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.
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- V. I. Arnold, Plane curves, their invariants, perestroikas and classifications, Adv. Sov. Math. 21 (1994) 33–91Google Scholar
- V. I. Arnold, Invariants and perestroikas of wave fronts on the plane, in: Singularities of smooth mappings with additional structures, Proc. V. A. Steklov Inst. Math. 209 (1995) 14–64 (in Russian)Google Scholar
- V. I. Arnold, Topological invariants of plane curves and caustis, University Lecture Series 5 (1994) AMS, Providence, RIGoogle Scholar
- V. I. Arnold, Exact lagrangian curves on a sphere —1: indices of points and of pairs of points with respect to hypersurfaces, Letter 4–1994, 8 April 1994Google Scholar
- V. I. Arnold, V. V. Goryunov, O. V. Lyashko and V. A. Vassiliev, Singularities I. Local and global theory, Encyclopaedia of Mathematical Sciences 6, Dynamical Systems VI, Springer Verlag, Berlin a.o., 1993Google Scholar
- V. I. Arnold, V. V. Goryunov, O. V. Lyashko and V. A. Vassiliev, Singularities II. Classification and Applications, Encyclopaedia of Mathematical Sciences 39, Dynamical Systems VIII, Springer Verlag, Berlin a.o., 1993Google Scholar
- N. Max and T. Banchoff, Every sphere eversion has a quadruple point, in: Contributions to Analysis and Geometry, The Johns Hopkins University Press, Baltimore and London, 1981, 191–209Google Scholar
- F. Pham, Introduction à l’étude topologique des sinqularités de Landau, Gauthier-Villars, Paris, 1967Google Scholar
- V. A. Vassiliev, Cohomology of knot spaces, Adv. Sov. Math. 1 (1990) 23–69, AMS, Providence, RIGoogle Scholar
- V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, AMS, Providence, RI, 1992Google Scholar
- O. Y. Viro, Some integral calculus based on Euler characteristics, LNM 1346 (1988) 127–138, SpringerGoogle Scholar
- O. Y. Viro, First degree invariants of generic curves on surfaces, Preprint 1994:21, Dept. of Maths., Uppsala UniversityGoogle Scholar