Return to Answer

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].

3. [TOP & DIFF] As John Klein points out in the comments, smoothing a topological manifold is in general a question formulated by first putting a combinatorial structure on the manifold (normally a handlebody structure or a PL structure). The examples of Kervaire, Ells & Kuiper and Tamura mentioned above yield topological manifolds having no differentiable structure. However, these are still PL manifolds.

The exotic $\mathbb{R}^{4}$'s mentioned earlier give an example of a topological manifold having uncountably many inequivalent differentiable structures.

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].

3. [TOP & DIFF] As John Klein points out in the comments, smoothing a topological manifold is in general a question formulated by first putting a combinatorial structure on the manifold (normally a handlebody structure or a PL structure). That is, $$\mathrm{DIFF}\rightarrow \mathrm{TOP}$$ is nothing but the composition of the previous two functors and is therefore neither surjective nor injective.

Indeed, the examples of Kervaire, Ells & Kuiper and Tamura mentioned above yield topological manifolds having no differentiable structure. However, these are still PL manifolds.

The exotic $\mathbb{R}^{4}$'s mentioned earlier give an example of a topological manifold having uncountably many inequivalent differentiable structures. Another example given above are Milnor's non-diffeomorphic $S^{7}$ spheres.

[Mil1961] _Milnor, J. W., Two complexes which are homeomorphic but combinatorially distinct, Ann. Math. (2) 74, 575-590 (1961). ZBL0102.38103.

[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.

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.

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.