It is said that Hurewicz proved that an aspherical is determined up to homotopy equivalence by its fundamental group. But I can only find the proofs to show that two aspherical CW complexes are homotopy equivalent if and only if their fundamental groups are isomorphic. I wonder if the statement is only true for the CW case, or it's also true for the general case?


It's not true for arbitrary spaces. For instance, let $X$ be a Warsaw circle, obtained by "closing up" a topologist's sine curve so that it is path-connected. Then $X$ has trivial (singular) homology and homotopy groups, and in particular is an aspherical space with the same fundamental group as a point, but $X$ is not homotopy equivalent to a point.

  • $\begingroup$ Thanks, I wonder why most papers would omit CW, is that because CW case is general enough? $\endgroup$ – 6666 Sep 11 '17 at 16:22
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    $\begingroup$ I would be somewhat surprised if you saw that exact statement in very many papers. What is true for arbitrary spaces is that an aspherical space is determined up to weak homotopy equivalence by its fundamental group. It's also possible that the papers you're reading state that all spaces they consider are assumed to be homotopy equivalent to CW complexes, or something like that. $\endgroup$ – Eric Wofsey Sep 11 '17 at 16:25

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