Even Stiefel-Whitney invariants for anti-hermitian quaternionic forms

Abstract

We extend all cohomological invariants of similarity classes of quadratic forms to anti-hermitian forms over a quaternion algebra. This uses the fact that such invariants can be lifted to Witt invariants, which can be described as combinations of $\lambda$-operations, and those $\lambda$-operations have recently been extended to hermitian forms over algebras with involution. In the article we present a detailed combinatoric description of invariants of quadratic forms which are partially diagonalized, and show how this combinatorics extend to anti-hermitian quaternionic forms. The methods developped here are intended to be later used for general algebras with involution.