The motivic lambda algebra and motivic Hopf invariant one problem

Balderrama, William; Culver, Dominic Leon; Quigley, J D

doi:10.2140/gt.2025.29.1489

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The motivic lambda algebra and motivic Hopf invariant one problem

William Balderrama, Dominic Leon Culver and J D Quigley

Geometry & Topology 29 (2025) 1489–1570

DOI: 10.2140/gt.2025.29.1489

Abstract

We investigate forms of the Hopf invariant one problem in motivic homotopy theory over arbitrary base fields of characteristic not equal to $2$ . Maps of Hopf invariant one classically arise from unital products on spheres, and one consequence of our work is a classification of motivic spheres represented by smooth schemes admitting a unital product.

The classical Hopf invariant one problem was resolved by Adams, following his introduction of the Adams spectral sequence. We introduce the motivic lambda algebra as a tool to carry out systematic computations in the motivic Adams spectral sequence. Using this, we compute the $E_{2}$ -page of the $ℝ$ -motivic Adams spectral sequence in filtrations $f \leq 3$ . This universal case gives information over arbitrary base fields.

We then study the $1$ -line of the motivic Adams spectral sequence. We produce differentials $d_{2} (h_{a + 1}) = (h_{0} + ρ h_{1}) h_{a}^{2}$ over arbitrary base fields, which are motivic analogues of Adams’ classical differentials. Unlike the classical case, the story does not end here, as the motivic $1$ -line is significantly richer than the classical $1$ -line. We determine all permanent cycles on the $ℝ$ -motivic $1$ -line, and explicitly compute differentials in the universal cases of the prime fields $𝔽_{q}$ and $ℚ$ , as well as $ℚ_{p}$ and $ℝ$ .

Keywords

motivic, Adams spectral sequence, Hopf invariant, lambda algebra

Mathematical Subject Classification

Primary: 55T15

Secondary: 14F42, 55Q25, 55Q45, 55S10

References

Publication

Received: 19 July 2022

Revised: 21 August 2023

Accepted: 13 October 2023

Published: 31 May 2025

Proposed: Marc Levine

Seconded: Stefan Schwede, Haynes R Miller

Authors

William Balderrama
Department of Mathematics University of Virginia Charlottesville, VA United States

Dominic Leon Culver
Max-Planck-Institut für Mathematik Bonn Germany

J D Quigley
Department of Mathematics Cornell University Ithaca, NY United States	Department of Mathematics University of Virginia Charlottesville, VA United States

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Volume 29, issue 3 (2025)