What is the homology of $\mathbb{R}^n-U$, where $U$ is a bounded open ball in $\mathbb{R}^n$?
This is a smaller question I have on my road to understand the local consistency condition on defining the orientation of an $n$-manifold based on their local $n$th homology groups
If $U$ is an open ball then $\Bbb R^n-U$ is homotopy equivalent to the $(n-1)$-sphere $S^{n-1}$, so in dimensions $\ne n-1$ its homology is zero, and in dimension $n-1$ its homology is $\Bbb Z$ (or whichever other coefficient ring you are using).