Editing Talk:Exotic sphere (section)

==Perelman or Poincare?==

The article, as of just before my edit, said that the triviality of omega3 required perelman's proof of the poincare conjecture.  Does it depend on the Poincare Conjecture itself (as implied by the statement at the end of the same section that the triviality <=>PC), which happened to have been proved by Perelman, or, as suggested by the pre-edit article, does the statement depend on some method or other result proved by Perelman in his mass of Ricci flow work (which is broader than PC)?  I write from total ignorance, but perhaps whoever added that statement can clarify.  If it really does depend on Perelman's work rather than merely depending on the bare statement of the Poincare Conj, it would be excellent to have a sentence or reference clarifying this relation.  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:David Farris|David Farris]] ([[User talk:David Farris|talk]] • [[Special:Contributions/David Farris|contribs]]) 21:04, 17 October 2007 (UTC)</small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

The previous definition of the groups theta_n was incorrect -- it was talking about cobordism classes of homotopy n-spheres. In dimension 3, all manifolds are cobordant so theta_3 is trivial by definition.  The actual definition of theta_n (due to Rene Thom) is that it is h-cobordism classes of homotopy n-spheres, with the connect-sum operation as the monoid structure.  3-manifolds have an essentially unique connect-sum decomposition and there are no inverses (Kneser and Milnor proved this).  So I reformulated the section: first define the monoid of exotic spheres, then define the monoid of homotopy spheres.  Then mention when n>4, they are the same.  Exotic spheres are homeomorphic to the standard n-sphere.  Homotopy spheres are homotopy-equivalent to the standard n-sphere.     I think the original author was taking the line of reasoning that if you use Perelman's result (the proof of the Poincare conjecture) then there is only one homotopy 3-sphere (the standard sphere).  I doubt the previous author was referring to anything more than truth of the Poincare conjecture. [[User:Rybu|Rybu]]  <small>—Preceding [[Wikipedia:Signatures|comment]] was added at 21:27, 17 October 2007 (UTC)</small><!--Template:Undated--> <!--Autosigned by SineBot-->