Spherical classes and the Dickson algebra

Nguyen H. V. Hu'ng and Franklin P. Peterson

We attack the conjecture that the only spherical classes in the homology 
of $Q_0S^0$ are Hopf invariant one and Kervaire invariant one elements. 
We do this by computing products in the $E^2$-term of the unstable Adams 
spectral sequence converging to $\pi_*(Q_0S^0)$
using results about the Dickson algebra and by studying the 
Lannes--Zarati homomorphism.