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May 26, 2023 at 16:46 comment added Tom Goodwillie Incidentally, if a volume-preserving diffeomorphism is homotopic to the identity then it is homotopic to the identity through volume-preserving diffeomorphisms.
May 26, 2023 at 3:51 vote accept ddorado
May 26, 2023 at 3:46 answer added Michael Albanese timeline score: 6
May 26, 2023 at 3:42 comment added Denis T Look at two-dimensional torus (as both $M$ and $N$) to get very clear "no" answer to the question in title.
May 26, 2023 at 3:25 history asked ddorado CC BY-SA 4.0