Let $M$ be an $n$-dimensional closed manifold. Choose $x \in M$. Using long exact sequence of pairs $(M,M - x)$, we have
$$H_k(M - x, \mathbb{Z}) \cong H_k(M, \mathbb{Z})$$
for $k<n-1$. For $k=n-1$, I see this is an isomorphism with $\mathbb{Q}$-coefficient. I am curious if there is an example that they are not isomorphic with integer coefficient.