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?**