|Location:||MSRI: Baker Board Room|
The Goodwillie tower of the identity, when specialized to odd dimensional spheres, has many wonderful properties. In particular, localized at a prime p, one gets a spectral sequence converging to the homotopy groups of the 2n+1 sphere which start from the stable homotopy groups of certain spaces L(k,n). When n=0, it has been long conjectured that the spectral sequence collapses at E^2.
This amounts to saying that certain non-infinite loop maps from
QL(k,0) to QL(k+1,0) assemble to give a long exact sequence in homotopy.
Meanwhile, infinite loop maps in the other direction appear in the statement of a conjecture of G. Whitehead from the late 1960's.
By calculating everything on primitives in mod p homology, I am able to show that these two sets of maps fit together in the best way possible.
This proves the conjecture about the Goodwillie tower at all primes (Behrens has a version when p=2), and simplifies my 1982 proof of the Whitehead Conjecture.
The Hecke algebras of type A may make an appearance. Then again, they may not.No Notes/Supplements Uploaded No Video Files Uploaded