The Hopf Rings for KO and KU

Abstract In this work we compute the Hopf rings for the spectra representing orthogonal and unitary K -theory, KO and KU . Specifically, we use Hopf ring properties to find and simplify the ordinary mod 2 homology groups for these spectra. Our main tool is the bar spectral sequence, which allows us to advance from one space to the next in the spectrum.