Given a closed, connected, spin, differentiable $n$-manifold $M$, $n\geq 3$, are there any restrictions on the integral singular homology groups of it? In other words, can one realize any finitely generated abelian group as a $k$-th singular homology group over $\mathbb Z$ of some closed, connected, spin, differentiable $n$-manifold $M$, where $1\leq k\leq n$, $n\geq 3$?