The Whitehead Conjecture, the Tower of S^1 Conjecture, and Hecke algebras of type A

TL;DR

This paper proves the Whitehead conjecture for all primes, links it to the Goodwillie tower of the identity, and reveals that Hecke algebras of type A govern key homology maps in infinite loopspaces.

Contribution

It provides a comprehensive proof of the Whitehead conjecture and establishes a novel connection between Steenrod algebra maps and Hecke algebras of type A.

Findings

Whitehead conjecture proven for all primes

Identity tower on the circle collapses as expected

Hecke algebra identities determine homology maps

Abstract

In the early 1980's the author proved G.W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990's, Arone and Mahowald showed that the Goodwillie tower of the identity had remarkably good properties when specialized to odd dimensional spheres. In this paper we prove that these results are linked, as has been long suspected. We give a state-of-the-art proof of the Whitehead conjecture valid for all primes, and simultaneously show that the identity tower specialized to the circle collapses in the expected sense. Key to our work is that Steenrod algebra module maps between the primitives in the mod p homology of certain infinite loopspaces are determined by elements in the mod p Hecke algebras of type A. Certain maps between spaces are shown to be chain homotopy contractions by using identities in these Hecke algebras.