Timeline for Proving that a space cannot be delooped.
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2013 at 20:57 | vote | accept | Anton Fetisov | ||
| Apr 13, 2013 at 1:01 | answer | added | Peter May | timeline score: 17 | |
| Apr 12, 2013 at 19:50 | answer | added | Neil Strickland | timeline score: 19 | |
| Apr 12, 2013 at 18:10 | comment | added | Sean Tilson | cohomology over the steenrod algebra from the nishida relation. Just pick some example space to start with where you know the product in homology and the action of the steenrod algebra. | |
| Apr 12, 2013 at 18:09 | comment | added | Sean Tilson | So let me rephrase my last statement, determining when something has an $E_n$ structure is not easy in general, as far as I can tell. I think it is probably the case that you might be able to classify all finite dimensional compact lie groups that are n-fold loop spaces. I know that there is work of Bauer, Pedersen, Notbohm, Grodal, Kitchloo and others that is concerned with when things are finite loop spaces and when they can be manifolds. Try looking at that seciont of the $H_{\infty}$ ring spectra volume, it could be that you can determine something about the module structure of the ... | |
| Apr 12, 2013 at 16:41 | comment | added | Anton Fetisov | @Dylan: Very good references, thank you! @Sean: Is the question any simpler for finite-dimensional spaces? Compact Lie groups? | |
| Apr 12, 2013 at 16:17 | comment | added | Sean Tilson | Also, this is not an easy question in general. | |
| Apr 12, 2013 at 16:16 | comment | added | Sean Tilson | Every n-fold loop space, by neglect of structure, is an $H_n$ space. This implies that there is an action of some bits of the Dyer-Lashof algebra on its homology. There is something called the nishida relation that tells you how this must relate to the action of the steenrod algebra. I would recommend at looking at Steinberger's chapter of math.uchicago.edu/~may/BOOKS/h_infty.pdf | |
| Apr 12, 2013 at 15:04 | comment | added | Dylan Wilson | You can also find explicit examples in Stasheff's book on H-spaces. | |
| Apr 12, 2013 at 15:04 | comment | added | Dylan Wilson | Key word is "obstruction theory." See Robinson: arxiv.org/pdf/1301.1572v1.pdf and Goerss-Hopkins: math.northwestern.edu/~pgoerss/spectra/obstruct.pdf For example. | |
| Apr 12, 2013 at 14:39 | history | asked | Anton Fetisov | CC BY-SA 3.0 |