Products in spinᶜ-cobordism

We calculate the mod 2 spinᶜ-cobordism ring up to uniform F-isomorphism (i. e. , inseparable isogeny). As a consequence we get the prime ideal spectrum of the mod 2 spinᶜ-cobordism ring. We also calculate the mod 2 spinᶜ-cobordism ring ``on the nose'' in degrees 33. We construct an infinitely generated nonunital subring of the 2-torsion in the spinᶜ-cobordism ring. We use our calculations of product structure in the spin and spinᶜ cobordism rings to give an explicit example, up to cobordism, of a compact 24-dimensional spin manifold which is not cobordant to a sum of squares, which was asked about in a 1965 question of Milnor.