3
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ Are you allowed boundaries? $\endgroup$
    – David Roberts
    Apr 2, 2014 at 1:43

1 Answer 1

8
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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