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surface-classification - Computational Topology(Jeff...

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Computational Topology (Jeff Erickson) Surface Classification Q: So you like the grass? A: Yeah. Well, actually I don’t care what surface I’m playing on. — Daniela Hantuchová (May 17, 2001) press conference at the Tennis Masters Series, Rome, Italy 6 Surface Classification In this lecture, I’ll 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 Möbius bands. A complete proof of this theorem relies on Kerékjártó and Rado’s 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 Brahana’s 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 Conway’s ‘zero irrelevancy’ proof [ 5 ] . 6.1 Attaching Handles and Möbius Bands To attach a handle to a surface Σ , find two disjoint closed disks on Σ , delete the interiors of the disks, and glue an annulus to the two boundary circles. The inverse operation—deleting an annulus and gluing disks onto each boundary circle—is called detaching a handle . Left to right: Attaching a handle. Right to left: Detaching a handle. To attach a Möbius band to a surface Σ , find a single closed disk in Σ , delete its interior, and glue a Möbius band to its boundary circle. The inverse operation—deleting a Möbius band and gluing a disk onto its boundary circle—is called detaching a Möbius band . Left to right: Attaching a Möbius band. Right to left: Detaching a Möbius band. Let Σ( g , h ) denote the surface obtained from the sphere by attaching g handles and h Möbius bands. (You should convince yourself that it doesn’t matter where the handles and Möbius bands are attached, or in which order.) For example, Σ( 0 , 0 ) is the sphere; Σ( 1 , 0 ) is the torus; Σ( 0 , 1 ) is the projective plane ; and Σ( 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 Möbius [ 9 ] . 1
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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. Let e be an edge in the graph G . If e separates two distinct faces f and f 0 , we can delete e to obtain a new graph G \ e (pronounced ‘ G without e ’). The two faces f and f 0 are merged into a single face in G \ e . If the embedding of G is represented by a rotation system, we simply delete both occurrences of e .
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