Abstract
We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds$\#^{g}S^{n}\times S^{n}$relative to a disc in a stable range, for$2n\geqslant 6$. Our calculation is also valid for$2n=2$assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.
Highlights
In the study of the cohomology of the mapping class group Γg of the genus g surface Σg, an important role is played by its normal subgroup Tg, the Torelli group, consisting of those diffeomorphisms which act trivially on H1(Σg; Z)
This is the kernel of the homomorphism Γg → Sp2g(Z) which sends a diffeomorphism to the induced map on H1(Σg; Z), and so is equipped with an outer action of Sp2g(Z)
In this paper we will study the generalization of this problem to all even dimensions 2n, replacing the surface of genus g by its 2n-dimensional analogue Wg := #g Sn × Sn
Summary
In the study of the cohomology of the mapping class group Γg of the genus g surface Σg, an important role is played by its normal subgroup Tg, the Torelli group, consisting of those diffeomorphisms which act trivially on H1(Σg; Z). (iii) Letting H(g) denote the local coefficient system on BDiff(Wg, D2n) given by the action of diffeomorphisms on Hn(Wg; Q), a key step in the proof of this theorem is to completely describe the bigraded cohomology ring H ∗(BDiff(Wg, D2n); H(g)⊗) in a stable range, together with its behaviour in the variable as a functor on the (signed) Brauer category.
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