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I've been trying to understand nontrivial torsion subgroups $G$ of the first homology group $H_1\cong\mathbb{Z}^s \times G$, where $s$ is the rank (first betti number) and G is a finite abelian group.

I've seen examples such as the Klein Bottle, $\mathbb{SO}(3)$, $\mathbb{RP}_2$ where $G=\mathbb{Z_2}$. From these, it's easy to construct cases where $G=(\mathbb{Z}_2)^n$, because the homology of the disjoint union is the direct sum of homologies.

Are there examples where $G$ is something besides $(\mathbb{Z}_2)^n$? Maybe $G=\mathbb{Z_3}$?

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  • $\begingroup$ What is your restriction on spaces? Just manifolds? Because with topological spaces you can get any group. $\endgroup$ Apr 16 at 20:11
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    $\begingroup$ The $n$-fold dunce cap has first homology group $\mathbb{Z}/n\mathbb{Z}$. Here you can see it for $n=3$: math.stackexchange.com/questions/3997767/… $\endgroup$
    – Mark
    Apr 16 at 20:12

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For $H_1 \approx \mathbb Z / n \mathbb Z$, take any manifold or CW complex space whose fundamental group is $\mathbb Z / n \mathbb Z$ (and apply the 1st Hurewicz theorem). You can use lens spaces, for example: start with $S^3$ as the unit sphere in $\mathbb C^2$, and mod out by the (free) action of $\mathbb Z / n \mathbb Z$ given by $$k \cdot (w,z) = (e^{2 \pi i k/n} w, e^{2 \pi i k m/n} z), \quad k \in \mathbb Z / n \mathbb Z $$ where $m$ is chosen to be relatively prime to $n$; for example, $m=1$ will do for your question.

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