I was reading Hatcher's algebraic topology book,I found when computing something,it's convenient to give the generator explictly,hence I was tring to figure out what is the generator for the local homology $H^n(X,X-\{x\})$ for a manifold $X$ with dimension $n$.
I do as follows:first if we can find the generator for $H^n(U,U-\{x\})$ for some neiborhood,since excision is induced by the inclusion,then generator for $H^n(U,U-\{x\})$ is the same as generator for $H^n(X,X-\{x\})$
Second homeomorphism induced isomorphism in homology,hence only need to find a generator for $H^n(\Bbb{R}^n,\Bbb{R^n - \{0\}})$.
Finally I use the result in Hatcher's book example2.23,which says generator for $H^n(\Delta_n,\partial\Delta_n)$ is the identity map.Hence the identity map $id:\Delta_n\to \Delta_n$ givens the generator for $H^n(\Delta_n,\partial\Delta_n)=H^n(\Bbb{R}^n,\Bbb{R}^n-\{0\})$ (since homotopy of the pair $(\Bbb{R}^n,\Bbb{R}^n-0)\sim(\Delta_n,\partial\Delta_n)$ )
Intuitively,the generator for the local homology is some n-simplex that maps onto $U$ by $\varphi:\Delta_n \to U$(which can be written down explicitly using chart map).Is my understanding correct?
Taking $n = 1$ for example,the local homology for the $H^1(\Bbb{R},\Bbb{R}- 0)$,the generator is some map $\Delta_1 \to \Bbb{R}$ that starting from $-1$ to $1$ or $1$ to $-1$.$n= 2$ for example the generator for $H^2(\Bbb{R}^2,\Bbb{R}^2- 0)$ is a map onto disk around the origin.
