Fundamental classes in motivic homotopy theory

By Frédéric Déglise, F. Jin, Adeel A. Khan. Journal of the European Mathematical Society (Print), 2018

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f67cc9d9610858f87313f5cf42c3c32cf6eb26f

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Citations in the corpus

EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS (2020, methodology): “Here we have used the lci pushforward $∗ : KO 0(Z) ρ ' KOL$ (Z)→ KO0(k) of Déglise–Jin–Khan [DJK18].”
Motivic infinite loop spaces (2017, background): “(1), as is the proof of (5) once we know that for E;E02SH(S), the presheaf ˇ 0 1 T Hom(E;E0) on Sm S extends to Corrfr(Sm S). This is indeed the case by the theory of fundamental classes developed in [DJK18]. Remark B.2.4. If S is the spectrum of a eld, one can also prove Proposition B.2.3(5) using ”
Perfection in motivic homotopy theory (2018, background): “First, we recall that the theory of fundamental classes developed in [12] applies in this more general setting (see 4.”