A note on the double quaternionic transfer and its f-invariant

TL;DR

This paper explores the double transfer in quaternionic geometry, deriving a formula for the f-invariant and applying it to quaternionic flag manifolds, advancing understanding in stable homotopy theory.

Contribution

It introduces a simple formula for the f-invariant in the quaternionic double transfer case and applies it to quaternionic flag manifolds.

Findings

Derived a formula for the f-invariant of quaternionic double transfer

Analyzed the f-invariant's role in the Adams-Novikov spectral sequence

Applied results to quaternionic flag manifolds

Abstract

It is well-known that for a line bundle over a closed framed manifold, its sphere bundle can also be given the structure of a framed manifold, usually referred to as a transfer. Given a pair of lines, the procedure can be generalized to obtain a double transfer. We study the quaternionic case, and derive a simple formula for the f-invariant of the underlying bordism class, enabling us to investigate its status in the Adams-Novikov spectral sequence. As an application, we treat the situation of quaternionic flag manifolds.