For 2n ≥ 6 and g ≥ 2, the rational cohomology of the Torelli group of W_g = #^g S^n × S^n is an algebraic representation of Sp_{2g} or O_{g,g}, and its classifying space is nilpotent.
Mapping class groups of highly connected $(4k+2)$-manifolds
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abstract
We compute the mapping class group of the manifolds $\sharp^g(S^{2k+1}\times S^{2k+1})$ for $k>0$ in terms of the automorphism group of the middle homology and the group of homotopy $(4k+3)$-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.
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The cohomology of Torelli groups is algebraic
For 2n ≥ 6 and g ≥ 2, the rational cohomology of the Torelli group of W_g = #^g S^n × S^n is an algebraic representation of Sp_{2g} or O_{g,g}, and its classifying space is nilpotent.