One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference is given to Milnor's paper, [*Differential Topology Forty-six Years Later*](http://www.ams.org/notices/201106/rtx110600804p.pdf) where this fact is indeed printed on the bottom of the first column of page 2. 


However, I could not follow the arguments that lead to this conclusion, nor find them in the litterature (eg, Freedman-Quinn's book).


Results of the 60s imply (Theorem 2 of Milnor's quoted paper) that any PL structure on $\mathbf R^4$ gives rise to a well-defined, compatible, smooth structure. Then Freedman classified topological closed 4-manifolds, and Donaldson proved that many of them cannot be smoothed. But how does this help constructing exotic PL structures on $\mathbf R^4$?



Piecewise linear (PL) structures on $\mathbf R^4$
There are three facts: 


1. existence of uncountably many non-diffeomorphic exotic $\mathbf R^4$'s.
2. any smooth manifold has a PL structure.
3. Any PL manifold of dimension $<7$ has a smooth structure which is unique up to diffeomorphism.


For the latter two facts see 1.5 and 1.8 in [this survey](http://arxiv.org/pdf/1212.0885) where explicit references are given.


