Real spin bordism and orientations of topological K-theory

We construct a commutative orthogonal C 2 C_2 -ring spectrum, M S p i n R c \mathrm {MSpin}^c_{\mathbb {R}} , along with a C 2 C_2 - E ∞ E_{\infty } -orientation M S p i n R c → K U R \mathrm {MSpin}^c_{\mathbb {R}} \to \mathrm {KU}_{\mathbb {R}} of Atiyah’s Real K-theory. Further, we define E ∞ E_{\infty } -maps M S p i n → ( M S p i n R c ) C 2 \mathrm {MSpin} \to (\mathrm {MSpin}^c_{\mathbb {R}})^{C_2} and M U R → M S p i n R c \mathrm {MU}_{\mathbb {R}} \to \mathrm {MSpin}^c_{\mathbb {R}} , which are used to recover the three well-known orientations of topological K \mathrm {K} -theory, M S p i n c → K U \mathrm {MSpin}^c \to \mathrm {KU} , M S p i n → K O \mathrm {MSpin} \to \mathrm {KO} , and M U R → K U R \mathrm {MU}_{\mathbb {R}} \to \mathrm {KU}_{\mathbb {R}} , from the map M S p i n R c → K U R \mathrm {MSpin}^c_{\mathbb {R}} \to \mathrm {KU}_{\mathbb {R}} . We also show that the integrality of the A ^ \hat {A} -genus on spin manifolds provides an obstruction for the fixed points ( M S p i n R c ) C 2 (\mathrm {MSpin}^c_{\mathbb {R}})^{C_2} to be equivalent to M S p i n \mathrm {MSpin} , using the Mackey functor structure of π _ ∗ M S p i n R c \underline {\pi }_*\mathrm {MSpin}^c_{\mathbb {R}} . In particular, the usual map M S p i n → M S p i n c \mathrm {MSpin} \to \mathrm {MSpin}^c does not arise as the inclusion of fixed points for any C 2 C_2 - E ∞ E_{\infty } -ring spectrum.