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Unformatted text preview: Computational Topology (Jeff Erickson) Surfaces There is nothing below the surface, my faithful friendabsolutely nothing . Letter 52, The Mahatma Letters to A. P. Sinnett (1882) 5 Surfaces For the next several lectures, we will move from the Euclidean plane to a wider class of topological spaces that locally resemble the plane. A topological space is called a 2manifold if for every point x , there is an open subset U such that x U and U is homeomorphic to R 2 . More succinctly, a 2manifold is a space that is locally homeomorphic to the plane. 2manifolds are also called surfaces . In this lecture, Ill describe a classical combinatorial description of 2manifolds that is useful both for abstract mathematical arguments and as the basis of concrete data structure. The same combinatorial structure can be described in two equivalent ways: Polygonal schema: A collection of polygons with edges glued together in pairs. Cellular graph embedding: An embedding of a graph G on a surface , such that every face of the embedding is a topological disk; 5.1 Polygonal Schemata A polygonal schema is a finite collection of polygons with oriented sides identified in pairs. More explicitly, let f 1 , f 2 ,..., f n denote a finite set of polygons in the plane, called faces , whose total number of sides is even. (I will always refer to the vertices of these polygons as corners , and their edges as segments .) Formally, an orientation of an edge e is a linear map from the unit interval [ 0,1 ] to e ; however, since there are only two such maps, we can think of an orientation of a side e as a permutation of its endpoints, or a labeling of its endpoints as head ( e ) and tail ( e ) . A polygonal schema assigns each side an orientation and a label from some finite set, such that each label is assigned exactly twice. (The standard choice of labels in illustrations are lowercase letters.) a b c d e a b c e d A polygonal schema with signature ( a bc adb )( cde e ) ; arrows indicate edge orientations. Each polygonal schema defines a topological space () by identifying pairs of sides with matching labels and orientations. The labels indicate which sides should be identified; the orientations indicate how the sides are to be identified. Each pair of identified sides becomes a single path in () , which we call an edge . The identification of sides in induces an identification of corners. Thus, several corners may be mapped to the same point in () , which we call a vertex . These vertices and edges define a graph G () embedded in () . We can encode any polygonal schema by listing the labels and orientations of sides in cyclic order around each polygon. Whenever we traverse a side along its orientation, we record its label; when we traverse an edge against its orientation, we record its label with a bar over it. Thus, in the example schema on the previous page, traversing each polygon counterclockwise, starting from its bottom left...
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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.
 Fall '08
 Staff

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