Huttinga, Zane2024-08-222024-08-222024https://hdl.handle.net/11299/265136University of Minnesota Ph.D. dissertation. 2024. Major: Mathematics. Advisor: Tyler Lawson. 1 computer file (PDF); vii, 234 pages + 2 supplementary files.I present an investigation of the multiplicative structure of H_*(SL_1 MU; F_2) as an algebra over F_2. This includes a list of generators in low degrees and the multiplication table for these generators. This ring has applications in topology related to orientability of vector bundles, as it is closely tied to the homology of the spectrum sl_1 MU. The ring H_*(SL_1 MU) is closely tied to the Hopf ring H_*(MU_2*) introduced by Ravenel and Wilson in 1977. All of the available multiplicative information on H_*(SL_1 MU) descends from H_*(MU_2*), and accordingly, I present a complete algebraic description of this Hopf ring.enAlgebraic topologyHomotopy theoryHopf ringSL_1 MUThom spectrumTopologyThesis or Dissertation