Homology groups of $R^n $ \ a closed subset which is homeomorphic to $R^k$ for a $k$

Question

How can i calculate the Homology groups of $R^n $ \ (a closed subset which is homeomorphic to $R^k$ for a $k$) ? (i mean $R^n $ with a closed subset which is homeomorphic to $R^k$ removed.

Answer 1

Alexander Duality should do the trick.

However, the answer depends on which open set you mean. For example $\mathbb R^2 \setminus I$, where I is the open unit interval has the homology of $S^1$, whereas $\mathbb R^2\setminus \mathbb R \times \{0\}$ has the homology of two points.

In particular, find a bounded open subset $U$ hoeomorphic $\tilde{H_j}(M) \cong \tilde{H}^{n-j-1}(S^n \setminus M)$ for $M$ a closed submanifold (excision will give the desired result.)