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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 2-manifold 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 2-manifold is a space that is locally homeomorphic to the plane. 2-manifolds are also called surfaces . In this lecture, Ill describe a classical combinatorial description of 2-manifolds 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 lower-case 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