# Piecewise linear (PL) structures on $\mathbf R^4$

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 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 $\mathbf R^4$?

• Would Taube's result that there are uncountably many exotic smooth structures on $\mathbb{R}^4$ (en.wikipedia.org/wiki/Exotic_R4) be the missing piece of your puzzle? – Benoît Kloeckner Feb 26 '15 at 12:04
• 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 that is unique up to diffeomorphism. For the latter two facts see 1.5 and 1.8 in the survey arxiv.org/pdf/1212.0885 where explicit references are given. – Igor Belegradek Feb 26 '15 at 15:10
• @IgorBelegradek Thanks! I had missed (2). If you write your comment as an answer, I'll be happy to accept it. – ACL Feb 26 '15 at 18:07

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