Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence $ 0\rightarrow Ext(H_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow H^{k}(X,\mathbb{Z})\rightarrow Hom(H_{k}(X,\mathbb{Z}),\mathbb{Z})\rightarrow 0 $ for $1\le k$. In particular a torsion element in $H_{1}(X,\mathbb{Z})$ induces a torsion element in $H^2(X,\mathbb{Z})$.
I would like to understand the statement above in the following situation. Consider a 2-torus (i.e. $(S^1)^2$) fibration $f:X\rightarrow Y$. Assume $H_{1}(Y,\mathbb{Z})=\pi_{1}(Y)=\mathbb{Z}/2\mathbb{Z}$ and monodromy representation is given by $(-id,-id)$ on $(S^1)^2$. Then the Spectral sequence associated with $f$ shows $ H_{1}(X,\mathbb{Z})=H_{1}(Y,\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}. $ Is it possible to "see" the torsion in $H^2(X,\mathbb{Z})$ induced by $H_{1}(X,\mathbb{Z})$?
One may want to use Poincare duality (to realize the torsion as a submanifold etc), so please take $Y$ in your favorite dimension.