Abstract
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.
Highlights
Let denote the 2 -dimensional manifold # × and Diff(, 2 ) denote the topological group of diffeomorphisms of fixing an open neighbourhood of a disc 2 ⊂ pointwise in the ∞topology
′ is a finite index subgroup of and an arithmetic subgroup associated to the algebraic group G ∈ {Sp2, O, }
It is a consequence of Margulis super-rigidity that they are almost algebraic representations of ′ ; that is, there is a finite index subgroup of ′ such that the restriction of the representation to this subgroup extends to a rational representation of the algebraic group G [42, 1.3.(9)]
Summary
Let denote the 2 -dimensional manifold # × and Diff( , 2 ) denote the topological group of diffeomorphisms of fixing an open neighbourhood of a disc 2 ⊂ pointwise in the ∞topology. In [33] the first author proved that the rational cohomology groups of Tor( , 2 ) are finitedimensional in each degree as long as 2 ≥ 6. It is a consequence of Margulis super-rigidity that they are almost algebraic representations of ′ ; that is, there is a finite index subgroup of ′ such that the restriction of the representation to this subgroup extends to a rational representation of the algebraic group G [42, 1.3.(9)]. Theorem A implies that its cohomology with coefficients in any algebraic ′ -representation (as usual, these are representations over the rationals) is independent of the choice of finite index subgroup in a stable range: Corollary B.
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