Timeline for Who discovered this definition of Stiefel-Whitney classes?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2022 at 12:36 | history | edited | Martin Sleziak |
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| Dec 21, 2016 at 16:42 | answer | added | Nicholas Kuhn | timeline score: 6 | |
| Dec 20, 2016 at 16:46 | vote | accept | user84144 | ||
| Dec 20, 2016 at 7:50 | comment | added | user84144 | What I mean is that I believe I ``know'' which class I want to pull back to get $w_n$ (in response to your very reasonable complaint that one needs to pin down what one means by "generators"). For instance, I'm optimistic that I could write down an explicit representative by a Schubert cell, if necessary. (We wouldn't know that it corresponds to a symmetric polynomial without doing more work.) Now I'm not suggesting that this is an efficient definition, but in my mind it illustrates a rough conceptual divide between a "classifying space definition" and "Steenrod square definition". | |
| Dec 19, 2016 at 23:46 | comment | added | Oscar Randal-Williams | @DenisNardin: Being homogeneous does not characterise generators; the cohomology of BO has many graded ring automorphisms. | |
| Dec 19, 2016 at 23:42 | comment | added | Oscar Randal-Williams | @user84144: Is this really a definition? The only way I can think of showing that map i) is injective, and ii) hits the symmetric polynomials, needs me to already have a definition of Stiefel-Whitney classes and have proved the sum formula for them. | |
| Dec 19, 2016 at 22:47 | comment | added | user84144 | Well, you give me too much credit. What I actually have in my head is a specific choice of generator labelled $w_n$, corresponding to a specific symmetric polynomial under the embedding $H^*(BO(n)) \hookrightarrow H^*(BO(1)^n)$. I lazily decided not to be more precise, thinking that nobody would care, but that seems to have been a wrong presumption. | |
| Dec 19, 2016 at 22:32 | comment | added | Denis Nardin | @OscarRandal-Williams I think he meant the homogeneous polynomial generators, which I believe in this case is in fact a characterization of the elements of $H^*(BO;\mathbb{Z}/2)$ we are talking about | |
| Dec 19, 2016 at 22:13 | comment | added | user84144 | Sorry, I purposefully did not say "the generators" but maybe should have said "certain generators", which I realize is not actually a definition either, but is probably enough indication of what I'm referring to. | |
| Dec 19, 2016 at 22:04 | answer | added | Carlo Beenakker | timeline score: 7 | |
| Dec 19, 2016 at 21:58 | answer | added | John Rognes | timeline score: 17 | |
| Dec 19, 2016 at 21:52 | comment | added | Oscar Randal-Williams | That is not a definition at all, as ``the generators" does not define elements of a ring. | |
| Dec 19, 2016 at 21:42 | history | asked | user84144 | CC BY-SA 3.0 |