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Unformatted text preview: Computational Topology (Jeff Erickson) Cell Complexes: Definitions One who does not realize his own value is condemned to utter failure. (Every kind of complex, superiority or inferiority, is harmful to man). — ‘Al¯ ı ibn Ab¯ ı T . ¯alib, Nahj al-Balagha [Peak of Eloquence] There once lived a man named Oedipus Rex. You may have heard about his odd complex. His name appears in Freud’s index ’Cause he loved his mother. — Tom Lehrer, “Oedipus Rex”, More of Tom Lehrer (1959) 15 Cell Complexes: Definitions With a few exceptions, we have focused exclusively on problems involving 2-manifolds, represented by polygonal schemata. Before we (finally!) consider problems involving more general spaces, we need to step back and consider how these more general spaces might be represented, as part of the input to an algorithm. This is a surprisingly subtle question, even if we restrict our attention to higher-dimensional manifolds . A large class of topological spaces of practical interest can be represented by a decomposition into subsets, each with simple topology, glued together ‘nicely’ along their boundaries. A decomposition of this form is commonly called a cell complex . Abstract graphs (as branched 1-manifolds) are examples of cell complexes, as are polygonal schemata, quadrilateral and hexahedral meshes, and 3d triangulations supporting normal surfaces. There are several different ways to formalize the intuitive notion of a ‘cell complex’, striking different balances between simplicity and generality. I’ll describe several different definitions in this lecture. The figure below shows the ‘cell complex’ of minimum complexity (number of cells) homeomorphic to the torus, for four standard definitions. The simplest CW complex, Δ-complex, regular Δ-complex, and simplicial complex homeomorphic to the torus. 15.1 Abstract Simplicial Complexes An abstract simplicial complex is a collection X of finite sets that is closed under taking subsets; that is, for any set S ∈ X and any subset T ⊆ S , we have T ∈ X . In particular, the intersection of any two sets in X is also a set in X , and the empty set ∅ is an element of every simplicial complex. The sets in X are called simplices . The vertices of X are the singleton sets in X (or their elements); the set of vertices of X is denoted X . The dimension of a simplex is one less than its cardinality; in particular, every vertex is a 0-dimensional simplex, and the empty set has dimension- 1. The dimension of a simplicial complex is the largest dimension of any simplex. The subsets of a simplex are called its faces . A simplex in X is maximal if it is not a proper face of any other simplex in X ; the maximal simplices in X are also called the facets of X . An abstract simplicial complex X is pure if every facet has the same dimension....
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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