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 $1\leq k\leq n$, $n\geq 3$? If this question can be answered only for one fixed $k$ and not for all $k$'s simultaneously, it will be also very helpful.

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 $1\leq k\leq n$, $n\geq 3$? If this question can be answered only for one fixed $k$ and not for all $k$'s simultaneously, it will be also very helpful. In particular, I wonder whether arbitrary torsion can be realized.

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 $1\leq k\leq n$, $n\geq 3$?

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 $1\leq k\leq n$, $n\geq 3$? If this question can be answered only for one fixed $k$ and not for all $k$'s simultaneously, it will be also very helpful.