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.

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.