The Whitehead conjecture, the tower ofS1conjecture, and Hecke algebras of type A

Abstract

In the early 1980s, the author proved G. W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990s, 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.