Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Are there examples of homotopy equivalent smooth, orientable manifolds $M$ and $N$ of the same dimension with non-isomorphic compactly supported cohomology rings?

share|cite|improve this question
Are you allowed boundaries? – David Roberts Apr 2 '14 at 1:43

1 Answer 1

up vote 7 down vote accepted

Let $M$ be a punctured torus and $N$ be a twice-punctured plane. Then $M$ and $N$ are homotopy equivalent, but their one-point compactifications are not (the first being a torus and the second having the homotopy type of $S^2\vee S^1\vee S^1$). In particular, $H_c^*(M)$ has a nontrivial cup product but $H^*_c(N)$ does not.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.