Talk:Piecewise linear manifold

Can someone clarify when a PL manifold admits (unique?) DIFF structure? Does this happens for sure, as stated in 4-manifolds article, if dimension is less than 6? Omar.zanusso (talk) 17:48, 31 May 2008 (UTC)


 * Maybe I'm wrong but there are exotic R4 overthere... so triangulate R4 in any way you want... it is PL but there are plenty (uncountable many) non diffeomorphic structures —Preceding unsigned comment added by 158.109.1.23 (talk) 08:27, 2 June 2010 (UTC)

Smoothing theory
I moved the reference to the Kirby-Sibenmann class since it was in a mis-leading place: the K-S class is the obstruction to giving a TOP manifold a PL structure.

Yes, I can confirm that every PL manifold of dimension less than or equal to 6 admits a unique smooth structure up to diffeomorphism. The standard reference is a book by Hirsch and Mazur, "Smoothings of piecewise linear manifolds", Annals of Mathematics Studies, No. 80. Princeton University Press,Djcrowley (talk) 17:44, 16 September 2009 (UTC)


 * A better alternative to PDIFF could be A, which is a larger category naturally containing both smooth and PL categories, and BA --> BPL is a product fibration Remarksen (talk) 06:11, 5 February 2012 (UTC)