The Structure of the Spin^h Bordism Spectrum

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.