Triple points and surgery of immersed surfaces

For a sufficiently general immersion of a smooth or polyhedral 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 involves elementary notions of modification of surfaces by surgery. Let f: M2 2E3 be an immersion in general position of a closed surface M2 into Euclidean 3-space so that there are finitely many N(f) triple points of /. The purpose of this note is to give a direct and elementary proof of the fact that N(f) x(M) (mod 2). The techniques used in this paper are related to the notion of surgery for surfaces, by which a pair of discs is replaced by a cylinder having the same boundary. The approach used in this paper was reported on at the American Mathematical Society Annual Meeting in Las Vegas, January 1972. With the added assumption of differentiability, it is possible to give a new proof of this same result using normal characteristic classes and singularities of projections, as in [l]. For an immersion /: M2 -. E3, set Gr(f) = lx in E3 f| -1(x) consists of precisely r points of M2}. The condition that f is in general position implies that if x e Gr(f) and f1(x) = tp1, P2,..., P,i then there are disjoint disc neighborhoods D(p1), D(p2),.9., D(pr) of these points in M and a homeomorphism w: B D3 of a ball neighborhood B of x to the unit ball in E3 so that (w of) (D(pi)) is the intersection of D3 with the plane orthogonal to the ith coordinate vector. If the immersion f is from a particular category, for example, differentiable or piecewise-linear, then we may assume that the homeomorphism w is in the same category, and in fact all the constructions which will be described can be altered in fairly standard ways so Presented to the Society January 18, 1972; received by the editors September 10, 1973. AMS(MOS) subject classfications(1970). Primary 57C35, 57D40, 57D65; Secondary 55A20.