Regular Homotopy

A regular homotopy between two immersions is a smooth one-parameter family of immersions for . Equivalently, it is a smooth map whose restriction to every slice is an immersion. Unlike an isotopy, a regular homotopy may pass through self-intersections. In the formulation, the maps are immersions and their spatial derivatives vary continuously with (Smale 1959, p. 281).

For immersed oriented closed curves in the plane, the Whitney-Graustein theorem classifies regular homotopy classes by the contour winding number of the unit tangent vector about the origin. In contrast, any two immersions of the 2-sphere in are regularly homotopic. In particular, the standard embedding is regularly homotopic to its reflection, giving a sphere eversion (Smale 1959, p. 281).