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 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 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.

arXiv comment45 pages, final version; to appear in JEMS

References in corpus (22)

• Triangulated categories of mixed motives
• A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
• Algebraic K-theory, A^1-homotopy and Riemann-Roch theorems
• Enhanced six operations and base change theorem for higher Artin stacks
• The class of Eisenbud--Khimshiashvili--Levine is the local A1-Brouwer degree
• Perfection in motivic homotopy theory
• Finite Chow-Witt correspondences
• On the algebraic cobordism spectra MSL and MSp
• Push-forwards for Witt groups of schemes
• Virtual Cartier divisors and blow-ups
• Orientation theory in arithmetic geometry
• MW-motivic complexes
• Borel-Moore motivic homology and weight structure on mixed motives
• The Milnor-Witt motivic ring spectrum and its associated theories
• Aspects of enumerative geometry with quadratic forms
• Virtual fundamental classes of derived stacks I
• Bivariant theories in motivic stable homotopy
• On the rational motivic homotopy category
• Künneth formulas for motives and additivity of traces
• The intrinsic stable normal cone
• Towards the dual motivic Steenrod algebra in positive characteristic
• Triangulated categories of relative 1-motives

Cited by in corpus (21)

• Perfection in motivic homotopy theory
• Modules over algebraic cobordism
• Aspects of enumerative geometry with quadratic forms
• Virtual fundamental classes of derived stacks I
• On the rational motivic homotopy category
• Bivariant theories in motivic stable homotopy
• Quadratic types and the dynamic Euler number of lines on a quintic threefold
• Framed transfers and motivic fundamental classes
• Künneth formulas for motives and additivity of traces
• An arithmetic count of the lines meeting four lines in P^3
• Algebraic G-theory in motivic homotopy categories
• An Example of Wild Ramification in an Enriched Riemann-Hurwitz Formula
• Borel isomorphism and absolute purity
• Absolute Poincaré duality in étale cohomology
• The quadratic Euler characteristic of a smooth projective same-degree complete intersection
• Combing a hedgehog over a field
• Introduction to Framed Correspondences
• A Gersten complex on real schemes
• Trace maps in motivic homotopy and local terms
• An Introduction to ${\mathbf{A}}^1$-Enumerative Geometry
• An introduction to six-functor formalisms