# Are there examples where torsion subgroup of the first homology group is not $(\mathbb{Z}_2)^n$?

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 $$G=\mathbb{Z_3}$$?

• What is your restriction on spaces? Just manifolds? Because with topological spaces you can get any group. Apr 16 at 20:11
• 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/…
– Mark
Apr 16 at 20:12

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.