a CW complex), there exists a (unique up to homotopy) map $f: Z\to B$ such that the pull-back bundle $f^*(\xi)$ is isomorphic to $P$. (In fact, weak contractibility of $EG$ implies the universality property.)

If you do not like this definition, you (or harambe, or somebody else) should specify which definition do you have in mind.

Now, with this definition, your question becomes:

Is $EG$ contractible?

(a priori, it is only weakly contractible). This is still not entirely well-defined since $EG$ is, a priori, unique only up to weak homotopy.

Assume now that $G$ is homotopy equivalent (not "homotopic") to a CW complex. (There are other interpretations of what kind of groups one should be considering here when working in the CW category, see Milnor's paper below.) Then Milnor's infinite join construction (as in On the construction of universal bundles II produces a space which is an infinite join of copies of $G$, which is a direct limit of finite joins. Now, up to homotopy, $G$ is a $CW$ complex $V$, hence, $EG$ is homotopy equivalent to an infinite join of $V$, which is a CW complex. We know that the latter is weakly contractible, hence, by Whitehead's theorem (which you can find in any textbook on algebraic topology, Hatcher is the standard reference these days), it is contractible. Hence, Milnor's $EG$ is contractible. The conclusion, therefore, is that:

A contractible classifying space $EG$ exists.

For all what I know, there are noncontractible (but weakly contractible) classifying spaces. Of course, maybe you are working in the CW category to begin with, then the entire question has trivial positive answer, since weakly contractible CW complexes are contractible.