Motivic Euler characteristics and 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.

arXiv commentThis is a final version, the paper is published online by the Nagoya Math. Journal

References in corpus (4)

• The six operations in equivariant motivic homotopy theory
• A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
• The class of Eisenbud--Khimshiashvili--Levine is the local A1-Brouwer degree
• The special linear version of the projective bundle theorem

Cited by in corpus (10)

• An Arithmetic Count of the Lines on a Smooth Cubic Surface
• An arithmetic enrichment of Bézout's Theorem
• Künneth formulas for motives and additivity of traces
• Applications to A1-enumerative geometry of the A1-degree
• An arithmetic count of the lines meeting four lines in P^3
• Computing A1-Euler numbers with Macaulay2
• Ulrich sheaves, the arithmetic writhe and algebraic isotopies of space curves
• Motivic Pontryagin classes and hyperbolic orientations
• Odd rank vector bundles in eta-periodic motivic homotopy theory
• Invariants of upper motives