Portal de Periódicos da CAPES

Planar surface immersions

1979; Duke University Press; Volume: 23; Issue: 4 Linguagem: Inglês

10.1215/ijm/1256047938

1945-6581

Louis H. Kauffman,

Adhesion, Friction, and Surface Interactions

In this paper immersions of surfaces with boundary into the plane, R2, will be classified up to an equivalence relation called image homotopy.When two immersions are image homotopic, there is a smooth deformation through immersion images of one image to the other.This deformation may be drawn or visualized.It gives the appearance of a motion of the immersion in time.Before proceeding further, the reader might enjoy viewing the long image homotopy shown in Figure 3.This image homotopy is an example of mod-2 planar phenomena that we shall deal with in greater detail.Planar surface immersions are a mixture of integral and mod-2 phenomena.For example, there are infinitely many image homotopy classes of immersions of a once punctured torus, but only two image homotopy classes for a surface of genus greater than one having a single boundary component.In the latter case, these two immersions are distinguished by a mod-2 quadratic form just as in [KB].In fact, our results are quite similar to those of [KB], where immersions into the sphere, S2, are classified up to image homotopy.Except for the use of quadratic forms, we do not assume familiarity with [KB].The paper is organized as follows:In Section 1 image homotopy is discussed and defined.Proposition 1.6 shows that ,9(N)-(N)/dI(N) where #(N) denotes image homotopy classes of immersions of N, denotes regular homotopy, and (N) is the mapping class group of N (acting on (N) by composition).Section 2 discusses the role of curves on the surface.The Whitney-Graustein Theorem [W] is recalled and used to compute (N).A boundary invariant, B(f), of an immersion f:N-l 2 is defined in terms of the boundary curves of N. Proposition 2.3 computes (N) for a k-holed disk in terms of the boundary invariant.Section 3 discusses the generators of the mapping class group (N) and then considers three important examples: (1) If N= T, a punctured torus, then L(N)SL(2,Z) and (N)=Z+.(2) If N= T#A (a torus with two holes), then the extra boundary component acts as a catalyst to reduce the toral part of the immersion modulo two. (3)If N T # T, a once-punctured double torus, then (N) contains no more than two elements.These examples reflect the way particular sorts of diffeomorphisms of N act on