#### Volume 23, issue 4 (2019)

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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
##### Abstract

We give a tool for understanding simplicial desuspension in ${\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}$–algebraic topology: we show that $X\to \Omega \left({S}^{1}\wedge X\right)\to \Omega \left({S}^{1}\wedge X\wedge X\right)$ is a fiber sequence up to homotopy in $2$–localized ${\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}$ algebraic topology for $X={\left({S}^{1}\right)}^{m}\wedge {\mathbb{G}}_{m}^{\wedge q}$ with $m>1$. It follows that there is an EHP spectral sequence

${ℤ}_{\left(2\right)}\otimes {\pi }_{n+1+i}^{{\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}}\left({S}^{2n+2m+1}\wedge {\left({\mathbb{G}}_{m}\right)}^{\wedge 2q}\right)⇒{ℤ}_{\left(2\right)}\otimes {\pi }_{i}^{{\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1},s}\left({S}^{m}\wedge {\left({\mathbb{G}}_{m}\right)}^{\wedge q}\right).$

##### Keywords
$A^1$ homotopy, motivic homotopy, simplicial EHP sequence, motivic homotopy of spheres
##### Mathematical Subject Classification 2010
Primary: 14F42
Secondary: 19G12, 55Q40
##### 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