The Structure of the Spin^h Bordism Spectrum

TL;DR

This paper computes the spin^h bordism groups at the prime 2 by analyzing the cohomology of the spin^h bordism spectrum, providing a detailed structural understanding and explicit decompositions up to degree 30.

Contribution

It offers a structure theorem for the cohomology of the spin^h bordism spectrum as a module over the mod 2 Steenrod algebra and provides a 2-local splitting of the spectrum.

Findings

Computed the spin^h bordism groups at prime 2.

Established a structure theorem for the cohomology as a Steenrod algebra module.

Explicitly decomposed cohomology in degrees up to 30.

Abstract

Spin$^h$ manifolds are the quaternionic analogue to Spin$^c$ manifolds. We compute the spin$^h$ bordism groups at the prime 2 by proving a structure theorem for the cohomology of the spin$^h$ bordism spectrum $\mathrm{MSpin}^h$ as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of $\mathrm{MSpin}^h$ as a wedge sum of familiar spectra. We also compute the decomposition of $H^*(\mathrm{MSpin}^h;\mathbb{Z}/2\mathbb{Z})$ explicitly in degrees up through 30 via a counting process.