Defintion: a $n$-manifold $M$ is said rigid if any homotopy equivalence $M\rightarrow N$ is homotopic to a homemorphism, where $N$ is an $n$-manifold.

The sphere $S^{3}$ and hyperbolic compact oriented $3$-manifold (without boundary) and the torus $T^{3}$ are rigid if I'm not wrong.

Question: More generally, suppose that $M$ is a compact oriented $3$-manifold (without boundary) is it rigid ?

  • 4
    $\begingroup$ The lens spaces L(7,1) and L(7,2) are homotopy-equivalent but not homeomorphic, if I remember the number correctly. $\endgroup$
    – Misha
    Jan 9 '15 at 22:52
  • 1
    $\begingroup$ I think that $M$ is rigid if its universal covering is contractible. This is predicted by the Borel conjecture, but probably it's known in dimension 3. $\endgroup$
    – YCor
    Jan 9 '15 at 23:30
  • 1
    $\begingroup$ @YCor: Yes, rigidity holds for aspherical 3-manifolds (post-Perelman). $\endgroup$
    – Misha
    Jan 9 '15 at 23:58
  • $\begingroup$ mathoverflow.net/questions/77400/… $\endgroup$
    – Misha
    Jan 10 '15 at 0:01
  • $\begingroup$ possible duplicate of Diffeomorphism of 3-manifolds $\endgroup$
    – Misha
    Jan 10 '15 at 0:04

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