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.
Dalian notes on rational Pontryagin classes
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abstract
The rational Pontryagin classes, evaluated on fiber bundles where the fiber is a 2n-dimensional euclidean space, can be nonzero in cohomology dimensions much greater than 4n. This makes a striking contrast with the Pontryagin classes of vector bundles.
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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.