The homotopy fixed points of Real spin bordism

TL;DR

This paper refines the 2-local splitting of Real spin bordism into a $C_2$-equivariant map, leading to a splitting of the homotopy fixed points and opening avenues for further exploration in the genuine setting.

Contribution

It introduces a $C_2$-equivariant refinement of the 2-local splitting of Real spin bordism, connecting it to higher connective covers and Eilenberg--Mac Lane spectra.

Findings

A $C_2$-equivariant map from Real spin bordism to a sum of connective covers of $ ext{ku}_ ext{R}$.

A 2-local splitting of the homotopy fixed points of Real spin bordism.

Discussion of prospects in the genuine setting.

Abstract

We show that the 2-local splitting of spin$^c$ bordism by Anderson--Brown--Peterson and Stong refines to a $C_2$-equivariant map in the category of spectra with $C_2$-action from Real spin bordism to a sum of (higher) connective covers of $\mathrm{ku}_{\mathbb{R}}$ and suspensions of mod 2 Eilenberg--Mac Lane spectra. We use this to deduce a corresponding 2-local splitting of the homotopy fixed points of Real spin bordism. We also discuss prospects that arise in the genuine setting.