Timeline for Natural setting for characteristic classes?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2010 at 14:52 | comment | added | Tom Goodwillie | @Daniel: In what sense does a Ranicki quadratic structure on a chain complex give a Spivak normal spherical fibration? I haven't quite got the hang of that approach. | |
| Apr 27, 2010 at 16:06 | comment | added | Daniel Moskovich | @Dev Actually I haven't looked at that sort of idea... thanks!!! | |
| Apr 27, 2010 at 7:34 | comment | added | Dev Sinha | Again very computationally, a weakened version of the splitting principle is that the cohomology of unitary groups injects as the invariants in the cohomology of their maximal tori. You certainaly know that already, but if you haven't tried that perspective it might help. | |
| Apr 27, 2010 at 6:17 | comment | added | Daniel Moskovich | Tell me about it! One reason I'm interested in understanding such things is that I've tried to prove a splitting principle in pretty simple-looking but non-classical setting, and had no real success so far, nor reaching an understanding of what exactly is going wrong. Understanding the splitting principle is another one of many things I would like to achieve in this lifetime. | |
| Apr 27, 2010 at 6:09 | comment | added | Dev Sinha | Yes - unfortunately I don't understand how far Ranicki has been able to algebraicize manifold theory. From the computability side, one thing which makes Chern classes computable which does generalize well is that there are maps G_n \times G_m ---> G_{n+m} which are homotopy associative and commutative on classifying spaces. This gives rise to the Whitney sum formula. (This is structure collaborators and I have exploited recently in the cohomology of symmetric groups). The other thing which makes them computable is the splitting principle, which seems tougher to generalize. | |
| Apr 27, 2010 at 5:48 | comment | added | Daniel Moskovich | You have notions of Poincare duality and Spivak normal fibration via a quadratic structure on a chain complex, a la Ranicki. Those concepts still make sense for any abelian category, I think. Also, why do you need G to be a group? Classifying spaces make sense for Grothendieck fibrations over categories- see ncatlab.org/nlab/show/classifying+space How much structure of Chern classes etc. could we expect at this level of generality- or rather, "what goes wrong"? | |
| Apr 27, 2010 at 5:09 | history | answered | Dev Sinha | CC BY-SA 2.5 |