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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.

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    @Ryan: are there also counterexamples for k>n? are the higher singular homology groups of the (n-1)-Hawaian earring known?2011-12-28

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