Thom isomorphisms in triangulated motivic categories

Ananyevskiy, Alexey

doi:10.2140/agt.2021.21.2085

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Thom isomorphisms in triangulated motivic categories

Alexey Ananyevskiy

Algebraic & Geometric Topology 21 (2021) 2085–2106

DOI: 10.2140/agt.2021.21.2085

Abstract

We show that a triangulated motivic category admits categorical Thom isomorphisms for vector bundles with an additional structure if and only if the generalized motivic cohomology theory represented by the tensor unit object admits Thom classes. We also show that the stable $𝔸^{1}$ –derived category does not admit Thom isomorphisms for oriented vector bundles and, more generally, for symplectic bundles. In order to do so we compute the first homology sheaves of the motivic sphere spectrum and show that the class in the coefficient ring of $𝔸^{1}$ –homology corresponding to the second motivic Hopf map $ν$ is nonzero, which provides an obstruction to the existence of a reasonable theory of Thom classes in $𝔸^{1}$ –cohomology.

Keywords

Thom isomorphisms, Thom classes, $\mathbb{A}^1$–cohomology, $\mathbb{A}^1$–derived category, triangulated motivic category

Mathematical Subject Classification

Primary: 14F42

Secondary: 14F45

References

Publication

Received: 5 May 2020

Revised: 16 August 2020

Accepted: 2 September 2020

Published: 18 August 2021

Authors

Alexey Ananyevskiy
St. Petersburg Department of Steklov Mathematical Institute PDMI RAS St. Petersburg Russia

Volume 21, issue 4 (2021)