Proof of a conjecture of Whitney

Let M be a closed, connected, nonorientable surface of Euler characteristic X which is smoothly embedded in Euclidean 4-space, R 4 , with normal bundle v. The Euler class of i>, denoted by e(v), is an element of the cohomology group H 2 (M; %) (the letter % denotes twisted integer coefficients). Since the group H%M; %) is infinite cyclic, e(>), is m times a generator for some integer m. In a paper presented to a Topology Conference held at the University of Michigan in 1940, H. Whitney studied the possible values that this integer m could take on for different embeddings of the given surface M. He gave examples to show that m can be nonzero (unlike the case for an orientable manifold embedded in Euclidean space) and proved that 1 m = 2X (mod 4) . Finally, he conjectured that m could only take on the following values: 2X -4, 2X, 2X + 4, , 4 -2X . It is the purpose of the present paper to give a proof of this conjecture of Whitney. The proof depends on a corollary of the Atiyah-Singer index theorem.