Skip to main content
17 events
when toggle format what by license comment
S Oct 18, 2016 at 11:44 history suggested uhbif19 CC BY-SA 3.0
added some spacers
Oct 18, 2016 at 11:18 review Suggested edits
S Oct 18, 2016 at 11:44
May 10, 2016 at 9:58 answer added Andrew Ranicki timeline score: 5
Jun 30, 2010 at 14:30 answer added Sean Tilson timeline score: 2
May 2, 2010 at 6:37 vote accept Daniel Moskovich
Apr 27, 2010 at 15:16 comment added Tim Perutz You might find Fulton-Lang's "Riemann-Roch algebra" helpful. They explain the universal target for Chern classes in algebraic geometry. The algebra - Grothendieck groups, $\lambda$-rings and all that - is somewhat portable.
Apr 27, 2010 at 13:45 comment added Tim Perutz Daniel, "characteristic classes" do appear elsewhere: Grothendieck's (?) Chern classes of algebraic vector bundles (see e.g. Fulton's "Intersection theory"); Milnor's SW classes for quadratic forms, valued in Milnor K-theory; Connes's Chern character from the K-theory of a $\mathbb{C}^*$-algebra to its cyclic homology. In each case, there are manifest parallels with topology, but I don't know whether they all do (or should) fit under the same categorical umbrella.
Apr 27, 2010 at 12:25 comment added Charlie Frohman I saw Chern give a lecture when I was a graduate student. He admonished all the topologists for thinking of characteristic classes as cohomology classes, as opposed to geometric objects. He said we were throwing out a lot of really important information by doing that. Maybe, the right framework for understanding characteristic classes is physical as opposed to homotopy theoretic. Just a thought. :)
Apr 27, 2010 at 10:29 comment added Daniel Moskovich @Tim Abelian categories are the most general setting in which homological algebra makes sense, I think, in that you have the five lemma, the snake lemma, and derived functors. It's no longer algebraic topology perhaps... but it's "the natural setting for the machine"... Am I missing something? My question is what the corresponding natural setting for characteristic classes should be... it bothers me not to know that. Surely it must be well-known? Do number theorists or algebraic geometers ever use characteristic classes the way they use fundamental groups, or various versions of homology?
Apr 27, 2010 at 10:10 answer added Jeffrey Giansiracusa timeline score: 7
Apr 27, 2010 at 8:56 comment added Tim Perutz Daniel, forgive me, but what do abelian categories have to do with the chain complexes in algebraic topology? Singular cohomology is the simplest generalized cohomology theory - that's one "right setting" - and singular cochains form an $E_\infty$-algebra which remembers homotopy type - that's another. But chain complexes over abelian categories...?
Apr 27, 2010 at 5:55 comment added Daniel Moskovich @Pete Thanks! I can imagine much more general settings in which there is a sensible notion of classifying space for fibrations or bundles though- see ncatlab.org/nlab/show/classifying+space What, if anything, goes wrong at that (rather over-ambitious) level of generality?
Apr 27, 2010 at 5:09 answer added Dev Sinha timeline score: 4
Apr 27, 2010 at 4:50 comment added Pete L. Clark For any topological group G, the classifying space BG exists and is universal for principal G-bundles on reasonable -- say paracompact -- spaces, i.e., isomorphism classes of G bundles on a space X correspond to homotopy classes of maps from X to BG. Thus you can define G-characteristic classes by pulling back cohomology classes on BG. This seems pretty general, no?
Apr 27, 2010 at 4:38 comment added Daniel Moskovich @Harry You're right- homotopy theory is yet another component of algebraic topology. Its natural setting is model categories. It's also intrinsic. Something unrelated to this question which I'm unclear about is how much of homotopy theory (on model categories) is recovered by homotopy of chain complexes (on abelian categories).
Apr 27, 2010 at 4:06 comment added Harry Gindi What about homotopical information like fibration sequences?
Apr 27, 2010 at 3:22 history asked Daniel Moskovich CC BY-SA 2.5