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Thom isomorphisms in triangulated motivic categories

Alexey Ananyevskiy

Algebraic & Geometric Topology 21 (2021) 2085–2106

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 𝔸1–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 𝔸1–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 𝔸1–cohomology.

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Thom isomorphisms, Thom classes, $\mathbb{A}^1$–cohomology, $\mathbb{A}^1$–derived category, triangulated motivic category
Mathematical Subject Classification
Primary: 14F42
Secondary: 14F45
Received: 5 May 2020
Revised: 16 August 2020
Accepted: 2 September 2020
Published: 18 August 2021
Alexey Ananyevskiy
St. Petersburg Department of Steklov Mathematical Institute
St. Petersburg