Timeline for Loop spaces and infinite braids
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 31, 2010 at 18:26 | vote | accept | Romeo | ||
| Oct 31, 2010 at 8:13 | comment | added | Romeo | @Rezk, if we restrict to the pure braids, does Cohen's theorem say anything? | |
| Oct 30, 2010 at 22:40 | comment | added | Charles Rezk | Closely related to this: there's a virtual vector bundle (classified by a map $B(B_\infty)\to BO$), such that the associated Thom spectrum is equivalent to the mod $2$ Eilenberg MacLane spectrum. That is, "mod 2 homolgy = braid bordism". See F. Cohen, "Braid orientations and bundles with flat connections", Invent. Math. 46 (1978). | |
| Oct 30, 2010 at 21:17 | comment | added | Oscar Randal-Williams | The homology of the pure braid groups does not stabilise in the usual sense, so the homology of $BP_\infty$ is going to be very large. However, you should look at the preprint "Representation theory and homological stability" by Church and Farb. | |
| Oct 30, 2010 at 21:11 | comment | added | Romeo | Ah, that's a great theorem! Do the pure braids fit in here? Fred Cohen has a result on the simplicial group $P_{\infty}$ containing a simplicial subgroup homotopy equivalent to $\Omega S^2$, but I'm not sure how/if that relates to your statement above. | |
| Oct 30, 2010 at 20:56 | comment | added | Oscar Randal-Williams | I learnt it from G. Segal "Configuration-spaces and iterated loop-spaces" Invent. Math. 21 (1973) for example, special case b) after Theorem 3. I suspect this is not the earliest proof. | |
| Oct 30, 2010 at 20:49 | comment | added | Romeo | Do you know a proof (or where one is found)? | |
| Oct 30, 2010 at 20:36 | history | answered | Oscar Randal-Williams | CC BY-SA 2.5 |