3-manifoldsat.algebraic-topologygt.geometric-topologysmoothing-theory
For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$,
$${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$
a weak homotopy equivalence? Equivalently, is the space of smooth structures on a topological 3-manifold contractible? (This is as opposed to just connected, which is the usual statement of Moise's theorem.)
I have come to believe that answering the questions I posted would be more enlightening if I try to provide an overview of the larger context that they are part of.
The literature treating and generalizing the topics mentioned in the post for surfaces is as extensive as it is interesting. The 1960's and 70's were times of very active research in this part of topology, and it still is today. Three wonderful resources are Kirby & Siebenmann's book Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Milnor's paper Differential Topology Forty-six Years Later, and A. Ranicki's slides. I will refer to these as [KS], [Mil2011] and [Ran], respectively. Also, a word on notation: uniqueness will mean up to PL, DIFF or TOP homeomorphism, depending on the category at hand. Unless otherwise stated, words like manifold or surface will have general meaning (i.e. possibly with boundary and possibly non-compact). Finally, a list of references is included at the bottom.
A given PL structure on a topological manifold may have compatible differentiable structures that are inequivalent. That is, $$\mathrm{DIFF}\rightarrow \mathrm{PL}$$ is not injective. In [Mil1956] J. Milnor gave an example of a manifold PL-homeomorphic to the usual 7-dimensional sphere $S^{7}$, but not diffeomorphic to it. In fact, it is known that for $n\neq 4$ a topological $n$-sphere admits a unique PL structure (For $n\le 3$ see [Moi1977] or [TL], for $n\ge 5$ is due to Smale and can be found in [Sma1962]. The case $n=4$ is an open question). Therefore, the inequivalent differentiable structures that Milnor constructed in [Mil1956] are all compatible with the usual PL structure on $S^{7}$.
Even More, on the topological manifold $\mathbb{R}^{4}$ it is possible to define uncountably many inequivalent PL or differentiable structures. An excellent account of this exotic $\mathbb{R}^{4}$'s can be found in Chapter XIV of [Kir1989].
The functor above is also not surjective. That is, there are PL manifolds that do not admit a compatible differentiable structure. M. Kervaire gave such an example in [Ker1960]. Later, J. Ells and N.H. Kuiper (see [EK1961]), and I. Tamura (see [Tam1961]) gave examples in dimension 8, the lowest possible.
In dimensions 7 or less, PL manifolds always admit compatible differentiable structure, and in dimensions 6 or less this happens in an a unique way (See Theorem 2 in [Mil2011] and Theorem 3.10.8 and Problems 3.10.19-20 in [TL] for dimension up to three). In this sense, DIFF=PL for manifolds of dimension $n\leq 6$, which means that the number of inequivalent differentiable structures on a topological 4-sphere is also unknown.
The obstruction to finding a differentiable structure on a given PL manifold is called the Munkres-Hirsch-Mazur obstruction (see the last paragraph on [Mil2011]).
The exotic $\mathbb{R}^{4}$'s mentioned above provide an example of a topological manifold having uncountably many inequivalent PL structures. This disproves the manifold version of the Haupvermutung. The non-manifold version of the Haupvermutung was disproven by J. Milnor ([Mil1961]), who found two homeomorphic compact simplicial complexes that are not PL homeomorphic.
In dimension $3$ or less, the Haupvermutung is true (see Chapters 35 & 36 in [Moi1977] or Thurston/Levy's book). In this sense PL=TOP for manifolds of dimension $n\le 3$. Moreover, as mentioned earlier, except possibly for $n=4$ there is only one $n$-dimensional PL sphere.
The obstruction to finding a PL structure on a given topological manifold culminated with the resuts of Kirby and Siebenmann (the Kirby–Siebenmann class). (see [KS] and Theorem 1 in [Mil2011]).
More striking is the E8 manifold which, not being triangulable, cannot have a differentiable structure. It provides and example of a topolgical manifold of dimension four that admits neither PL nor differentiable structures.
The exotic $\mathbb{R}^{4}$'s mentioned earlier give an example of a topological manifold having uncountably many inequivalent differentiable structures.
In dimension 3 or less the results above yield DIFF=PL=TOP.
