Local invariants of mappings of surfaces into three-space
Following Arnol’s and Viro’s approach to order 1 invariants of curves on surfaces [1, 2, 3, 20], we study invariants of mappings of oriented surfaces into Euclidean 3-space. We show that, besides the numbers of pinch and triple points, there is exactly one integer invariant of such mappings that depends only on local bifurcations of the image. We express this invariant as an integral similar to the integral in Rokhlin’s complex orientation formula for real algebraic curves. As for Arnold’s J + invariant [1, 2, 3], this invariant also appears in the linking number of two legendrian lifts of the image. We discuss a generalization of this linking number to higher dimensions.
Our study of local invariants provides new restrictions on the numbers of different bifurcations during sphere eversions.
KeywordsBifurcation Diagram Triple Point Double Point Oriented Surface Plane Curf
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