12
$\begingroup$

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 $\mathbf R^4$?

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ 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? $\endgroup$
    – Benoît Kloeckner
    Commented Feb 26, 2015 at 12:04
  • 3
    $\begingroup$ 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. $\endgroup$
    – Igor Belegradek
    Commented Feb 26, 2015 at 15:10
  • $\begingroup$ @IgorBelegradek Thanks! I had missed (2). If you write your comment as an answer, I'll be happy to accept it. $\endgroup$
    – ACL
    Commented Feb 26, 2015 at 18:07

1 Answer 1

Reset to default
13
$\begingroup$

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 where explicit references are given.

Improve this answer
$\endgroup$
4
  • $\begingroup$ For dimension $<7$, is it even correct to say that the category of smooth manifolds and the category of PL manifolds are equivalent? $\endgroup$
    – Student
    Commented Nov 3, 2021 at 8:37
  • $\begingroup$ @Student: sorry don't know much about categories. Can you rephrase the question without using the word "category"? $\endgroup$
    – Igor Belegradek
    Commented Nov 3, 2021 at 13:04
  • $\begingroup$ Basically I want to find a correspondence on the maps too. More precisely (related discussion), let $X, Y$ be smooth closed $4$-manifolds, and $X', Y'$ be the corresponding PL closed $4$-manifolds. Can we find an assignment $\Phi$ that sends any smooth map $f: X \to Y$ to a PL map $g: X' \to Y'$ and $\Psi$ that goes in another direction, such that $\Psi \Phi (f)$ is smoothly homotopic to $f$, and that $\Phi \Psi (g)$ is PL homotopic to $g$? $\endgroup$
    – Student
    Commented Nov 3, 2021 at 21:44
  • $\begingroup$ @Student: To me it seems the functors should be PL or smooth homeomorphisms (not merely maps). I would look at "Obstructions to the Smoothing of Piecewise-Differentiable Homeomorphisms" by Munkres who deals with this kind of issues, and "Obstruction theories for smoothing manifolds and maps" by Hirsch. $\endgroup$
    – Igor Belegradek
    Commented Nov 3, 2021 at 21:58

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .