Timeline for Simplicial replacements in smoothing theory
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 16 at 7:13 | comment | added | Ryan Budney | Apologies for taking so long to accept your answer. I got busy with other things. | |
| Feb 16 at 7:12 | vote | accept | Ryan Budney | ||
| Oct 25, 2013 at 0:56 | history | edited | Tom Goodwillie | CC BY-SA 3.0 |
added 1994 characters in body
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| Oct 25, 2013 at 0:27 | comment | added | Tom Goodwillie | Ryan, I will edit my answer to sketch a proof that $Aut^{PL}$ as a subspace of $Aut^{Top}$ has the wrong homotopy type. | |
| Oct 24, 2013 at 23:29 | comment | added | Peter May | I remember Terry Wall telling me very long ago that if you give the set of PL homeomorphisms of the n-sphere the subspace topology in Top(n), the inclusion is a homotopy equivalence. I don't remember him telling me a proof. | |
| Oct 24, 2013 at 19:09 | comment | added | Ricardo Andrade | (continued) This parametric lifting property is often required. It holds if one uses, for example, simplicial sets instead of topological spaces: since filtered colimits of sets commute with finite limits, a map from a finite simplicial set to a filtered colimit $\operatorname{colim}_i X_i$ necessarily lifts to one of the simplicial sets $X_i$. Further, I believe this is one of the reasons why quasi-topological spaces are used in proofs of the h-principle: quasi-topological spaces also admit this parametric lifting property for filtered colimits, at least when the source is compact Hausdorff. | |
| Oct 24, 2013 at 19:05 | comment | added | Ricardo Andrade | @Ryan: I have a vague comment concerning topologies on spaces of germs. The natural topology on a space of germs is the colimit or final topology. Obviously, a point in the filtered colimit $\operatorname{colim}_i X_i$ lifts to one of the spaces $X_i$. But this does not hold parametrically: a map into the filtered colimit does not usually factor through any of the spaces $X_i$. (to be continued) | |
| Oct 24, 2013 at 12:53 | comment | added | Ryan Budney | Thanks Tom, this might be starting to help. Is there a good example of how the compact-open topology on the PL automorphisms of $\mathbb R^n$ isn't appropriate, that $B(Aut^{PL}(\mathbb R^n))$ would not classify PL structures on topological manifolds? Orthogonally, is it true that there is no known useful topology on spaces of germs, appropriate for setting up smoothing theory? | |
| Oct 24, 2013 at 12:47 | history | answered | Tom Goodwillie | CC BY-SA 3.0 |