Ivan Panin and Charles Walter: Quaternionic Grassmannians and Pontryagin classes in algebraic geometry

paniniv@gmail.com, Charles.Walter@unice.fr

Submission: 2010, Nov 2

The quaternionic Grassmannian HGr(r,n) is the affine open subscheme of the ordinary Grassmannian parametrizing those 2r-dimensional subspaces of a 2n-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular there is HP^{n} = HGr(1,n+1). For a symplectically oriented cohomology theory A, including oriented theories but also hermitian K-theory, Witt groups and symplectic and special linear algebraic cobordism, we have A(HP^{n}) = A(pt)[p]/(p^{n+1}). We define Pontryagin classes for symplectic bundles. They satisfy a splitting principle and the Cartan sum formula, and we use them to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Pontryagin classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Pontryagin classes.

2000 Mathematics Subject Classification: 14F42, 14F99, 19E20, 19G38, 19G12

Keywords and Phrases: Symplectic bundles, Pontryagin classes, Thom classes, quaternionic grassmannians, symplectically oriented cohomology theories.

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