Timeline for Why is BG infinite dimensional for G finite ?
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7 events
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| Sep 6, 2013 at 1:39 | comment | added | Omar Antolín-Camarena | Oh, thanks, @IanAgol, I hadn't noticed Andreas Thom's comment. | |
| Sep 5, 2013 at 22:49 | comment | added | Ian Agol | Omar, yes, this result only applies when the complex is the base is a finite CW-complex, as pointed out by Andreas Thom. | |
| Sep 5, 2013 at 22:12 | comment | added | Omar Antolín-Camarena | What result for multiplicativity of the Euler characteristic are you using? I know of two neither of which seems to apply here. Both require homology of the base, fibre and total space to be finite dimensional (which is the case here); one additionally assumes the base is a finite CW-complex, the other result assumes the action of the fundamental group of the base on the homology of the fibre is trivial. Both of those results can be found in Serre's Homologie Singulière Des Espaces Fibrés. | |
| Jan 10, 2012 at 6:42 | comment | added | Andreas Thom | Nice, but the first argument only shows that $BG$ cannot be finite. The question was, if it can be finite-dimensional. I think one cannot prove that it cannot be finite-dimensional without actually computing the cohomology of cyclic groups. | |
| Nov 3, 2011 at 3:57 | comment | added | Elizabeth S. Q. Goodman | This simple idea is even nicer to me if we change the question: that is, it gives a proof that "a finite-dimensional contractible space can't have a free action of G". In other words, a fixed-point theorem. | |
| Nov 1, 2011 at 19:20 | comment | added | JSE | This is also the core of Akita's lovely argument that the Torelli group has infinite-dimensional cohomology groups. | |
| Nov 1, 2011 at 19:10 | history | answered | Ian Agol | CC BY-SA 3.0 |