Fundamental classes in motivic homotopy theory

TL;DR

This paper develops a theory of fundamental classes and bivariant theories in motivic homotopy theory, integrating intersection theory tools and extending classical concepts like Euler classes and the Gauss-Bonnet formula.

Contribution

It introduces fundamental classes and bivariant theories in motivic homotopy, extending intersection theory and Euler classes to this setting, including applications to Milnor-Witt spectra.

Findings

Constructed bivariant theories for motivic spectra.

Extended intersection theory tools to motivic homotopy.

Proved a motivic Gauss-Bonnet formula.

Abstract

We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fulton's intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess intersections, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by D\'eglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Barge-Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss-Bonnet formula, computing Euler characteristics in the motivic homotopy category.