The cohomology of Torelli groups is algebraic
Forum of Mathematics, Sigma, Tome 8 (2020)
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The Torelli group of $W_g = \#^g S^n \times S^n$ is the group of diffeomorphisms of $W_g$ fixing a disc that act trivially on $H_n(W_g;\mathbb{Z} )$. The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $\text{Sp}_{2g}(\mathbb{Z} )$ or $\text{O}_{g,g}(\mathbb{Z} )$. In this article we prove that for $2n \geq 6$ and $g \geq 2$, they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent.
@article{10_1017_fms_2020_41,
author = {Alexander Kupers and Oscar Randal-Williams},
title = {The cohomology of {Torelli} groups is algebraic},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {8},
year = {2020},
doi = {10.1017/fms.2020.41},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2020.41/}
}
Alexander Kupers; Oscar Randal-Williams. The cohomology of Torelli groups is algebraic. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.41
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