Timeline for Cohomology map induced by the group actions on homogeneous vector bundles
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2009 at 4:21 | vote | accept | algori | ||
| Nov 19, 2009 at 14:02 | comment | added | Tyler Lawson | Absolutely this is usually only a statement up to filtration/extension problems, but in this case the spectral sequence being acted on has no indeterminacy whatsoever. When K is U(n), the Eilenberg-Moore spectral sequence for the homology of $E_0$ is concentrated in $Ext^0$ and $Ext^1$, with no possible additive extensions, and the product of elements in $Ext^r$ and $Ext^s$ is in $Ext^{r+s}$ up to indeterminacy from higher filtrations - so the Yoneda product really tells the whole story. For more general Lie groups the cohomology of T may be a more complicated module over that of K. | |
| Nov 19, 2009 at 5:57 | comment | added | algori | Thanks, Tyler! There is one thing though that is not clear to me. A spectral sequence gives the (co)homology only up to a filtration. How does one prove that the Yoneda pairing is indeed the homology of $K$ acting on the homology of $E_0$, and not just one Eilenberg-Moore spectral sequence acting on the other one? E. g. the Leray spectral sequences corresponding to the maps to $K/T$ fail to detect the non-trivial part of the (co)action. | |
| Nov 19, 2009 at 0:12 | history | edited | Tyler Lawson | CC BY-SA 2.5 |
more escaping stars, argh.
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| Nov 18, 2009 at 23:45 | history | edited | Tyler Lawson | CC BY-SA 2.5 |
underscores+stars
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| Nov 18, 2009 at 16:06 | history | answered | Tyler Lawson | CC BY-SA 2.5 |