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.
Highlights
We develop the theory of fundamental classes in the setting of motivic homotopy theory
The theory of l-adic sheaves developed in SGA4 [SGA4] was the first complete incarnation of the six functors formalism, and for a long time the only one available in algebraic geometry
A key aspect of the six functor formalism that was highlighted in the seminar SGA5 [SGA5] is the absolute purity property
Summary
It follows from [Ayo07, 1.7.3] that the family of orientations ηf for f smooth (Definition 2.3.5) forms a system of fundamental classes for the class of smooth s-morphisms This system is stable under (arbitrary) base change: explicit homotopies ∆∗(ηf ) ≃ ηg as in Definition 2.3.6(iv) are provided by the deformation to the normal cone space, as in the proof of [Deg18a, Lem. 2.3.13] (where the right-hand square (3) can be ignored). In view of the construction of the Gysin map, the claim follows directly from the facts that the morphism ηp : ThX (p∗E) → p!(SX ) is invertible, and that the functor p∗ : SH (X) → SH (E) is fully faithful (Paragraph 2.1.3). The Thom isomorphism satisfies the properties of compatibility with base change and with direct sums (that is, φE⊕F/X ≃ φE⊕F/F ◦φF/X for vector bundles E and F over X) These follow respectively from the compatibility of the Gysin morphisms p! Under the assumptions of Proposition 2.5.4(ii), the following diagrams commute: f!Σef f ∗v! ∼ f!Σef u!g∗
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