The Whitney-Grauert theorem states that regular curves in R^2 (i.e. immersions of S^1) are classified up to regular homotopy by the winding number of the derivative. I will present Eliashberg and Geiges’s simple proof of this theorem, in which regular plane curves are realized as projections of curves in R^3 satisfying a certain geometric condition (they’ll be Legendrian curves in the standard contact structure). This is one of the simplest examples of the general pattern of lifting a purely topological problem to an equivalent but simpler problem in contact/symplectic geometry. No knowledge of contact geometry is assumed; the only prerequisite is differential forms on R^3.

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Last updated: 28 May 2024