Timeline for Are the Stiefel-Whitney classes of a vector bundle the only obstructions to its being invertible?
Current License: CC BY-SA 2.5
12 events
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| Oct 20, 2010 at 14:32 | vote | accept | Georges Elencwajg | ||
| Oct 18, 2010 at 18:32 | comment | added | Charles Rezk | Here's a more direct proof of what I need. Claim: If $V$ and $W$ are real vector bundles over $BG$ of dimensions $m$ and $n$ (with $G$ any finite group), then $V\oplus W$ trivial implies $V$ trivial. Proof: a trivialization of $V\oplus W$ gives a map $BG\to G(m+n,m)$, where $G(m+n,m)$ is the Grassmannian of $m$-planes in $R^{m+n}$. By Miller's theorem (aka the Sullivan conjecture), any map from $BG$ to a finite CW complex (such as the Grassmannian) is homotopic to a constant map. Thus $V$ and $W$ are equivalent to trivial bundles. | |
| Oct 18, 2010 at 18:27 | comment | added | Charles Rezk | @Mariano: good point. I've fixed the answer to address it. | |
| Oct 18, 2010 at 18:26 | history | edited | Charles Rezk | CC BY-SA 2.5 |
Last fix, promise
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| Oct 18, 2010 at 18:11 | history | edited | Charles Rezk | CC BY-SA 2.5 |
Fixed claim one more time
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| Oct 18, 2010 at 18:09 | comment | added | Mariano Suárez-Álvarez | You say «so non-trivial bundles over BG which come from representations cannot have inverses»: couldn't the inverse not come from a representation? | |
| Oct 18, 2010 at 18:07 | comment | added | Charles Rezk | Well, I keep removing errors each time too. The claim that "bundles over BG" are the same as G-representations is too strong; it's not true, as was already known to Atiyah in 1961. | |
| Oct 18, 2010 at 18:00 | history | edited | Charles Rezk | CC BY-SA 2.5 |
Softened claim about bundles over BG
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| Oct 18, 2010 at 16:57 | comment | added | Georges Elencwajg | Wow! Not only is your answer great but it improves from minute to minute! It will take me a little time to digest, since I didn't know this method of using representation theory, but I have a fantastic teacher now...Thanks again, Charles. | |
| Oct 18, 2010 at 16:23 | comment | added | Charles Rezk | I've added to my answer, giving a counterexample to the Pontryagin class version of the statement. | |
| Oct 18, 2010 at 16:22 | history | edited | Charles Rezk | CC BY-SA 2.5 |
Added pontryagin example, and fixed errors
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| Oct 18, 2010 at 14:18 | history | answered | Charles Rezk | CC BY-SA 2.5 |