Timeline for Why torsion is important in (co)homology ?
Current License: CC BY-SA 2.5
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| May 5, 2010 at 16:34 | comment | added | Dan Ramras | The fact that both the trivial and non-trivial bundle over S admit a flat connection takes more work. Melissa Liu and Nan-Kuo Ho have some papers about this. | |
| May 5, 2010 at 16:33 | comment | added | Dan Ramras |
These statements are true for bundles of any dimension. The fact about bundles over surfaces can be seen in several ways; here's one: Note that the classifying map $S\to BU(n)$ can be assumed to land in the 2-skeleton of $BU(n)$, which is just $S^2$, regardless of n (I'm thinking of the standard CW structure on the Grassmannian). So the classifying map really lands in $CP^\infty$, and hence bundles over S all have the form $L\oplus \epsilon^k$, where L is a line and `$\epsilon^k$ is trivial.
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| May 5, 2010 at 9:23 | comment | added | Qfwfq | You mean line bundles? | |
| May 5, 2010 at 5:12 | history | answered | Dan Ramras | CC BY-SA 2.5 |