# Homology of subsets of $\mathbb R^n$

Let $E \subset \mathbb R^n$. Must the homology groups $H_k (E)$ be trivial for $k \geq n$? How about just for $k > n$? If not, whats an example?

Thanks.

-
What have you tried so far? –  Jesse Madnick Dec 28 '11 at 4:34
Your question is answered here: logic.info.waseda.ac.jp/~eda/index.html –  Ryan Budney Dec 28 '11 at 4:55
@Ryan: Nice counterexample! I would have thought that the ‘dimension’ of a subspace would be bounded by the ‘dimension’ of the space, but I guess pathological subspaces can behave badly... –  Zhen Lin Dec 28 '11 at 6:00
@Ryan: are there also counterexamples for $k>n$? are the higher singular homology groups of the (n-1)-Hawaian earring known? –  user17786 Dec 28 '11 at 8:16