Vol. 31, No. 1, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 332: 1  2
Vol. 331: 1  2
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
Proof of a conjecture of Whitney

William Schumacher Massey

Vol. 31 (1969), No. 1, 143–156
Abstract

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
Milestones
Received: 15 May 1969
Published: 1 October 1969
Authors
William Schumacher Massey