Coming back to surfaces I want to point out that Theorem 8.3 in [Moi1977] shows that every surface is triangulable. At the begining of the proof it is shown that triangulations and PL structures are equivalent structures on a surface. Moreover, Theorem 8.5 is the Haupvermutung for surfaces. Therefore, a complete classification of non-compact surfaces (with boundary) seems to have been achieved by the results contained and mentioned in Prishlyak and Mischenko's paper.
Finally, I want to point out that the result that two smooth surfaces are diffeomorphic iff they are homeomorphic is due to J. Munkre's and can be found in his dissertation Some Applications of Triangulation Theorems, U. of Michigan, 1955. The proof uses the triangulation theorems proven by E.E. Moise, who was Munkres' advisor.
REFERENCES:
The references [KS], [Mil2011] and [Ran] are listed (with links) right at the beginning, in the second paragraph above.
[Cai1934] Cairns, S. S., On the triangulation of regular loci, Ann. Math. (2) 35, 579-587 (1934). ZBL0012.03605.
[Cai1936] Cairns, S. S., Polyhedral approximations to regular loci., Ann. Math., Princeton, (2) 37, 409-415 (1936). ZBL62.0806.04.
[Cai1961] Cairns, S. S., A simple triangulation method for smooth manifolds, Bull. Am. Math. Soc. 67, 389-390 (1961). ZBL0192.29901.
[Ker1960] Kervaire, M.A., A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34, 257-270 (1960). ZBL0145.20304.
[Kir1989] Cairns, S. S., A simple triangulation method for smooth manifolds, Bull. Am. Math. Soc. 67, 389-390 (1961). ZBL0192.29901.
[EK1961] Eells, J.; Kuiper, N.H., Manifolds which are like projective planes, Publ. Math., Inst. Hautes 'Etud. Sci. 14, 181-222 (1962). ZBL0109.15701.
[Mil1956] _Milnor, J. W., On Manifolds Homeomorphic to the $7$-Sphere, Ann. Math. (2) 64, 399-405 (1956). ZBL0102.38103.
[Mil1961] _Milnor, J. W., Two complexes which are homeomorphic but combinatorially distinct, Ann. Math. (3) 74, 575-590 (1961). ZBL0102.38103.
[Moi1977] _Moise, E. E., Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics. 47. New York - Heidelberg - Berlin: Springer-Verlag. X, 262 p. (1977). ZBL0349.57001.
[Mun1960] _Munkres, J.R., Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. Math. (2) 72, 521-554 (1960). ZBL0108.18101.
[Mun1966] _Munkres, J.R., Elementary differential topology. Lectures given at Massachusetts Institute of Technology, Fall, 1961. Revised ed, Annals of Mathematics Studies. 54. Princeton, N.J.: Princeton University Press. XI, 112 p. (1966). ZBL0161.20201.
[Sma1962] _Smale, S., On the structure of manifolds, Am. J. Math. 84, 387-399 (1962). ZBL0109.41103.
[Tam1961] _Tamura, I., 8-manifolds admitting no differentiable structure, J. Math. Soc. Japan 13, 377-382 (1962). ZBL0109.16302.
[TL] _Thurston, W. P., Three-dimensional geometry and topology. Vol. 1. Ed. by Silvio Levy, Princeton Mathematical Series. 35. Princeton, NJ: Princeton University Press. x, 311 p. (1997). ZBL0873.57001.
[Whi1940] _Whitehead, J.H.C., On $C^1$-complexes, Ann. Math. (2) 41, 809-824 (1940). ZBL0025.09203.
I recommand Rob Kirby's video of his first lecture in Edinburgh: http://www.maths.ed.ac.uk/~aar/kirby.htm his lecture begins with Alexander's trick.
Best Answer
$\mathsf{Diff}(S^3)$ is homotopy equivalent to $O(4)$, by Hatcher's solution to the Smale Conjecture: Hatcher, Allen E. A proof of the Smale conjecture, Diff(S3)≃O(4). Ann. of Math. (2) 117 (1983), no. 3, 553–607.
Cerf proved that the Smale conjecture implies that $\mathsf{Diff}(M) \to \mathsf{Homeo}(M)$ is a weak equivalence for all $3$--manifolds $M$, in J. Cerf, Sur les difféomorphismes de la sphère de dimension trois ( $\Gamma_4$ = 0 ), Lecture Notes in Math., vol. 53, Springer-Verlag, Berlin and New York, 1968, I think. See Linearization in 3-Dimensional Topology by Hatcher.