The Simplicial EHP Sequence in A1-Algebraic Topology

TL;DR

This paper develops a new tool for understanding simplicial desuspension in A1-algebraic topology, establishing fiber sequences and spectral sequences that connect A1-homotopy groups of certain spheres and algebraic groups.

Contribution

It introduces a fiber sequence up to homotopy in 2-localized A1-algebraic topology and derives an EHP spectral sequence for A1-homotopy groups of spheres and algebraic groups.

Findings

Established a fiber sequence up to homotopy for specific A1-spectra.

Derived an EHP spectral sequence relating A1-homotopy groups.

Provided computational tools for A1-algebraic topology.

Abstract

We give a tool for understanding simplicial desuspension in A1-algebraic topology: we show that $X \to \Omega (S^1 \wedge X)\to \Omega (S^1 \wedge X \wedge X)$ is a fiber sequence up to homotopy in 2-localized A1-algebraic topology for $X = (S^1)^m \wedge G_m^{\wedge q}$ with $m>1$. It follows that there is an EHP sequence spectral sequence $$Z_{(2)} \otimes \pi_{n+1+i}^{A1} (S^{2n+2m+1} \wedge (G_m)^{\wedge 2q}) \Rightarrow Z_{(2)} \otimes \pi_{i}^{A1, s} (S^{m} \wedge (G_m)^{\wedge q}).$$