Timeline for Euler characteristic with compact support of spaces of Euclidean lattices
Current License: CC BY-SA 4.0
11 events
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| Jul 23, 2020 at 23:44 | comment | added | sadok kallel | Hi Fernando. Yes it is (up to sign depending on the dimension of the manifold). I guess I meant to say in my response that $\chi$ is not necessarily zero in odd dimensions because the manifold is non-compact. | |
| Jul 22, 2020 at 22:03 | comment | added | Fernando Muro | Hi, Sadok. By Poincaré duality, this would be the same as the ordinary Euler characteristic, computed from homology, right? | |
| Jul 22, 2020 at 18:18 | answer | added | Ian Agol | timeline score: 2 | |
| Jul 22, 2020 at 7:03 | comment | added | sadok kallel | Yes. For $n=2$, this space is homeomorphic to the complement in $S^3$ of a trefoil knot, and so $\chi_c=0$. | |
| Jul 22, 2020 at 6:30 | comment | added | YCor | Do you know the value for small $n$? at least $n=2$? | |
| Jul 21, 2020 at 23:28 | comment | added | Ian Agol | Ah, I see. Somehow I was thinking you wanted the euler characteristic of the compactification (which I think was shown to exist by Borel-Serre). | |
| Jul 21, 2020 at 23:12 | comment | added | sadok kallel | Many thanks Ian. My question is about $\chi_c$ (compact support). So odd dimensional (non-compact) it is non zero. However your answer and reference are useful. My more general interest is actually finding a stratification of this space into a finite number of -interesting-strata, possibly open cells. In case you also have a thought on this, thank you in advance for sharing it. | |
| Jul 21, 2020 at 22:55 | comment | added | Ian Agol | For $n$ even, this will be odd dimensional, and hence have euler characteristic 0. In general, this will be a bundle over the symmetric space $X$ of symmetric positive definite matrices of determinant 1 (unimodular positive definite quadratic forms or metrics on $R^n$), with fiber $SO(n)$. Hence if the base or fiber has odd dimension, then the Euler characteristic will be 0. For the other dimensions, the volume has been computed and should be proportional to the euler characteristic by applying Chern-Gauss Bonnet to the base. arxiv.org/abs/math/0402085 | |
| Jul 21, 2020 at 16:18 | review | First posts | |||
| Jul 21, 2020 at 16:23 | |||||
| Jul 21, 2020 at 16:16 | history | edited | YCor | CC BY-SA 4.0 |
edited tags; edited tags
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| Jul 21, 2020 at 16:14 | history | asked | sadok kallel | CC BY-SA 4.0 |