Let $M^3$ and $N^3$ be two oriented closed $3$-manifolds which are simple homotopy equivalent. Are $M^3$ and $N^3$ diffeomorphic? I was told that this follows from the geometrization conjecture, but I cannot find a reference.
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1$\begingroup$ There are very few $3$-manifolds which can be homotopy equivalent without being diffeomorphic. I think the only examples are lens spaces (and perhaps some connected sums). So it should be enough to check that non-diffeomorphic lens spaces are not simply homotopy equivalent, which I think follows from computing Reidemeister torsions. $\endgroup$– ThiKuCommented Mar 21, 2019 at 21:49
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$\begingroup$ See Ian Agol's answer here. The answer is yes and it is due to Turaev once one knows the geometrization conjecture. $\endgroup$– mmeCommented Mar 21, 2019 at 22:29
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