The structure of the spin^{ℎ} bordism spectrum

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