Whitney Formula In Higher Dimensions

. The classical Whitney formula relates the algebraic number of times that a generic immersed plane curve cuts itself to the index ("rotation number") of this curve. Both of these invariants generalize to higher dimension for the immersions of an n-dimensional manifold into an open (n+1)-manifold with the null-homologous image. We give a version of the Whitney formula if n is even. We pay special attention to immersions of S 2 into R 3 . In this case the formula is stated in the same terms which were used by Whitney for immersions of S 1 into R 2 . 1. Introduction Let f : S 1 ! R 2 be a generic immersion (i.e. an immersion without triple points and self-tangencies). The index of f is the degree of the Gauss map (which maps S 1 to the direction of df(v) where v is a tangent vector field positive with respect to the standard orientation of S 1 ). Whitney in [7] showed that the index is the only invariant of f up to deformation in the class of immersions. Fix a generic..