Isotropic stable motivic homotopy groups of spheres

TL;DR

This paper develops an isotropic stable motivic homotopy theory framework, constructing an Adams spectral sequence to compute isotropic stable homotopy groups of spheres, revealing deep algebraic structures akin to classical topology.

Contribution

It introduces the isotropic stable motivic homotopy category, constructs an isotropic Adams spectral sequence, and establishes an isomorphism with classical Steenrod algebra cohomology, including higher structures.

Findings

Isotropic stable homotopy groups of spheres are isomorphic to Ext-groups of the topological Steenrod algebra.

The isotropic Adams spectral sequence converges and computes these groups.

The isomorphism respects ring and higher product structures.

Abstract

In this paper we explore the isotropic stable motivic homotopy category constructed from the usual stable motivic homotopy category, following the work of Vishik on isotropic motives (see [29]), by killing anisotropic varieties. In particular, we focus on cohomology operations in the isotropic realm and study the structure of the isotropic Steenrod algebra. Then, we construct an isotropic version of the motivic Adams spectral sequence and apply it to find a complete description of the isotropic stable homotopy groups of the sphere spectrum, which happen to be isomorphic to the $Ext$-groups of the topological Steenrod algebra. At the end, we will see that this isomorphism is not only additive but respects higher products, completely identifying, as rings with higher structure, the cohomology of the classical Steenrod algebra with isotropic stable homotopy groups of spheres.