Boy's surface [4] is a self-transverse immersion of the real projective plane into Euclidean 3-space with a single triple point. J. F. Hughes has observed [10] that for each positive integer n there is a self-transverse immersion of a closed 2w-manifold in Euclidean 3«-space with an odd number of triple points. An example is obtained by taking the cartesian product of n copies of Boy's immersion and making it selftransverse using a regular homotopy. These examples are all non-orientable. In this paper we consider when orientable examples can be found.
THEOREM. There is a self-transverse immersion M 2n q'^U Zn of some closed orientable 2n-manifold M with an odd number of triple points if and only ifn is even and n > 4.This is a simple consequence of the more technical Theorem 1.2 stated below. For n = 1 it is well known (see, for example, [2]) that a self-transverse immersion of a closed orientable surface in U 3 has an even number of triple points. Hughes observes that the same is true for orientable 4-manifolds in U 6 .Our results are obtained using the approach of [6] which is based on ideas of U. Koschorke and B. Sanderson [12] (also developed independently by P. Vogel [18] and J . Lannes [13], and by A . Sziics [17]). The method is homotopy theoretic; explicit manifolds and immersions are not constructed.In order to state the main result it is necessary to review some of these ideas and to explain how the number of triple points of an immersion can be viewed as a bordism invariant. Further details may be found in [5,6,7].Given a self-transverse immersion /: M 2 "^ <*+ |R 3n of a closed oriented 2w-manifold, compactness ensures that the number of triple points is finite. If n is even a sign may be attached to each triple point by comparing the standard orientation of IR 3n with that provided by the orientation of the three normal planes at that point. Since the normal planes are unordered this is impossible if n is odd. We write 0 3 (i) for the number of triple points of the immersion, taking the number modulo 2 if n is odd and counting the points in a signed way if n is even.Recall that the Pontrjagin-Thom construction gives an isomorphism from the bordism group of closed oriented 2«-manifolds immersed in IR 3n to the stable homotopy group n$ n (MSO(n)) where MS0(n) is the Thorn complex of a universal
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