Intersection theory and cobordism with a quadratic twist
Final Report Abstract
In this project we obtain new results in algebraic geometry via methods stemming from A1-homotopy theory. Using this theory, classical invariants with integral values can be refined into quadratic forms-valued ones, which carry new arithmetic information. The results obtained concern three (related) topics: (a) Cohomology theories for algebraic varieties involving quadratic information: we constructed operations in Hermitian K-theory, a theory describing vector bundles equipped with a quadratic form. We also studied the notion of orientation for theories where the motivic Hopf map eta is inverted. (b) Isotropic motivic categories: those lie between the motivic (algebrogeometric) and classical (topological) theories. (c) Nisnevich classifying spaces of algebraic groups: their study permits to better understand invariants of the associated algebraic structures. This leads to generalisations of Smirnov-Vishik “subtle Stiefel-Whitney classes” of quadratic forms to the context of other linear algebraic groups.
Publications
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Isotropic stable motivic homotopy groups of spheres. Advances in Mathematics, 383, 107696.
Tanania, Fabio
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Cellular objects in isotropic motivic categories. Geometry & Topology, 27(5), 2013-2048.
Tanania, Fabio
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ODD RANK VECTOR BUNDLES IN ETA-PERIODIC MOTIVIC HOMOTOPY THEORY. Journal of the Institute of Mathematics of Jussieu, 1-32.
Haution, Olivier
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A Serre-type spectral sequence for motivic cohomology. Algebraic Geometry, 386-420.
Tanania, Fabio
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Motivic cohomology of the Nisnevich classifying space of even Clifford groups. Documenta Mathematica, 29(1), 191-208.
Tanania, Fabio
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The stable Adams operations on Hermitian K-theory. Geometry & Topology, 29(1), 127-169.
Fasel, Jean & Haution, Olivier
