In Hatcher's Algebraic Topology, section $3.3$, he defines:
A local orientation of a manifold $M$ at a point $x$ is a choice of generator $\mu_x$ of the infinite cyclic group $H_n(M,M\setminus\{x\})$.
My question is why is $H_n(M, M\setminus\{x\})$ necessarily cyclic?
I can see this for $M=\mathbb{R}^n$, because by the relative homology sequence we get: $$H_n(\mathbb{R}^n,\mathbb{R}^n\setminus\{x\})\simeq H_{n-1}(\mathbb{R}^n\setminus\{x\})\simeq H_{n-1}(\mathbb{S}^{n-1})\simeq \mathbb{Z}$$ This case is easy because $\{x\}$ and $\mathbb{R}^n$ have trivial homology for $n>0$.
How am I supposed to see this in the general case?