A generalization of Banchoff's triple point theorem

Consider an immersion of a surface into $S^{3}$. Banchoff's theorem states that the parity of the number of triple points and the parity of the Euler characteristic of the surface coincide. Here we generalize this theorem to codimension 1 immersions of arbitrary even dimensional manifolds in spheres. The proof is an analogue of a proof of Banchoff's theorem circulated in preprint form due to R. Fenn and P. Taylor in 1977.