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 ${\mathrm{M}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^h$ as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of ${\mathrm{M}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^h$ as a wedge sum of familiar spectra. We also compute the decomposition of $H^{\ast }({\mathrm{M}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^h;\mathbb{Z}\mathrm{/}2\mathbb{Z})$ explicitly in degrees up through 30 via a counting process.