Motivic Euler characteristics and Witt-valued characteristic classes

TL;DR

This paper develops a motivic framework for Euler characteristics and characteristic classes in Witt cohomology, extending classical transfer and splitting principles to compute these classes for vector bundles.

Contribution

It introduces a motivic version of the Becker-Gottlieb transfer and refines splitting principles, enabling the calculation of Witt-valued characteristic classes for vector bundles.

Findings

Established a motivic Becker-Gottlieb transfer.

Refined splitting principles for SL and SL2 bundles.

Developed a calculus for Witt-valued characteristic classes.

Abstract

This paper examines a number of related questions about Euler characteristics and characteristic classes with values in Witt cohomology. We establish a motivic version of the Becker-Gottllieb transfer, generalizing a construction of Hoyois. Ananyevskiy's splitting principle reduces questions about characteristic classes of vector bundles in $\text{SL}$-oriented, $\eta$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\text{SL}_2$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.