Computational Methods for Higher Real K-Theory With Applications to tmf

We begin by present a new Hopf algebra which can be used to compute the tmf homology of a space or spectrum at the prime 3. Generalizing work of Mahowald and Davis, we use this Hopf algebra to compute the tmf homology of the classifying space of the symmetric group on three elements. We also discuss the Es Tate spectrum of tmf at the prime 3.

We then build on work of Hopkins and his collaborators, first computing the Adams-Novikov zero line of the homotopy of the spectrum eo, at 5 and then gener-alizing the Hopf algebra for tmf to a family of Hopf algebras, one for each spectrum eop-1 at p. Using these, and using a K(p-1)-local version, we further generalize the Davis-Mahowald result, computing the eo, homology of the cofiber of the transfer map BƩp → S.

We conclude by computing the initial computations needed to understand the homotopy groups of the Hopkins-Miller real K-theory spectra for heights large than p-1 at p. The basic computations are supplemented with conjectures as to the collapse of the spectral sequences used herein to compute the homotopy.