On vanishing of all fourfold products of the Ray classes in symplectic cobordism

TL;DR

This paper revisits the proof that all fourfold products of Ray classes vanish in symplectic cobordism, using transfer computations related to projective bundles of Spin vector bundles.

Contribution

It provides a revised proof of the vanishing of fourfold Ray class products in symplectic cobordism through transfer calculations.

Findings

All fourfold products of Ray classes are zero in symplectic cobordism.

Transfer computations are used to establish the main result.

The proof revises previous arguments with new computational techniques.

Abstract

This note provides certain computations with transfer associated with projective bundles of Spin vector bundles. One aspect is to revise the proof of the main result of \cite{B} which says that all fourfold products of the Ray classes are zero in symplectic cobordism.