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.
Asked
Active
Viewed 646 times
4
-
You can use alexander duality for example – Mykola Pochekai Jun 30 '18 at 23:54
1 Answers
0
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.)
Andres Mejia
- 20,977
-
In the question, the set removed from $\mathbb R^n$ is supposed to be closed. So it won't be the open unit interval in $\mathbb R^2$. – Andreas Blass Jul 01 '18 at 00:59
-