2 Surface Topology. 2.1 Topological Type. Computational Topology Surface Topology Afra Zomorodian

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1 Klein ottle for rent inquire ithin. Anonymous 2 Surfce Topology Lst lecture, e spent considerle mount of effort defining mnifolds. We like mnifolds ecuse they re loclly Eucliden. So, een though it is hrd for us to reson out them glolly, e kno ht to do in smll neighorhoods. It turns out tht this ility is ll e relly need. This is rther fortunte, ecuse e suddenly he spces ith more interesting structure thn the Eucliden spces to study. Recll tht topology, like Eucliden geometry, is study of the properties of spces tht remin inrint (do not chnge) under fixed set of trnsformtions. In topology, e expnd the trnsformtions tht re lloed from rigid motions (Eucliden geometry) to homeomorphisms: ijectie i-continuous mps. In this lecture, e sk hether e my clssify mnifolds under this set of trnsformtions, nd e see tht such clssifiction is possile for to-dimensionl mnifolds or surfces. 2.1 Topologicl Type To egin ith, e should indicte ht e men y clssifiction. This notion hs nice mthemticl definition, hich you my he seen in high school. Definition 2.1 (prtition) A prtition of set is decomposition of the set into susets (cells) such tht eery element of the set is in one nd only one of the susets. We ish to prtition the set of mnifolds ccording to their connectiity. We re forced to look t different prtitioning schemes in our serch for one hich is computtionlly fesile. Ech scheme depends on n equilence reltion. Definition 2.2 (equilence) Let S e nonempty set nd let e reltion eteen elements of S tht stisfies the folloing properties for ll,, c S: 1. (Reflexie). 2. (Symmetric) If, then. 3. (Trnsitie) If nd c, c. Then, the reltion is n equilence reltion on S. It is cler from the definition of homeomorphism tht it is n equilence reltion. The folloing theorem llos us to derie prtition from n equilence reltion. Theorem 2.1 Let S e nonempty set nd let e n equilence reltion on S. Then, yields nturl prtition of S, here ā = {x S x }. ā represents the suset to hich elongs to. Ech cell ā is n equilence clss. As homeomorphism is n equilence reltion, e my use it to prtition mnifolds y this theorem. If to mnifolds re plced in the sme suset, they re connected the sme y, nd e sy tht they he the sme topologicl type. One of the fundmentl questions in topology is hether this prtition is computle. In this lecture, e focus on the solution to this prolem in to dimensions. 1

2 2.2 Bsic 2-Mnifolds Before clssifying 2-mnifolds, hoeer, it ould e nice to meet fe of them. In this section, e look t fe sic 2-mnifolds. () {x R 3 x = 1} () Identify oundry to Figure 1. The sphere S 2 (c) Instructions for flt sphere The sphere. Topologiclly, the sphere S 2 is the simplest surfce. We re most comfortle ith the implicit surfce definition in Figure 1(), tht defines the unit sphere s suspce of R 3. The sphere my e defined, hoeer, using digrm in Figure 1(), hich sks us to mke the entire oundry of disc to single point. This process is clled identifiction: this mens tht ll the points in the oundry should e treted s if they ere the sme point. The identifiction here gies us topologicl sphere. We my lso mke sphere out of pper s shon in Figure 1(c). Pper hs no curture, so it hs flt geometry, nd e get flt sphere. The strct sphere defined y the digrm (), long ith the flt sphere, highlight the difference eteen the sphere s topologicl concept, nd sphere s geometric entity. It is importnt for you to consider the difference crefully. We only cre out connectiity in topology, nd ht is connected like the geometric sphere is sphere, no mtter its geometry. () Donut surfce () Digrm (c) Instructions for flt torus Figure 2. The torus T 2 The torus. The torus is fmilir to us s the surfce of gel or donut, s shon in Figure 2(). We my descrie torus s suspce of R 3 geometriclly. For exmple, torus of reolution is creted hen e seep circle round the z-xis: T (u, ) = ((1 + cos u) cos, (1 + cos u) sin, sin(u)). The torus my lso e descried i digrm in Figure 2(), in hich the edges re glued ccording to their direction of their rros. Finlly, e cn uild flt torus using the directions in Figure 2(c) The Möius strip. Figure 3() shos n emedded Möius strip: 2-mnifold ith oundry. It is esy to construct one y gluing one end of strip of pper to the other end ith single tist, s shon in the digrm in Figure 3(). This mnifold is not orientle. The notes for lst lecture included definition of orientility for smooth mnifolds in n ppendix. We ill see nother forml definition of orientility in the next lecture. For no, orientility mens tht the surfce hs to sides. In Figure 3(c), M. C. Escher estlishes tht the Möius strip is one-sided y mrching nts on the strip Note tht the oundry of the Möius strip is single cycle. This cycle corresponds to the to unglued edges in the digrm 3() hich e my no glue ith or ithout tist. The projectie plne. If e put non-mtching rros on the remining to edges of the Möius digrm s in Figure 4(), e get the projectie plne RP 2. This ction corresponds to gluing the oundry of disk to the oundry 2

