Real spin bordism and orientations of topological $\mathrm{K}$-theory

Abstract: We construct a commutative orthogonal $C_2$-ring spectrum, $\mathrm{MSpin}^c_{\mathbb{R}}$, along with a $C_2$-$E_{\infty}$-orientation $\mathrm{MSpin}^c_{\mathbb{R}} \to \mathrm{KU}{\mathbb{R}}$ of Atiyah's Real K-theory. Further, we define $E{\infty}$-maps $\mathrm{MSpin} \to (\mathrm{MSpin}^c_{\mathbb{R}})^{C_2}$ and $\mathrm{MU}{\mathbb{R}} \to \mathrm{MSpin}^c{\mathbb{R}}$, which are used to recover the three well-known orientations of topological $\mathrm{K}$-theory, $\mathrm{MSpin}^c \to \mathrm{KU}$, $\mathrm{MSpin} \to \mathrm{KO}$, and $\mathrm{MU}{\mathbb{R}} \to \mathrm{KU}{\mathbb{R}}$, from the map $\mathrm{MSpin}^c_{\mathbb{R}} \to \mathrm{KU}{\mathbb{R}}$. We also show that the integrality of the $\hat{A}$-genus on spin manifolds provides an obstruction for the fixed points $(\mathrm{MSpin}^c{\mathbb{R}})^{C_2}$ to be equivalent to $\mathrm{MSpin}$, using the Mackey functor structure of $\underline{\pi}*\mathrm{MSpin}^c{\mathbb{R}}$. In particular, the usual map $\mathrm{MSpin} \to \mathrm{MSpin}^c$ does not arise as the inclusion of fixed points for any $C_2$-$E_{\infty}$-ring spectrum.