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Mathematics B‑class High‑priority

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Does anyone have a reference for the following claim maid in this article. (I thought it is still open).

Take a sum of m copies of II_{1,1} and 2n copies of E_8. If m \le 2n, then "Donaldson and Furuta proved" that no smooth structure exists. Katzmik 09:23, 9 May 2007 (UTC)

This is the 10/8 theorem, theorem 2 of a paper of Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifolds topology, J. Diff. Geom. 26 (1987), 397–428.. Orthografer 17:59, 30 May 2007 (UTC)

I looked up that theorem, and it does not say this. That theorem assumes the positive part has rank 1 or 2. There is a relevant theorem of Furuta, but it's not quite as strong as stated. --Dylan Thurston (talk) 20:45, 8 September 2009 (UTC)

My mistake, Furuta's theorem is precisely what's written. Reference added. --Dylan Thurston (talk) 00:20, 10 September 2009 (UTC)

Changed priority to top because our universe is at least 4 manifold! We need to understant this Daniel de França (talk) 19:21, 12 March 2008 (UTC)

Can someone give a reference about statement "For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way"? Omar.zanusso (talk) 15:10, 30 May 2008 (UTC)

I can give you a meta-reference: Thurston's book mentions this fact and should have some references for it. --C S (talk) 23:15, 31 July 2008 (UTC)

That book only mentions results for dimensions 3 or less. Kirby and Siebenmann's book is a better guess; however, their techniques only apply in dimensions 5 and higher, so are not useful for this article. They refer to Cerf's 1959 paper for dimensions 4 and less, but that seems to be a mistake as far as I can tell. --Dylan Thurston (talk) 20:45, 8 September 2009 (UTC)

Classification of homeomorphic manifolds

Article currently has a huge hole in it: a discussion of the non-classifiability of 4-manifolds, the thm by Markov (more precisely, "there is no algorithm for classifying 4-D manifolds") see e.g. http://www.mathkb.com/Uwe/Forum.aspx/math/13873/4-manifold-classification for references. Current article is even misleading: right from the get-go, just a few sentences in, it starts talking about "Freedman's classification", which seems to imply that they are classifiable! Should state first, that, in general, they're not, but that there are certain special cases which are (i.e. the Freedman case). linas (talk) 17:56, 26 November 2011 (UTC)

PROOF OF THE 4-DIMENSIONAL SMOOTH POINCARÉ CONJECTURE IN PDE's ALGEBRAIC TOPOLOGY

The 4-dimensional smooth Poincaré conjecture has been proved in the following work:

A. Prástaro, Exotic PDE's. Mathematics Without Boundaries: Surveys in Interdisciplinary Research. (Eds. P. M. Pardalos and Th. M. Rassias.) Springer-Heidelberg New York Dordrecht London (2014), 471--532. ISBN 978-1-4939-1124-0 (print) 978-1-4939-1124-0 (eBook). DOI: 10.1007/978-1-4939-1124-0. arXiv: 1101.0283.

For complementary information see also the following works:

A. Prástaro, Exotic heat PDE's. Commun. Math. Anal. 10(1)(2011), 64-81. arXiv: 1006.4483.

A. Prástaro, Exotic heat PDE's.II. Essays in Mathematics and its Applications. (Dedicated to Stephen Smale.) (Eds. P. M. Pardalos and Th. M. Rassias.) Springer-Heidelberg New York Dordrecht London (2012), 369--419. ISBN 978-3-642-28820-3 (Print) 978-3-28821-0 (Online). DOI: 10.1007/978-3-642-28821-0. arXiv: 1009.1176.

The method used is the Algebraic Topology of PDE's formulated by A. Prástaro in a series of early published works on the geometric theory of PDE's and the characterization of their global solutions by means of suitable bordism groups.

In particular in and it is proved the following theorem.

Theorem The Ricci flow equation for n-dimensional Riemannian manifolds, admits that starting from a n-dimensional sphere S^n, we can dynamically arrive, into a finite time, to any n-dimensional homotopy sphere M. When this is realized with a smooth solution, i.e., solution with characteristic flow without singular points, then S^n is diffeomorphic to M. The other homotopy spheres \Sigma^n, that are homeomorphic to S^n only, are reached by means of singular solutions. In particular, for n belonging equal to 1,2,3,4,5,6, one has also that any smooth n-dimensional homotopy sphere M is diffeomorphic to S^n. In particular, the case n=4, is related to the proof that the smooth Poincaré conjecture is true.

Agostino.prastaro (talk) 19:01, 26 October 2014 (UTC)

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