The homotopy fixed points of Real spin bordism
Hassan H. Abdallah, Yigal Kamel
TL;DR
This work extends the classic ABP/Stong 2-local splittings of spin and spin^c bordism to the $C_2$-equivariant setting, yielding a refined 2-local splitting of Real spin bordism $\mathrm{MSpin}^c_{\mathbb{R}}$ into higher connective covers of $\mathrm{ku}_{\mathbb{R}}$ and shifts of $\mathrm{H}\mathbb{Z}/2$, with the underlying map matching the classical ABP splitting on spectra. It then deduces a corresponding 2-local splitting for the homotopy fixed points $(\mathrm{MSpin}^c_{\mathbb{R}})^{hC_2}$ and provides explicit computations of its homotopy groups in terms of $\mathrm{ko}$-connective pieces and mod 2 data, together with the mod 2 Borel cohomology contributions. A central contribution is showing that each ABP component refines to a $C_2$-equivariant map in $\mathrm{Sp}^{\mathrm{B}C_2}$, enabling the $hC_2$-analysis; the paper also discusses obstructions to a genuine (non-Borel) refinement and proposes new $C_2$-spectra $\mathrm{ku}_{\mathbb{R}}\langle {4n,2} \rangle$ to address odd-index obstructions. The results clarify the relationship between Real spin bordism and equivariant $K$-theory, enabling concrete calculations of $(\mathrm{MSpin}^c_{\mathbb{R}})^{hC_2}$ and guiding future work on genuine splittings in the $C_2$-equivariant 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.