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Whitney formula in higher dimensions. (English) Zbl 0937.53008
Summary: 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 are generalized 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 \(\mathbb{R}^3\). In this case the formula is stated in the same terms which were used by Whitney for immersions of \(S^1\) into \(\mathbb{R}^2\).

MSC:
53A05 Surfaces in Euclidean and related spaces
57R42 Immersions in differential topology
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