Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder whether the surgeons' key problem has been solved. Is every simple homotopy equivalence between smooth, closed 3-manifolds homotopic to a diffeomorphism?


In related vein, it follows from J.H.C. Whitehead's theorem that a map of closed, connected smooth 3-manifolds is a homotopy equivalence if it has degree $\pm 1$ and induces an isomorphism on $\pi\_1$. Is there a reasonable criterion for such a homotopy equivalence to be simple? One could, for instance, ask about maps that preserve abelian torsion invariants (e.g. Turaev's).



Diffeomorphism of 3-manifolds
Turaev [defined a simple-homotopy invariant](https://mathscinet.ams.org/mathscinet-getitem?mr=970081) which is a complete invariant of homeomorphism type (originally assuming geometrization).


Here is the Springer link if you have a subscription:
[Towards the topological classification of geometric 3-manifolds](https://doi.org/10.1007/BFb0082780)


He claims in the paper that a map between closed 3-manifolds is a
homotopy equivalence if and only if it is a simple homotopy equivalence,
but he says that the proof of this result will appear in a later paper. I'm
not sure if this has appeared though (I haven't searched through his
later papers on torsion, and there's no MathScinet link).


