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For a transversal smooth immersion of a closed 2-dimensional surface into Euclidean 3-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach of this paper includes an introduction to normal Euler classes of smoothly immersed manifolds by means of singularities of projections. In this note we prove that for a (transversal) smooth immersion of a closed 2-dimensional surface into Euclidean 3-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach used here includes an introduction to the normal Euler class of a smoothly immersed manifold and is related to the theory of Stiefel-Whitney classes in terms of singularities of projections as developed in [2]. The main result of this paper is correct also for manifolds which are not smooth, and such a proof has been carried out using surgery techniques in [3]. 1. Singularities of projections and the Euler characteristic. Let f: M2 -_ R3 be a smooth immersion of a closed 2-dimensional surface into real 3-dimensional space with a coordinate system (x, y, z). Let rrz: R3 + R(z) denote the orthogonal projection into the z-axis and let SP) be the set of critical points of ruz 0 f: M2 -_ R(z), i.e. the set of points where the tangent plane to f(M) is orthogonal to the z-axis. Let rxz : R3 -p R(x, z) denote the orthogonal projection into the xz-plane, and let SXZQ) be the set of singularities of 0xz ? f: M2 -_ R(x, z), i.e. the set of points where the tangent plane to f(M) is orthogonal to the xz-plane. We assume that the coordinate system in 3-space is so chosen that the set SP) consists of a finite number of points, consisting of mo(rz 0 f) local minima, m1(7z ? f) nondegenerate saddle points, and m 2(Z Tf) local maxima. Received by the editors November 15, 1973. AMS (MOS) subject classifications (1970). Primary 57D45, 57D20, 57D35.

Triple points and singularities of projections of smoothly immersed surfaces