The bordism class of a quasi-symplectic manifold

Abstract

It is not known which unoriented bordism classes contain Spmanifolds (the stable tangent bundle admitting a reduction to the symplectic group). As an approximation, a class of smooth manifolds is introduced, containing all Sp-manifolds and quaternionic projective spaces HP(n), and its image in the unoriented bordism ring 9Z determined. Let T(M) denote the tangent bundle of a smooth manifold. If t is a right quaternionic vector bundle, the conjugate t* is a left quaternionic vector bundle; thus for right quaternionic vector bundles t and q, the tensor product ?H n* is a real vector bundle. A closed manifold M will be said to be quasi-symplectic if for some k and finite collection of right quaternionic vector bundles (i, vi