The Bousfield-Kuhn functor and Topological Andre-Quillen cohomology

TL;DR

This paper establishes a natural equivalence between the Bousfield-Kuhn functor and Topological Andre-Quillen cohomology for spheres, enabling new computations of unstable v_n-periodic homotopy groups of spheres via algebraic geometry of Lubin-Tate formal groups.

Contribution

It constructs a natural transformation linking the Bousfield-Kuhn functor to Topological Andre-Quillen cohomology and demonstrates its equivalence for spheres, providing a novel computational approach.

Findings

Natural transformation from Bousfield-Kuhn functor to TAQ cohomology.

Equivalence of the transformation for spheres.

Method for computing unstable v_n-periodic homotopy groups.

Abstract

We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological Andre-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable v_n-periodic homotopy groups of spheres from their Morava E-cohomology (as modules over the Dyer-Lashof algebra of Morava E-theory). We relate the resulting algebraic computations to the algebraic geometry of isogenies between Lubin-Tate formal groups.