All curves homologous to zero implies space is simply connected?

Question

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.

score 7 · Accepted Answer · answered Mar 25 '22 at 02:49

7

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.

answered Mar 25 '22 at 02:49

Compacto

2,200

And I guess we have constructions to make reasonable spaces with whatever (reasonable) fundamental groups we want? – paul garrett Mar 25 '22 at 02:50
It seems that every group can in fact be realized as the fundamental group of a 2-dim CW complex. See http://home.iitk.ac.in/~gabhi/MTH648Presentation.pdf – No-one Mar 25 '22 at 02:57
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The Poincare Homology Sphere is a classic example. – Cheerful Parsnip Mar 25 '22 at 03:25

All curves homologous to zero implies space is simply connected?

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