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The cohomology of Torelli groups is algebraic
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Abstract
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.
Cambridge University Press (CUP)
Title: The cohomology of Torelli groups is algebraic
Description:
Abstract
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.
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