Timeline for Compelling evidence that two basepoints are better than one
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2011 at 3:45 | vote | accept | Daniel Moskovich | ||
| Oct 18, 2010 at 8:58 | comment | added | Jeffrey Giansiracusa | @Tom, good point! Of course 1) this is using the additional fact that the components of the diffeomorphism group are contractible, 2) It is not so easy to make the submanifolds of $\mathbb{R}^\infty$ model compatible with the operad structure | |
| Oct 14, 2010 at 17:24 | comment | added | Tom Church | @Jeff: This is a situation where passing to homotopy classes, as you did in your very first sentence, makes the action less clear. The classifying space $B\text{Diff}^+(\Sigma_{g,n})$ (which is the same as $B\Gamma_{g,n}$) is the space of submanifolds of $\mathbb{R}^\infty$ diffeomorphic to $\Sigma_g$, together with a sequence of $n$ points lying on the submanifold. The action of $S_n$ on the classifying space is just to permute these points; this is a perfectly nice action. | |
| Oct 8, 2010 at 16:39 | comment | added | Ryan Budney | Thanks Jeff. Okay, this is a place where it's not immediately clear how one could do without groupoids. It appears my role as devil's advocate has been punctured. | |
| Oct 4, 2010 at 21:02 | comment | added | Jeffrey Giansiracusa | @Ryan - I don't think a configuration space model for the bar construction helps here. The symmetric group should act via its action on the labels, but as I said, in this case it only acts by outer automorphisms. Acting on a group $G$ by outer automorphisms is the same as acting on $BG$ up to homotopy by unbased maps, and the problem is to rectify this action to a strict action on some model for $BG$. | |
| Oct 4, 2010 at 13:08 | comment | added | Daniel Moskovich | == This is nice. | |
| Oct 3, 2010 at 22:08 | comment | added | Ryan Budney | One way to see the action of $\Sigma_n$ on $B\Gamma_{g,n}$ would be to use flashy explicit bar construction models, like Paolo Salvatore's "Configuration spaces with summable labels". | |
| Oct 3, 2010 at 22:04 | history | answered | Jeffrey Giansiracusa | CC BY-SA 2.5 |