Timeline for Cohomology operations on unoriented cobordism
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 5, 2016 at 17:22 | comment | added | Dylan Wilson | @TylerLawson Great! And tracing through the references, it looks like Gilmour shows in her thesis that the map HF_2--->MO isn't H_infty. (She only states the theorem as "the map isn't E_infty", but the proof just shows that the map on homology can't respect the Dyer-Lashof operations.) It's Proposition 6.8 here: theses.gla.ac.uk/3788/1/2006GilmourPhD.pdf | |
| Jul 5, 2016 at 15:52 | comment | added | Tyler Lawson | @Dylan the map definitely doesn't lift to a map of commutative ring spectra (prop 5.2 in Baker-Richter's "Some properties of the Thom spectrum...") | |
| Jul 5, 2016 at 10:46 | answer | added | Mark Grant | timeline score: 7 | |
| Jun 25, 2016 at 3:37 | comment | added | user51223 | I think some papers of Eccles and Grant have things similar to what you ask for. For instance, you may look at Eccles, P.J.; Grant, M. Self-intersections of immersions and Steenrod operations. (English) Zbl 1299.57018 Acta Math. Hung. 137, No. 4, 272-281 (2012). | |
| Jun 23, 2016 at 19:57 | comment | added | Dylan Wilson | These Steenrod ops the OP is referring to are the power operations... So they definitely depend on the multiplicative structure, not just the additive structure. But you're right, I guess I don't know whether that map is anything more than E_2... It's likely not E_infty but I would be surprised if it wasn't H_infty (which is what we're asking). I dunno how I'd prove it... The analog for MU and BP is false I think, but with HF_2 I feel more optimistic. | |
| Jun 23, 2016 at 17:55 | comment | added | Denis Nardin | @DylanWilson Sorry for the misattribution (and I agree that it is an additive statement, but cohomology operations are the homotopy of the endomorphism ring anyway, so they depend only on the additive structure) Is it true it is a commutative HF_2-algebra? Do you have a reference? I thought that the map HF_2->MO was only E_2 (so no control on power operations). | |
| Jun 23, 2016 at 16:51 | comment | added | Dylan Wilson | Denis: that was proved long before Quillen, and is an additive statement in any case... That said, it's true that MO is actually an HF_2 algebra so it is indeed the case that all its power operations come from the Dyer-Lashof algebra for HF_2. | |
| Jun 23, 2016 at 16:48 | comment | added | vareps | Could it be shown more directly, e.g. that these operations act on $N^*(pt)$ trivially? | |
| Jun 23, 2016 at 16:28 | comment | added | Denis Nardin | I'm not sure I follow. The spectrum $MO$ is a wedge of Eilenberg-MacLane spectra (that's Quillen's theorem), so all its cohomology operations come from Steenrod squares. | |
| Jun 23, 2016 at 16:06 | review | First posts | |||
| Jun 23, 2016 at 16:16 | |||||
| Jun 23, 2016 at 16:00 | history | asked | vareps | CC BY-SA 3.0 |