Hopf Rings and the Ando-Hopkins-Strickland Theorem

In this dissertation, we give a new proof of the main results of Ando, Hopkins, and Strickland regarding the generalized homology of the even connective covers of BU. In particular, we prove that the so-called 'symmetry' and 'cocycle' relations hold in the E-homology of BU<2k> for any complex-orientable E and that these relations are the defining relations whenever E=HQ or k=1, 2, or 3. This new proof avoids the algebro-geometric perspective of Ando, Hopkins, and Strickland and instead uses the work of Ravenel, Wilson, and Yagita on the unstable homology of the truncated Brown-Peterson spectra, as well as the relationship between these spectra and BU<2k>. This approach allows for a somewhat simpler proof of the classic Ando-Hopkins-Strickland Theorem, clarifies its relation to the E-homology Hopf ring of ku, and shows how it fits into a broader algebraic picture.