On P^1-stabilization in unstable motivic homotopy theory

Author:

Aravind Asok, Tom Bachmann, Michael J. Hopkins

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Algebraic Topology (math.AT), K-Theory and Homology (math.KT)

journal:

date:

2023-06-06 16:00:00

Abstract

We analyze stabilization with respect to ${\mathbb P}^1$ in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity in various motivic homotopy categories taking into account both the simplicial and Tate circles. Under suitable cellularity hypotheses, we refine the Whitehead theorem by showing that a map of nilpotent motivic spaces can be seen to be an equivalence if it so after taking (Voevodsky) motives. We then establish a version of the Freudenthal suspension theorem for ${\mathbb P}^1$-suspension, again under suitable cellularity hypotheses. As applications, we resolve Murthy's conjecture on splitting of corank $1$ vector bundles on smooth affine algebras over algebraically closed fields having characteristic $0$ and compute new unstable motivic homotopy of motivic spheres.

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