A generalization of symplectic Pontrjagin classes to vector bundles with structure Sp(n)\(\cdot Sp(1)\) (Q1064581)

The authors define a set of polynomial generators of \(H^*(B(Sp(n)\cdot Sp(1));{\mathbb{R}})\) and show that these generators have integral preimages, which they call ''generalized symplectic Pontryagin classes''. Relations are found between these and ordinary Pontryagin classes. All this is applied to the determination of the Pontryagin classes (both ''generalized symplectic'' and ''ordinary'') of the quaternionic projective space. The authors' definitions and calculations are based on the Chern-Weil isomorphism. \{Reviewer's remark: It is possible to define the same classes and prove the same relations in a more straightforward (and rather evident) way, just considering the correlation between the cohomology of B(Sp(n)\(\cdot Sp(1))\) and that of BSp(n)\(\times BSp(1)\); so the Chern-Weil construction is in a sense beside the point.\}