Bordism Groups of Immersions and Classes Represented by Self-Intersections

We prove a geometrical version of Herbert’s theorem by considering the self-intersection immersions of a self-transverse immersion up to bordism. This generalises Herbert’s theorem to additional cohomology theories and gives a commutative diagram in the homotopy of Thom complexes. The proof uses Koschorke and Sanderson’s operations and the fact that bordism of immersions gives a functor on the category of smooth manifolds and immersions.