Fundamental classes in motivic homotopy theory

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 twisted bivariant theory, extending the formalism of Fulton and MacPherson. We import the tools of Fulton’s intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess of intersection, 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 Déglise–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.

A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

“…of Déglise-Jin-Khan [DJK18]. If moreover X is proper then ̟ * (1) also coincides with π * z * z * (1), where z : X → V is the zero section (see Corollary 5.19, Corollary 5.17 and Proposition 5.18).…”

“…While it can be convenient to (not) pass to duals here (as in e.g. [DJK18]), we do not do this, since it confuses the first named author terribly.…”

Morel Homotopy Modules and Milnor-Witt Cycle Modules

“…We prove that the functor Ê is a Milnor-Witt cycle premodule [Fel18,Definition 3.1]. Indeed, most axioms are immediate consequence of the general theory [DJK18]. Moreover, in Theorem 2.16 we prove a ramification theorem of independent interest that can be applied to prove rule (R3a).…”

“…In order to prove this theorem, we study the cohomology theory associated with a motivic spectrum. This notion is naturally dual to the bivariant theory developed in [DJK18] and recalled in Section 2 (see Theorem 2.18). A motivic spectrum E leads to a functor Ê from the category of finitely generated fields over k to the category of graded abelian groups (be careful that the grading is not Z but is given by the category of virtual vector spaces).…”

“…In Section 2, we follow [DJK18] and define the bivariant theory associated to a motivic spectrum. We extend the main results for the associated cohomology theory.…”