One-Sided Surfaces and Orientability

Abstract

John W. Woll, Jr., is a Professor and Graduate Advisor at Western Washington State College in Bellingham, Washington. He has published several articles in physics and mathematics journals and is the author of a textbook on functions of several variables, published in 1964. In this article we discuss the concepts of orientability and sidedness and show that they are not the same. In fact we will show that a 2-torus imbedded in the right 3-manifold may have only one side, whereas objects like the Klein bottle and the Mobius band, which are generally considered to be one sided, may have two sides. We will be able to demonstrate, by using the classification scheme for 2-manifolds, that the 2-sphere is the only 2-manifold which does not admit both one-sided and two-sided imbeddings in a 3-manifold. The corresponding statement is not true in higher dimensions, and we reduce the problem of the number of sides an n-manifold may have when imbedded in an (n+ 1)-manifold to a question in group theory. The article was designed to (1) display on an elementary level the possibilities of cross-fertilization between different mathematical disciplines, in this case topology and group theory; (2) serve as an eye opener to the rich structure of manifolds; and (3) introduce selected topics from algebraic topology. As a consequence all concepts and arguments used have a high intuitive appeal, and there is almost an overabundance of examples, especially of 3-manifolds. It is pitched at an introductory survey level and, except for a few points, no outside information is used. The pace is leisurely until the end where it is stepped up a trifle.