The simplicial EHP sequence in 𝔸1–algebraic topology

Wickelgren, Kirsten; Williams, Ben

doi:10.2140/gt.2019.23.1691

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The simplicial EHP sequence in $\mathbb{A}^{1}$–algebraic topology

Kirsten Wickelgren and Ben Williams

Geometry & Topology 23 (2019) 1691–1777

DOI: 10.2140/gt.2019.23.1691

Abstract

We give a tool for understanding simplicial desuspension in $A^{1}$ –algebraic topology: we show that $X \to Ω (S^{1} \land X) \to Ω (S^{1} \land X \land X)$ is a fiber sequence up to homotopy in $2$ –localized $A^{1}$ algebraic topology for $X = {(S^{1})}^{m} \land G_{m}^{\land q}$ with $m > 1$ . It follows that there is an EHP spectral sequence

ℤ_{(2)} \otimes π_{n + 1 + i}^{A^{1}} (S^{2 n + 2 m + 1} \land {(G_{m})}^{\land 2 q}) \Rightarrow ℤ_{(2)} \otimes π_{i}^{A^{1}, s} (S^{m} \land {(G_{m})}^{\land q}) .

Keywords

$A^1$ homotopy, motivic homotopy, simplicial EHP sequence, motivic homotopy of spheres

Mathematical Subject Classification 2010

Primary: 14F42

Secondary: 19G12, 55Q40

References

Publication

Received: 14 December 2015

Revised: 5 June 2018

Accepted: 11 November 2018

Published: 17 June 2019

Proposed: Jesper Grodal

Seconded: Haynes R Miller, Mark Behrens

Authors

Kirsten Wickelgren
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

Ben Williams
Department of Mathematics University of British Columbia Vancouver, BC Canada

Volume 23, issue 4 (2019)