Vol. 31, No. 1, 1969

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Proof of a conjecture of Whitney

William Schumacher Massey

Vol. 31 (1969), No. 1, 143–156

Let M be a closed, connected, nonorientable surface of Euler characteristic χ which is smoothly embedded in Euclidean 4-space, R4, with normal bundle ν. The Euler class of ν, denoted by e(ν), is an element of the cohomology group H2(M;𝒵) (the letter 𝒵 denotes twisted integer coefficients). Since the group Hz(M;𝒵) is infinite cyclic, e(ν), 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

m ≡ 2χ ( mod 4).

Finally, he conjectured that m could only take on the following values:

2χ − 4,2χ,2χ + 4,⋅⋅⋅ ,4 − 2χ.

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.

Mathematical Subject Classification
Primary: 57.30
Received: 15 May 1969
Published: 1 October 1969
William Schumacher Massey