surface-homotopy - Computational Topology (Jeff Erickson)...

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Computational Topology (Jeff Erickson) Homotopy of Curves on Surfaces The infinite possibilities that each day holds should stagger the mind. The sheer number of experiences I could have is uncountable, breathtaking, and I’m sitting here refreshing my inbox. We live trapped in loops, reliving a few days over and over, and we envision only a few paths laid out ahead of us. We see the same things each day, respond the same way, we think the same thoughts, every day a slight variation on the last, every moment smoothly following the gentle curves of societal norms. We act like if we just get through today, tomorrow our dreams will come back to us. — Randall Munroe, “Dreams”, http://xkcd.com/137/ 7 Homotopy of Curves on Surfaces In this lecture, we’ll see efficient algorithms to determine whether a given loop on a given 2-manifold Σ is contractible. (We previously considered this problem for polygons with holes in the plane.) To make the problem concrete, we assume that Σ is represented by a polygonal schema Π with complexity n , and is a closed walk of length k in the induced embedded graph G or its dual G * . After developing some necessary mathematical tools, I will describe a classical algorithm of Dehn [ 3 ] for the special case when G is a system of loops, which runs in O ( k ) time. Dehn’s algorithm and its later generalizations are the foundation of geometric group theory . Next I will describe a simple extension of Dehn’s algorithm to more general schemata that runs in O ( gn + g 2 k ) time for surface of genus g . Finally, following a result of Dey and Guha [ 4 ] , I will show how to improve the running time to O ( n + k ) . Let Σ be a fixed, compact, connected 2-manifold. For most of the lecture, for reasons that will become clear, I will assume that χ (Σ) < 0; thus, Σ is not the sphere, the projective plane, the torus, or the Klein bottle. I will briefly reconsider these surfaces at the end of the lecture. 7.1 The Fundamental Group Recall the following definitions from Lecture 2. A path in Σ is a continuous function π : [ 0,1 ] Σ . The concatenation π · σ of two paths π and σ with π ( 1 ) = σ ( 0 ) is the path ( π · π 0 )( t ) : = ( π ( 2 t ) if t 1 / 2, σ ( 2 t - 1 ) if t 1 / 2. The reversal of a path π is the path π ( t ) : = π ( 1 - t ) . We easily observe that π · σ = σ · π . A loop is a path π whose endpoints coincide; this common endpoint is the loop’s basepoint . The concatenation of two loops is a loop, and the reversal of a loop is a loop. A (path) homotopy between paths π and π 0 is a continuous function h : [ 0,1 ] × [ 0,1 ] Σ such that h ( 0, t ) = π ( t ) and h ( 1, t ) = π 0 ( t ) for all t , and h ( s ,0 ) = π ( 0 ) = π 0 ( 0 ) and h ( s ,1 ) = π ( 1 ) = π 0 ( 1 ) for all s [ 0,1 ] . Two paths π and π 0 are homotopic , written π π 0 , if there is a homotopy between them. A loop is contractible if it is homotopic to a constant path at its basepoint. Homotopy is an equivalence relation; we write
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This note was uploaded on 01/24/2012 for the course CS 598 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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surface-homotopy - Computational Topology (Jeff Erickson)...

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