The title says it all. Is it true that if $X$ is a topological space on which all loops are homologous to zero then $X$ is simply connected, ie all loops are homotopic to zero?

I know that the result is true if $X$ is a domain in $\mathbb{C}$, but I wonder if it also holds for arbitrary topological spaces.

Also, please notice that I am aware that, in general, a loop can be homologous to zero without being homotopic to zero. Here I am asking what happens in the case when all loops are homologous to zero.


1 Answer 1


Not when the fundamental group is a perfect group, because the first homology group is equal to zero if and only if the fundamental group is equal to its commutator.

  • $\begingroup$ And I guess we have constructions to make reasonable spaces with whatever (reasonable) fundamental groups we want? $\endgroup$ Mar 25, 2022 at 2:50
  • $\begingroup$ It seems that every group can in fact be realized as the fundamental group of a 2-dim CW complex. See home.iitk.ac.in/~gabhi/MTH648Presentation.pdf $\endgroup$
    – No-one
    Mar 25, 2022 at 2:57
  • 4
    $\begingroup$ The Poincare Homology Sphere is a classic example. $\endgroup$ Mar 25, 2022 at 3:25

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