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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.

Davide F.
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    If I am understanding you correctly, a Googling of the phrase "exotic spheres" might answer your question. – Xander Henderson Oct 24 '17 at 16:45
  • This answered indeed to my question. And I wonder: in lower dimension is still false? Normally if the counterexample provided is in such strange space is because in low dimension it's true. – Davide F. Oct 24 '17 at 16:48
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    What do you mean "with the same covering space"? Universal cover? 3-manifolds are indeed largely determined by their fundamental group, though lens spaces (e.g. L(7,1), L(7,2)) give counterexamples. Past 3 dimensions there are massive numbers of counterexamples. Fake tori (I think you can even change the homeomorphism type!) abound in dimensions $n \geq 5$. –  Oct 24 '17 at 16:51
  • Yes, sorry, I was speaking about the universal cover indeed. A lapsus. – Davide F. Oct 24 '17 at 16:59
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    Dimension $3$ is complicated, but manifolds with multiple smooth structures are plentiful in dimension $4$: https://math.stackexchange.com/questions/2129843/uncountable-differential-structures-on-4-manifolds – anomaly Oct 24 '17 at 17:02

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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).

anomaly
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