Nonorientable surfaces in 4-space
Seiichi Kamada · Osaka City University (Osaka City University) · 1989
A. Kawauchi, T. Shibuya and S. Suzuki proved that any closed connected oriented surface piecewise-linearly and locally-flatly embedded in Euclidean 4space /2 4 can be deformed into a surface with a special configuration called a normal form [5].In this paper we difine normal forms for closed connected non-orientable surfaces in R* and prove that any closed connected non-orientable surface pieceaise-linearly and locally-flatly embedded in /2 4 is deformed into a normal form (Theorem 1.3).It is known that the Euler number of a closed connected non-orientable surface in R* can only take on the following values: 2%-4, 2%, 2%+4, •••, 4-2%, where % is the Euler characteristic of the surface.This was conjectured by H. Whitney in 1940 [10] and proved by W.S. Massey in 1969 [8] using the Atiyah-Singer index theorem.We give, as an application of Theorem 1.3, a geometrical proof to it.We prepare some definitions and state the main theorem (Theorem 1.3) in Section 1 and prove it in Section 2. In Section 3 we study the relationship between the Euler number and the normal form.Section 4 concerns unknotted non-orientable surfaces in Λ 4 .The above mentioned proof of the Whitney and Massey theorem are given in Section 5.Throughout this paper, we work in the piecewise linear category.For the notation, we refer to K-S-S [5].