Real spin bordism and orientations of topological K-theory

We construct a commutative orthogonal C₂-ring spectrum, MSpinᶜR, along with a C₂-E∞-orientation MSpinᶜR → KUR of Atiyah's Real K-theory. Further, we define E∞-maps MSpin → (MSpinᶜR)C₂ and MUR → MSpinᶜR, which are used to recover the three well-known orientations of topological K-theory, MSpinᶜ → KU, MSpin → KO, and MUR → KUR, from the map MSpinᶜR → KUR. We also show that the integrality of the Â-genus on spin manifolds provides an obstruction for the fixed points (MSpinᶜR)C₂ to be equivalent to MSpin, using the Mackey functor structure of π_*MSpinᶜR. In particular, the usual map MSpin → MSpinᶜ does not arise as the inclusion of fixed points for any C₂-E∞-ring spectrum.