Optional Lecture: A terse introduction to simplicial complexes 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. Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences University of Iowa

v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 2-simplex = oriented face = (v 1, v 2, v 3 ) 1-simplex = oriented edge = (v 1, v 2 ) e v1v1 v2v2 0-simplex = vertex = v Building blocks for oriented simplicial complex

2-simplex = oriented face = f = (v 1, v 2, v 3 ) = (v 2, v 3, v 1 ) = (v 3, v 1, v 2 ) – f = (v 2, v 1, v 3 ) = (v 3, v 2, v 1 ) = (v 1, v 3, v 2 ) v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 Building blocks for oriented simplicial complex

3-simplex = σ = (v 1, v 2, v 3, v 4 ) = (v 2, v 3, v 1, v 4 ) = (v 3, v 1, v 2, v 4 ) = (v 2, v 1, v 4, v 3 ) = (v 3, v 2, v 4, v 1 ) = (v 1, v 3, v 4, v 2 ) = (v 4, v 2, v 1, v 3 ) = (v 4, v 3, v 2, v 1 ) = (v 4, v 1, v 3, v 2 ) = (v 1, v 4, v 2, v 3 ) = (v 2, v 4, v 3, v 1 ) = (v 3, v 4, v 1, v 2 ) – σ = (v 2, v 1, v 3, v 4 ) = (v 3, v 2, v 1, v 4 ) = (v 1, v 3, v 2, v 4 ) = (v 2, v 4, v 1, v 3 ) = (v 3, v 4, v 2, v 1 ) = (v 1, v 4,v 3, v 2 ) = (v 1, v 2, v 4, v 3 ) = (v 2, v 3, v 4, v 1 ) = (v 3, v 1, v 4, v 2 ) = (v 4, v 1, v 2, v 3 ) = (v 4, v 2, v 3, v 1 ) = (v 4, v 3, v 1, v 2 ) Building blocks for oriented simplicial complex v4v4 v3v3 v1v1 v2v2

v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 2-simplex = oriented face = (v 1, v 2, v 3 ) 1-simplex = oriented edge = (v 1, v 2 ) Note that the boundary of this edge is v 2 – v 1 e v1v1 v2v2 0-simplex = vertex = v Note that the boundary of this face is the cycle e 1 + e 2 + e 3 = (v 1, v 2 ) + (v 2, v 3 ) – (v 1, v 3 ) = (v 1, v 2 ) – (v 1, v 3 ) + (v 2, v 3 ) Building blocks for oriented simplicial complex

3-simplex = (v 1, v 2, v 3, v 4 ) = tetrahedron boundary of (v 1, v 2, v 3, v 4 ) = – (v 1, v 2, v 3 ) + (v 1, v 2, v 4 ) – (v 1, v 3, v 4 ) + (v 2, v 3, v 4 ) n-simplex = (v 1, v 2, …, v n+1 ) v4v4 v3v3 v1v1 v2v2 Building blocks for oriented simplicial complex

v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 2-simplex = face = {v 1, v 2, v 3 } Note that the boundary of this face is the cycle e 1 + e 2 + e 3 = {v 1, v 2 } + {v 2, v 3 } + {v 1, v 3 } 1-simplex = edge = {v 1, v 2 } Note that the boundary of this edge is v 2 + v 1 e v1v1 v2v2 0-simplex = vertex = v Building blocks for an unoriented simplicial complex using Z 2 coefficients

3-simplex = {v 1, v 2, v 3, v 4 } = tetrahedron boundary of {v 1, v 2, v 3, v 4 } = {v 1, v 2, v 3 } + {v 1, v 2, v 4 } + {v 1, v 3, v 4 } + {v 2, v 3, v 4 } n-simplex = {v 1, v 2, …, v n+1 } v4v4 v3v3 v1v1 v2v2 Building blocks for a simplicial complex using Z 2 coefficients