Homology of manifolds

Question

Does it hold that $H_n(M) \cong \mathbb Z$ if $M$ is an $n$-manifold? I know that it is true for relative homology $H_n (M, M-p)$.

Answer 1

If the manifold is compact without boundary (i.e., "closed"), as well as connected, and you use $\mathbb{Z}$ coefficients in homology, then the isomorphism you want will hold if and only if $M$ is orientable.

Answer 2

It is true precisely when M is orientable; if there is a top-dimensional cycle, i.e., if we can orient M to get a top cycle C, then C cannot bound, because there are no (n+1)-cycles--remember than $|\partial M^n|$=$n-1$. If your manifold is not orientable, then we cannot find any coherent n-cycle, and we conclude the homology is zero.