Whitney’s formulas for curves on surfaces

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney’s formula to curves on an oriented punctured surface Σm, n, obtaining a family of identities indexed by elements of π1(Σm, n). To define analogs of the rotation number and the index of a base point of a curve γ, we fix an arbitrary vector field on Σm, n. Similar formulas are obtained for non-based curves.