# Homotopy type vs. weak homotopy type, and repercussions for EG

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?

• The title of the question should be "homotopy type of $EG$." By definition it has the weak homotopy type of a point. Apr 1, 2014 at 4:33
• 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 $BG\times W$. Apr 2, 2014 at 1:49
• 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 2, 2014 at 4:13
• 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. Jul 3, 2017 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 3, 2017 at 21:08