Optional Lecture: A terse introduction to simplicial complexes

Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences University of Iowa http://www.math.uiowa.edu/~idarcy/AppliedTopology.html Optional Lecture: A terse introduction to simplicialcomplexes in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, computer science, cosmology, engineering, imaging, mathematics, neurology, physics, statistics, etc.

Building blocks for oriented simplicial complex 0-simplex = vertex = v e 1-simplex = oriented edge = (v1, v2) 2-simplex = oriented face = (v1, v2, v3) v2 e1 e2 v1 v1 v3 v2 e3

Building blocks for oriented simplicial complex 2-simplex = oriented face = f = (v1, v2, v3) = (v2, v3, v1) = (v3, v1, v2) – f= (v2, v1, v3) = (v3, v2, v1) = (v1, v3, v2) v2 v2 e1 e2 e1 e2 v1 v3 v1 v3 e3 e3

Building blocks for oriented simplicial complex 3-simplex = σ = (v1, v2, v3, v4) = (v2, v3, v1, v4) = (v3, v1, v2, v4) = (v2, v1, v4, v3) = (v3, v2, v4, v1) = (v1, v3, v4, v2) = (v4, v2, v1, v3) = (v4, v3, v2, v1) = (v4, v1, v3, v2) = (v1, v4, v2, v3) = (v2, v4, v3, v1) = (v3, v4, v1, v2) – σ = (v2, v1, v3, v4) = (v3, v2, v1, v4) = (v1, v3, v2, v4) = (v2, v4, v1, v3) = (v3, v4, v2, v1) = (v1, v4,v3, v2) = (v1, v2, v4, v3) = (v2, v3, v4, v1) = (v3, v1, v4, v2) = (v4, v1, v2, v3) = (v4, v2, v3, v1) = (v4, v3, v1, v2) v2 v4 v3 v1

Building blocks for oriented simplicial complex 0-simplex = vertex = v e 1-simplex = oriented edge = (v1, v2) Note that the boundary of this edge is v2 – v1 2-simplex = oriented face = (v1, v2, v3) Note that the boundary of this face is the cycle e1 + e2 + e3 = (v1, v2) + (v2, v3) –(v1, v3) = (v1, v2) –(v1, v3)+ (v2, v3) v2 e1 e2 v1 v1 v3 v2 e3

Building blocks for oriented simplicial complex 3-simplex = (v1, v2, v3, v4) = tetrahedron boundary of (v1, v2, v3, v4) = – (v1, v2, v3) + (v1, v2, v4) – (v1, v3, v4) + (v2, v3, v4) n-simplex = (v1, v2, …, vn+1) v2 v4 v1 v3

Building blocks for an unorientedsimplicial complex usingZ2coefficients e 0-simplex = vertex = v 1-simplex = edge = {v1, v2} Note that the boundary of this edge is v2+ v1 2-simplex = face = {v1, v2, v3} v2 Note that the boundary of this face is the cycle e1 + e2 + e3 = {v1, v2} + {v2, v3} +{v1, v3} e1 e2 v1 v1 v3 v2 e3

Building blocks for a simplicial complex using Z2coefficients 3-simplex = {v1, v2, v3, v4} = tetrahedron boundary of {v1, v2, v3, v4} = {v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4} n-simplex = {v1, v2, …, vn+1} v2 v4 v1 v3

Optional Lecture: A terse introduction to simplicial complexes