As in the title, I was wondering if that's true or not. If it's true it seems quite odd, but on the other hand I'm not able to find a counterexample.
2026-04-01 11:37:57.1775043477
Two smooth manifolds without boundaries with the same universal cover and fundamental group are diffeomorphic?
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The space $S^2 \times S^3$ is a double cover of $M = S^2\times \mathbb{RP}^3$ and $N = \mathbb{RP}^2 \times S^3$, but they're not even homotopy equivalent (they have different $H^2$, for example).