This is another basic question.

I'm aware that weak homotopy equivalence is strictly weaker than homotopy equivalence. For one thing, there is a weak homotopy equivalence from $S^1$ to a four-point topological space $X$ such that the only continuous maps $X \to S^1$ are constant. For another, the Warsaw circle is weakly contractible, but not contractible.

I'm also aware that in the category of CW complexes, a weak homotopy equivalence is an equivalence.

Other examples are given here: (weak) homotopy equivalence.

Given a topological group $G$, I've seen $EG$ defined by the condition that it is weakly contractible and either (1) admits a free, continuous $G$-action or (2) is the total space of a principal $G$-bundle.

When can and can't one promote (1) to (2)? In other words, when isn't $EG$ in the first definition locally trivial over $BG$?

If one can find a CW structure on $G$, one can find one on a standard construction of $EG$ and then $EG$ is contractible, but there should be other model $EG$s.

Are there noncontractible $EG$? (Guess: yes.) Are there easy/canonical constructions of weakly contractible spaces satisying (1) or (2) such that there don't exist homotopy equivalences between them?

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  • The title of the question should be "homotopy type of $EG$." By definition it has the weak homotopy type of a point. – Qiaochu Yuan Apr 01 '14 at 04:33
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    For your second question: what happens if you consider $EG\times W$ where $W$ is the Warsaw circle, and then define $g(e,w) = (ge,w)$? This should give be a principal $G$-bundle over $BG\times W$. – Jason DeVito Apr 02 '14 at 01:49
  • Jason, that is certainly true, if somehow disappointing. The product $BG \times W$ is certainly weakly homotopy equivalent to $BG$, but do you have an example over $BG$ itself? – jdc Apr 02 '14 at 04:13
  • Take $G$ itself (with action by multiplication) with the indiscrete topology. That seems to satisfy (1) and the quotient by $G$ is a single point, a poor substitute for $BG$. I think (1) must be missing something. – Omar Antolín-Camarena Jul 03 '17 at 15:26
  • Ah, yes, I've seen this example since, but had forgotten about this question. I guess this *is* an answer, but makes me wonder if this is quite the question I meant. One thing that has always bothered me is that the "definition" of $EG$ with "there exists a bundle structure" seems very nonintrinsic. I think what I wanted might have been something along the lines of this: "$BG$ seems to be indifferently defined as a homotopy type and as a weak homotopy type, with little attention paid to the distinction. Do situations actually arise where this distinction makes a difference?" – jdc Jul 03 '17 at 21:08

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