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

Abstract

We construct a commutative orthogonal $C_2$-ring spectrum, ${\mathrm{MSpin}}_{\mathbb{R}}^c$, along with a $C_2$-$E_{\infty }$-orientation ${\mathrm{MSpin}}_{\mathbb{R}}^c\to {\mathrm{KU}}_{\mathbb{R}}$ of Atiyah's Real K-theory. Further, we define $E_{\infty }$-maps $\mathrm{MSpin}\to ({\mathrm{MSpin}}_{\mathbb{R}}^c{)}^{C_2}$ and ${\mathrm{MU}}_{\mathbb{R}}\to {\mathrm{MSpin}}_{\mathbb{R}}^c$, 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 ${\mathrm{MSpin}}_{\mathbb{R}}^c\to {\mathrm{KU}}_{\mathbb{R}}$. We also show that the integrality of the $\hat{A}$-genus on spin manifolds provides an obstruction for the fixed points $({\mathrm{MSpin}}_{\mathbb{R}}^c{)}^{C_2}$ to be equivalent to $\mathrm{MSpin}$, using the Mackey functor structure of ${\underline{\pi }}_{\ast }{\mathrm{MSpin}}_{\mathbb{R}}^c$. 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.

v2: 27 pages, a few minor edits; accepted version, to appear in Transactions of the American Mathematical Society