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

## 1 Answer

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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.