2018/05/31 by Frédéric Déglise, Fangzhou Jin, Adeel A. Khan · 55 citations
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Characteristic class #Cohomology #Euler's formula #Homotopy #Homotopy and Cohomology in Algebraic Topology #Intersection (aeronautics) #Intersection theory #Mathematical analysis #Mathematics #Pure mathematics #math.AG #math.KT
paper · pdf · open access · doi:10.4171/jems/1094
published in Journal of the European Mathematical Society 23(12), 3935-3993 (European Mathematical Society) · 45 pages, final version; to appear in JEMS
arxiv created 2021/01/29 · openalex publication_date 2021/05/25 · arxiv updated 2021/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.