Proof of a conjecture of Whitney
William S. Massey · Pacific Journal of Mathematics · 1969
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(ι>), 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.This corollary is concerned with manifolds with an orientation preserving involution; an elementary proof of the corollary has recently been given by K. Janϊch and E Ossa,[5].The author is grateful to G. Bredon and W. Browder for helpful discussions of the Atiyah-Singer theorem.The precise statement of the theorem which was conjectured by Whitney is contained in the next section.In order to remove the ambiguity in the sign of the integer m, it is necessary to give a rather thorough discussion of some basic notions regarding questions of orientation, local coefficient systems, etc.Although this material is more or less known, it is nowhere published in a form convenient for our purposes; hence it has been relegated to the appendix of this paper.2* Precise statement of the theorem* We will assume that M is a closed, connected, nonorientable surface which is embedded smoothly