Simple example of $X$ with torsion in $H^1(X,\mathbb{Z})$?

Question

Question: Is there a simple example of a space $X$ possessing torsion in its first integral cohomology group $H^1(X,\mathbb{Z})$?

For reasonable spaces $X$, e.g. CW-complexes, one has $H^1(X,\mathbb{Z}) \cong [X,\mathbb{T}]$, the group of homotopy classes of maps from $X$ to the circle group $\mathbb{T}$. I would be happiest if I could also see an explicit mapping $u : X \to \mathbb{T}$ whose homotopy class is torsion.

Answer 1

By the universal coefficient theorem $H^1(X)\cong\operatorname{Hom}(H_1(X),\Bbb Z)$ as $H_0$ is always free. Morphism groups to torsion-free groups are always torsion-free.