Let $M$ be a compact, connected $n$-manifold. Say that $M$ is orientable if there is a class $\alpha$ in $H_n(M)$ such that the reduced homology map $H_n(M)\rightarrow H_n(M,M\setminus\{p\})$ takes $\alpha$ to a generator of $H_n(M,M\setminus\{p\})\cong \mathbb Z$ for each $p\in M$. I am trying to show that this is equivalent to $H_n(M)\cong \mathbb Z$.
I know the map $H_n(M)\rightarrow H_n(M,M\setminus\{p\})$ fits into the long exact sequence of a pair
$$H_n(M\setminus\{p\}) \rightarrow H_n(M)\rightarrow H_n(M,M\setminus\{p\}) .$$
The group $H_n(M\setminus\{p\}) $ is 0 by Proposition 3.29 in Hatcher ($M\setminus\{p\}$ is a non-compact connected $n$-manifold). So, the map $H_n(M)\rightarrow H_n(M,M\setminus\{p\})$ is injective, which means $H_n(M)\cong\mathbb Z$ if $M$ is orientable. However, I am having trouble with the converse. Does anyone have any suggestions?
