  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.
