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Unformatted text preview: Computational Topology (Jeff Erickson) Surface Classification Q: So you like the grass? A: Yeah. Well, actually I dont care what surface Im playing on. Daniela Hantuchov (May 17, 2001) press conference at the Tennis Masters Series, Rome, Italy 6 Surface Classification In this lecture, Ill give a proof of the most fundamental result in 2-manifold topology: Surface Classification Theorem. Every compact connected 2-manifold can be constructed from the sphere by attaching either a finite number or handles or a finite number of Mbius bands. A complete proof of this theorem relies on Kerkjrt and Rados proof that any compact 2-manifold has a triangulation, but the classification of triangulated 2-manifolds is much older. Different sources attribute the first proof to Brahana [ 1 ] , Dehn and Heegard [ 2 ] , and Dyck [ 3 ] . 1 Brahanas proof is the one that appears in most topology textbooks, thanks to its appearance in an early textbook of Siefert and Threlfall. Several other proofs are known; we refer in particular to a completely self-contained proof by Thomassen [ 10 ] and Conways zero irrelevancy proof [ 5 ] . 6.1 Attaching Handles and Mbius Bands To attach a handle to a surface , find two disjoint closed disks on , delete the interiors of the disks, and glue an annulus to the two boundary circles. The inverse operationdeleting an annulus and gluing disks onto each boundary circleis called detaching a handle . Left to right: Attaching a handle. Right to left: Detaching a handle. To attach a Mbius band to a surface , find a single closed disk in , delete its interior, and glue a Mbius band to its boundary circle. The inverse operationdeleting a Mbius band and gluing a disk onto its boundary circleis called detaching a Mbius band . Left to right: Attaching a Mbius band. Right to left: Detaching a Mbius band. Let ( g , h ) denote the surface obtained from the sphere by attaching g handles and h Mbius bands. (You should convince yourself that it doesnt matter where the handles and Mbius bands are attached, or in which order.) For example, ( 0,0 ) is the sphere; ( 1,0 ) is the torus; ( 0,1 ) is the projective plane ; and ( 0,2 ) is the Klein bottle . 1 However, the classification of orientable surfaces was previously stated in various forms by Riemann, Klein, Jordan [ 6 ] , and Mbius [ 9 ] . 1 Computational Topology (Jeff Erickson) Surface Classification 6.2 Edge Surgery My proof of the classification theorem starts with a polygonal schema for the 2-manifold and modifies its graph through a series of edge contractions and edge deletions . Both operations remove edges from the graph, but in different ways....
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