A well-known feature of topological K-theory is the presence of Thom isomorphisms for specific types of structured vector bundles and the orientations that result from the corresponding bordism spectra. For example, KU is oriented by spin^c bordism, KO by spin bordism, and Atiyah’s C_2-equivariant “Real” K-theory, KR, is oriented by Real bordism. In this talk, I will introduce a C_2-equivariant E_∞-ring spectrum called Real spin bordism which simultaneously factors the Real and spin orientations of KR. In particular, the Real spin orientation of KR recovers all other orientations of topological K-theory spectra as E_∞-maps. I will also discuss an obstruction for the C_2-fixed points of Real spin bordism to be equivalent to spin bordism coming from the integrality of the A-roof genus on spin manifolds combined with formal properties of homotopy Mackey functors. This is joint work with Zach Halladay.
