Geometric representation of cohomology classes for the Lie groups Spin(7) and Spin(8)

By constructing concrete complex-oriented maps we show that the eight-fold of the generator of the third integral cohomology of the spin groups Spin(7) and Spin(8) is in the image of the Thom morphism from complex cobordism to singular cohomology, while the generator itself is not in the image. We thereby give a geometric construction for a nontrivial class in the kernel of the differential Thom morphism of Hopkins and Singer for the Lie groups Spin(7) and Spin(8). The construction exploits the special symmetries of the octonions.