I'm currently studying the homology of manifolds (so it's mainly orientation and duality questions) and I was wondering how to explicitly compute homology of spaces and particularly $n$-spheres.
Indeed, most of the computations of homology groups stem from either Mayer-Vietoris sequence or excision isomorphism. So, I tried to explicit these isomorphisms but they're actually really complicated since they themselves stem from the barycentric division repeated enough (maybe that's the key but I find harsh to visualise in high dimensions).
So here is my question : how can I find a generator for $H_n(\mathbb{S}^{n})$ ? And in general, how can I imagine generators of homology groups obtained by M-V sequence/excision thm ? How can I explicit the isomorphisms that we always use for computations ?
And all these questions arise from the fact that I was trying basic questions about orientation like showing that the $n$-sphere is orientable but I can't even represent myself $H_n(\mathbb{S}^n)$, I don't understand the isomorphisms $H_n(M,M \setminus \{x\}) \simeq H_n(\mathbb{R}^n,\mathbb{R}^n \setminus \{0\})$ etc. Maybe there is no easy way to see it and one just applies theorem without having explicited the objects but I find it easier this way. The exact thing I can't represent myself is :
To see the geometric significance of this group, choose a chart around $p$. In that chart there is a neighborhood of $p$ which is an open ball $B$ around the origin $O$. By the excision theorem, ${\displaystyle H_{n}\left(M,M\setminus \{p\};\mathbb {Z} \right)}$ is isomorphic to ${\displaystyle H_{n}\left(B,B\setminus \{O\};\mathbb {Z} \right)}$. The ball $B$ is contractible, so its homology groups vanish except in degree zero, and the space $B \setminus O$ is an $(n − 1)$-sphere, so its homology groups vanish except in degrees $n − 1$ and 0. A computation with the long exact sequence in relative homology shows that the above homology group is isomorphic to ${\displaystyle H_{n-1}\left(S^{n-1};\mathbb {Z} \right)\cong \mathbb {Z} }$. A choice of generator therefore corresponds to a decision of whether, in the given chart, a sphere around $p$ is positive or negative. A reflection of $\mathbb{R}^n$ through the origin acts by negation on ${\displaystyle H_{n-1}\left(S^{n-1};\mathbb {Z} \right)}$, so the geometric significance of the choice of generator is that it distinguishes charts from their reflections.
Thank you very much :)