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

TL;DR

This paper constructs a new Real K-theory oriented spectrum related to spin bordism, explores its properties, and reveals obstructions to certain fixed point equivalences using equivariant homotopy theory.

Contribution

It introduces a commutative orthogonal C2-ring spectrum for Real spin^c bordism with orientations linking to classical K-theories, and analyzes fixed point structures and obstructions.

Findings

Constructed a C2-equivariant spin^c bordism spectrum with Real K-theory orientation.

Reconstructed classical K-theory orientations from the new spectrum.

Identified obstructions to fixed point equivalences via the -genus integrality.

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.