SURFACE TOPOLOGYCS 468 – Lecture 210/2/2Afra Zomorodian – CS 468 Lecture 2 - Page 1OVERVIEW• Last lecture:– Manifolds: locally Euclidean– Homeomorphisms: bijective bi-continuous maps– Topology studies invariant properties– Classification?• This lecture:– Topological Type– Basic 2-Manifolds– Connected Sum– Classification Theorem– Conway’s ZIP proofAfra Zomorodian – CS 468 Lecture 2 - Page 2PARTITIONS• A partition of a set is a decomposition of the set into subsets (cells)such that every element of the set is in one and only one of thesubsets.• Let ∼ be a relation on a nonempty set S so that for all a, b, c ∈ S:1. (Reflexive) a ∼ a.2. (Symmetric) If a ∼ b, then b ∼ a.3. (Transitive) If a ∼ b and b ∼ c, a ∼ c.Then, ∼ is an equivalence relation on S.• Homeomorphism is an equivalence relation.Afra Zomorodian – CS 468 Lecture 2 - Page 3TOPOLOGICAL TYPE• (Theorem) Let S be a nonempty set and let ∼ be an equivalencerelation on S. Then, ∼ yields a natural partition of S, where¯a = {x ∈ S | x ∼ a}. ¯a represents the subset to which a belongs to.Each cell ¯a is an equivalence class.• Homeomorphism partitions manifolds with the sametopological type.• Can we compute this?– n = 1: too easy– n = 2: yes (this lecture)– n = 3: ?– n ≥ 4: undecidable! [Markov 1958]Afra Zomorodian – CS 468 Lecture 2 - Page 4BASIC 2-MANIFOLDS:SPHERE S2vAfra Zomorodian – CS 468 Lecture 2 - Page 5BASIC 2-MANIFOLDS:TORUS T2vvvva abbAfra Zomorodian – CS 468 Lecture 2 - Page 6BASIC 2-MANIFOLDS:M¨OBIUS STRIPwvvwa abbAfra Zomorodian – CS 468 Lecture 2 - Page 7BASIC 2-MANIFOLDS:PROJECTIVE PLANE RP2vvvbba avAfra Zomorodian – CS 468 Lecture 2 - Page 8BASIC 2-MANIFOLDS:MODELS OF RP2(a) Cross cap (b) Boy’s Surface (c) Steiner’s RomanSurfaceAfra Zomorodian – CS 468 Lecture 2 - Page 9BASIC 2-MANIFOLDS:KLEIN BOTTLE K2vvvvbba aAfra Zomorodian – CS 468 Lecture 2 - Page 10BASIC 2-MANIFOLDS:IMMERSION OF K2(a) Klein Bottle (b) M¨obius StripAfra Zomorodian – CS 468 Lecture 2 - Page 11CONNECTED SUM• The connected sum of two n-manifolds M1, M2isM1# M2= M1−˚Dn1[∂˚Dn1=∂˚Dn2M2−˚Dn2,where Dn1, Dn2are n-dimensional closed disks in M1, M2,respectively.=#Afra Zomorodian – CS 468 Lecture 2 - Page 12CLASSIFICATION THEOREM• (Theorem) Every closed compact surface is homeomorphic to asphere, the connected sum of tori, or connected sum of projectiveplanes.• Known since 1860’s• Seifert and Threlfall proof• Conway’s Zero Irrelevancy Proof or ZIP (1992)• Francis and Weeks (1999)Afra Zomorodian – CS 468 Lecture 2 - Page 13CONWAY’S ZIP:CAPAfra Zomorodian – CS 468 Lecture 2 - Page 14CONWAY’S ZIP:CROSSCAPAfra Zomorodian – CS 468 Lecture 2 - Page 15CONWAY’S ZIP:HANDLEAfra Zomorodian – CS 468 Lecture 2 - Page 16CONWAY’S ZIP:CROSS HANDLEAfra Zomorodian – CS 468 Lecture 2 - Page 17CONWAY’S ZIP:PERFORATIONSAfra Zomorodian – CS 468 Lecture 2 - Page 18CONWAY’S ZIP:ORDINARY SURFACES• Every (compact) surface is homeomorphic to a finite collection ofspheres, each with a finite number of handles, crosshandles,crosscaps, and perforations.• That is, all surfaces are ordinary.Afra Zomorodian – CS 468 Lecture 2 - Page 19CONWAY’S ZIP:ZIP-PAIRSAfra Zomorodian – CS 468 Lecture 2 - Page 20CONWAY’S ZIP:LEMMA 1: XHANDLE = 2 XCAPSAfra Zomorodian – CS 468 Lecture 2 - Page 21CONWAY’S ZIP:LEMMA 2: XHANDLE + XCAP = HANDLE + XCAP[Dyck 1888]Afra Zomorodian – CS 468 Lecture 2 - Page 22CONWAY’S ZIP:PROOF• Otherwise, we have handles, crosshandles, and crosscaps• Lemma 1: crosshandle = 2 crosscaps• Lemma 2: crosshandle + crosscap = handle + crosscap• So, handle + crosscap = 3 crosscaps• We get sphere, sphere with handles, or sphere with crosscaps. QEDAfra Zomorodian – CS 468 Lecture 2 - Page 23SPHERE EVERSIONS• Sphere is orientable (two-sided)• So, turn it inside out!• Smale 1957• Morin 1979• “Turning a Sphere Inside Out” [Max 1977]• “Outside In” [Thurston 1994]• “The Optiverse” [Sullivan 1998]Afra Zomorodian – CS 468 Lecture 2 - Page 24TWO EVERSIONSOutside In [Thurston 94]• The Optiverse [Sullivan 98]Afra Zomorodian – CS 468 Lecture 2 - Page
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