Real spin bordism and orientations of topological K-theory

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