George K. Francis and Jeffrey R. Weeks- Conway’s ZIP Proof

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  Conway’s ZIP Proof George K. Francis Jeffrey R. Weeks December 28, 2000 Surfaces arise naturally in many different forms, in branches of mathematics ranging from complex analysis to dynamical systems. The Classification Theorem, known since the 1860’s, asserts that all closed surfaces, despite their diverse origins and seemingly diverse forms, are topologically equivalent to spheres with some number of handles or crosscaps (to be defined below). The proofs found in most modern textbooks follow that
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  Conway’s ZIP Proof  George K. Francis Jeffrey R. WeeksDecember 28, 2000 Surfaces arise naturally in many different forms, in branches of mathematicsranging from complex analysis to dynamical systems. The Classification Theo-rem, known since the 1860’s, asserts that all closed surfaces, despite their diversesrcins and seemingly diverse forms, are topologically equivalent to spheres withsome number of handles or crosscaps (to be defined below). The proofs foundin most modern textbooks follow that of Seifert and Threlfall [5]. Seifert andThrelfall’s proof, while satisfyingly constructive, requires that a given surfacebe brought into a somewhat artificial standard form. Here we present a com-pletely new proof, discovered by John H. Conway in about 1992, which retainsthe constructive nature of [5] while eliminating the irrelevancies of the standardform. Conway calls it his Zero Irrelevancy Proof, or “ZIP proof”, and asks thatit always be called by this name, remarking that “otherwise there’s a real dangerthat its srcin would be lost, since everyone who hears it immediately regardsit as the obvious proof”. We trust that Conway’s ingenious proof will replacethe customary textbook repetition of Seifert-Threlfall in favor of a lighter, fat-free nouvelle cuisine approach that retains all the classical flavor of elementarytopology.We work in the realm of topology, where surfaces may be freely stretched anddeformed. For example, a sphere and an ellipsoid are topologically equivalent,because one may be smoothly deformed into the other. But a sphere and adoughnut surface are topologically different, because no such deformation ispossible. All the figures in the present article depict deformations of surfaces.For example, the square with two holes in Figure 1A is topologically equivalentto the square with two tubes (1B), because one may be deformed to the other.More generally, two surfaces are considered equivalent, or homeomorphic , if andonly if one may be mapped onto the other in a continuous, one-to-one fashion.That is, it’s the final equivalence that counts, whether or not it was obtainedvia a deformation.1  Figure 1. HandleFigure 2. Crosshandle Let us introduce the primitive topological features in terms of zippers or“zip-pairs”, a zip being half a zipper. Figure 1A shows a surface with two bound-ary circles, each with a zip. Zip the zips, and the surface acquires a handle (1D).If we reverse the direction of one of the zips (2A), then one of the tubes must“pass through itself” (2B) to get the zip orientations to match. Figure 2B showsthe self-intersecting tube with a vertical slice temporarily removed, so the readermay see its structure more easily. Zipping the zips (2C) yields a crosshandle (2D). This picture of a crosshandle contains a line of self-intersection. The self-intersection is an interesting feature of the surface’s placement in 3-dimensionalspace, but has no effect on the intrinsic topology of the surface itself. Figure 3. CapFigure 4. Crosscap If the zips occupy two halves of a single boundary circle (Figure 3A), andtheir orientations are consistent, then we get a cap (3C), which is topologicallytrivial (3D) and won’t be considered further. If the zip orientations are in-consistent (4A), the result is more interesting. We deform the surface so that2  corresponding points on the two zips lie opposite one another (4B), and beginzipping. At first the zipper head moves uneventfully upward (4C), but uponreaching the top it starts downward, zipping together the “other two sheets”and creating a line of self-intersection. As before, the self-intersection is merelyan artifact of the model, and has no effect on the intrinsic topology of the sur-face. The result is a crosscap (4D), shown here with a cut-away view to makethe self-intersections clearer.The preceding constructions should make the concept of a surface clear tonon-specialists. Specialists may note that our surfaces are compact, and mayhave boundary. Comment. A surface is not  assumed to be connected. Comment. Figure 5 shows an example of a triangulated surface. All surfacesmay be triangulated, but the proof [4] is difficult. Instead we may consider theClassification Theorem to be a statement about surfaces that have already beentriangulated. Definition. A perforation  is what’s left when you remove an open disk from asurface. For example, Figure 1A shows a portion of a surface with two perfora-tions. Definition. A surface is ordinary  if it is homeomorphic to a finite collectionof spheres, each with a finite number of handles, crosshandles, crosscaps, andperforations. Classification Theorem (preliminary version) Every surface is ordinary.Proof: Begin with an arbitrary triangulated surface. Imagine it as a patchworkquilt, only instead of imagining traditional square patches of material held to-gether with stitching, imagine triangular patches held together with zip-pairs(Figure 5). Unzip all the zip-pairs, and the surface falls into a collection of triangles with zips along their edges. This collection of triangles is an ordinarysurface, because each triangle is homeomorphic to a sphere with a single perfo-ration. Now re-zip one zip to its srcinal mate. It’s not hard to show that theresulting surface must again be ordinary, but for clarity we postpone the detailsto Lemma 1. Continue re-zipping the zips to their srcinal mates, one pair ata time, noting that at each step Lemma 1 ensures that the surface remains or-dinary. When the last zip-pair is zipped, the srcinal surface is restored, and isseen to be ordinary. 2 3  Figure 5. Install a zip-pair alongeach edge of the triangulation,unzip them all, and then re-zipthem one at a time.Figure 6. These zips only par-tially occupy the boundary cir-cles, so zipping them yields ahandle with a puncture. Lemma 1 Consider a surface with two zips attached to portions of its bound-ary. If the surface is ordinary before the zips are zipped together, it is ordinary afterwards as well.Proof: First consider the case that each of the two zips completely occupies aboundary circle. If the two boundary circles lie on the same connected com-ponent of the surface, then the surface may be deformed so that the boundarycircles are adjacent to one another, and zipping them together converts theminto either a handle (Figure 1) or a crosshandle (Figure 2), according to theirrelative orientation. If the two boundary circles lie on different connected com-ponents, then zipping them together joins the two components into one.Next consider the case that the two zips share a single boundary circle, whichthey occupy completely. Zipping them together creates either a cap (Figure 3)or a crosscap (Figure 4), according to their relative orientation.Finally, consider the various cases in which the zips needn’t completely oc-cupy their boundary circle(s), but may leave gaps. For example, zipping togetherthe zips in Figure 6A converts two perforations into a handle with a perfora-tion on top (6B). The perforation may then be slid free of the handle (6C,6D).Returning to the general case of two zips that needn’t completely occupy theirboundary circle(s), imagine that those two zips retain their normal size, whileall other zips shrink to a size so small that we can’t see them with our eyeglassesoff. This reduces us (with our eyeglasses still off!) to the case of two zips that do completely occupy their boundary circle(s), so we zip them and obtain ahandle, crosshandle, cap, or crosscap, as illustrated in Figures 1–4. When weput our eyeglasses back on, we notice that the surface has small perforations aswell, which we now restore to their srcinal size. 2 The following two lemmas express the relationships among handles, crosshan-dles, and crosscaps.4
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