Skip to main content
6 events
when toggle format what by license comment
Dec 3, 2012 at 0:37 comment added Qfwfq I see; thank you for the explanations.
Dec 3, 2012 at 0:22 comment added Misha ...which are all different and that's OK: Different categories address different needs, even though, DIFF, so far, is the most (but not exclusively) useful one for geometry and analysis.
Dec 3, 2012 at 0:18 comment added Misha @Qfwfq: This is what I was trying to get to in my discussion of the Lipschitz category and history. The problem (as I tried to explain it) is lack of analytical tools in the topological setting, since Lipschitz does not quite cut it. (Also, in dimension 4 Lipschitz is not equivalent to TOP, as proven by Donaldson and Sullivan.) Lastly, TOP does not completely kill exoticity in the sense that there are homotopy-equivalences which are not homotopic to homeomorphisms. One can also argue for Poincar\'e complexes, and so on. My viewpoint is that there are different manifold-like categories ...
Dec 2, 2012 at 23:47 comment added Qfwfq @Misha: « Since there are smooth manifolds which are homeomorphic but not diffeomorphic, one should replace the classical concept of a smooth manifold with a different (unspecified) categorical one. » - If the purpose is to "kill exoticity", wouldn't this "unspecified categorical concept" just concide with that of topological manifold?
Dec 2, 2012 at 23:14 comment added Mozibur Ullah Agreed, the generalised smooth spaces have to prove its worth (I'm guessing that it will), but that doesn't diminish the importance of the usual smooth manifolds. I guess an analogy would be the discovery of the complex plane and the realisation of its importance (it had to prove its worth - it wasn't enough to know that it was there), but that doesn't diminish the importance of the real line.
Dec 2, 2012 at 22:57 history answered Misha CC BY-SA 3.0