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.


[edit] I was assuming orientability. In the long exact sequence with $\mathbb{Z}$-coefficient
$$H\_n(M) \rightarrow H\_n(M, M-x) \rightarrow H\_{n-1}(M-x) \rightarrow H\_{n-1}(M) \rightarrow 0,$$
the first map is an isomorphism so the third is an isomorphism. Sorry for stupid question.



homology of punctured manifolds
This is also an isomorphism with $\mathbb{Z}$-coefficients. One way to see it is to use the "open-closed" exact sequence
$$H^0(M,\mathbb{Z})\rightarrow H^0(\{x\} ,\mathbb{Z})\rightarrow H^1\_c(M-x,\mathbb{Z})\rightarrow H^1(M,\mathbb{Z})\rightarrow 0$$
and Poincaré duality which identifies the last (nontrivial) arrow with $H\_{n-1}(M-x,\mathbb{Z})\rightarrow H\_{n-1}(M,\mathbb{Z})$. 


