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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
##### Keywords:
rotation number; Whitney formula; index; immersion
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