3 () Emedded () Digrm (c) Escher s Möius Strip II Figure 3. The Möius strip is non-orientle mnifold ith oundry. () Digrm () Instructions for flt RP 2 Figure 4. The projectie plne RP 2 of the Möius strip. This mnifold hs this nme ecuse of its ssocition ith projectie geometry used in rt nd computer grphics for representing ht e see on flt cns. For exmple, e kno tht rily lines neer intersect, s they re prllel. When e look t them in rel life, hoeer, e see tht they come together t the horizon, or t infinity. They lso intersect t horizon ehind us. We ould like ny to lines to intersect t most once, so e identify the to intersecting points s the sme point. Imgine the oundry of the digrm in 4() is the horizon. The rros on the digrm identify point nd its reflected imge round the origin. These points re clled nti-podl points. This mnifold is non-orientle s it contins Möius strip. It cnnot e emedded in R 3, so e he to e content ith immersions. Figure 5 shos three immersions of the projectie plne, ll of hich self-intersect. These immersions re fmous s they contin interesting geometry in ddition to their shred topology. To mke n pper model, e he to cut the pper to llo for the self-intersection. The Klein ottle. If e glue the free edges of the Möius strip in the sme direction, e get the Klein ottle K 2, s shon in Figure 6(). The Klein ottle is therefore equilent to gluing to Möius strips to ech other long their oundry. Like the projectie plne, it is closed non-orientle surfce. It is not emeddle in R 3, nd its immersions in Figures 6() nd 6(c) self-intersect ith the intersecting tringle colored in red. Once gin, e need to cut pper in order to mke flt model, s shon in FIgure 6(d). () Cross cp () Boy s Surfce (c) Steiner s Romn Surfce Figure 5. Models of the projectie plne RP 2 3

4 () Digrm () An immersion (c) Cut in hlf ( Möius strip) (d) Instructions for flt K 2 Figure 6. The Klein ottle K Connected Sum We my use the surfces e just defined to form lrger mnifolds. To do this, e form connected sums. Definition 2.3 (connected sum) The connected sum of to n-mnifolds M 1, M 2 is M 1 # M 2 = M 1 D 1 n M 2 D 2 n, D n 1 = D n 2 here D n 1, D n 2 re n-dimensionl closed disks in M 1, M 2, respectiely. In other ords, e cut out to disks nd glue the mnifolds together long the oundry of those disks using homeomorphism. In Figure 7, for exmple, e connect to tori to form sum ith to hndles. # = Figure 7. The connected sum of to tori is genus 2 torus. 2.4 The Clssifiction Theorem We re no le to stte result tht gies complete clssifiction of compct 2-mnifolds. Theorem 2.2 (clssifiction of compct 2-mnifolds) Eery closed compct surfce is homeomorphic to sphere, the connected sum of tori, or connected sum of projectie plnes. We ill see in the next lecture tht this clssifiction is esily computle. In the reminder of this lecture, e ill look t Cony s ZIP proof [2] of this theorem. The pper is proided on the esite s the notes for the rest of the lecture. The theorem nsers the homeomorphism question for compct mnifolds in to dimensions. After lerning out groups, e ill see tht this question is undecidle for dimensions four nd higher. This prolem is still open in three dimensions, nd three-mnifold topology is n ctie re of reserch. For ery ccessile oerie, see Weeks [3]. Acknoledgments The instruction for mking flt 2-mnifolds re from Firy nd Grdiner [1]. I rendered the models of projectie plne in Figure 5 in POV-Ry using descriptions y Tore Nordstrnd. Figures 6() nd 6(c) re from [4]. 4

5 References [1] FIRBY, P. A., AND GARDINER, C. F. Surfce Topology. Ellis Horood Limited, Chichester, Englnd, [2] FRANCIS, G. K., AND WEEKS, J. R. Cony s ZIP proof. Americn Mth. Monthly 106 (My 1999). [3] WEEKS, J. R. The Shpe of Spce. Mrcel Dekker, Inc., Ne York, NY, [4] ZOMORODIAN, A., AND EDELSBRUNNER, H. Fst Softre for Box Intersection. In Proc. 16th Ann. ACM Sympos. Comput. Geom. (2000), pp