Diffeomorphism of 3-manifolds 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).
 A: This doesn't actually answer the question, but concerns Tim's comment:-
"The core of the question - I think! - is whether group theory, plus a bit of extra topological input, recognizes the geometric pieces of a 3-manifold."
It's certainly true that the fundamental group sees a lot of the geometry.  Scott and Swarup proved that you can reconstruct the JSJ (torus) decomposition of an irreducible 3-manifold from the fundamental group.  If your manifold isn't irreducible then the Kneser--Milnor decomposition corresponds exactly to the Grushko decomposition of π1.  And π1 also determines the geometry of geometric pieces---Seifert-fibered pieces have normal cyclic subgroups etc.
(Of course, you need the Poincare Conjecture to know that you didn't connect sum with a fake 3-sphere!)
A: About the elliptic case $S^3/G$: elliptic 3-manifolds are classified up to homeomorphism by their $\pi_1$'s, except for the lens spaces. For the lens spaces the simple homotopy type classification is equivalent to the homeomorphism classification (see e.g Milnor, Whitehead torsion, Bulletin AMS, 72) but differs from the homotopy classification.
A: Regarding your first question, in 1953 Moise proved the (manifold) Hauptvermutung for 3-manifolds (Ann. of Math. 58, pp. 458-480). One way to state his result is that every homeomorphism (diffeomorphism) between compact 3-manifolds is homotopic to a PL homeomorphism.
A: Turaev defined a simple-homotopy invariant 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
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).
A: I don't know in general, so I'll just the more obvious cases. For hyperbolic 3-manifolds, this is implied by Mostow's rigidity theorem, which states that a homotopy equivalence of hyperbolic manifolds $n$-manifolds is homotopic to an isometry. It's also true for $S^3$, since both $Diff(S^3)$ and $Aut^h(S^3)$ have two components.
A: Waldhausen proved that homotopy equivalence is homotopic to  homeo (and hence diffeo) for Haken 3-manifolds. Perelman extends that to irreducible/infinite pi_1.
It's an old conjecture that the Whitehead group of any torsion free group is trivial. 
Irreducible 3-manifolds either have finite or torsion free pi_1, so given Perelman again  only S^3/G have potentially non-simple homotopy equivalences.  
