On P^1-stabilization in unstable motivic homotopy theory

TL;DR

This paper investigates P^1-stabilization in unstable motivic homotopy theory, introducing refined cellularity concepts, and proves key theorems leading to applications like Murthy's conjecture and motivic sphere computations.

Contribution

It introduces a refined notion of cellularity in motivic homotopy categories and extends classical theorems under these new conditions, advancing the understanding of P^1-stabilization.

Findings

Refined Whitehead theorem for nilpotent motivic spaces

A version of the Freudenthal suspension theorem for P^1

Resolution of Murthy's conjecture and new motivic sphere computations

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 (a.k.a., biconnectivity) 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.