Discrete Morse theory and the geometry of nonpositively curved simplicial complexes

Understanding the conditions under which a simplicial complex collapses is a central issue in many problems in topology and combinatorics. Let K be a simplicial complex endowed with the piecewise Euclidean geometry given by declaring edges to have unit length, and satisfying the property that every 2-simplex is a face of at most two 3-simplices in K. Our main theorem is that if |K| is nonpositively curved (in the sense of CAT(0)) then K simplicially collapses to a point. The main tool used in the proof is Forman's discrete Morse theory (see section 2.2), a combinatorial version of the classical smooth theory. A key ingredient in our proof is a combinatorial analog of the fact that a minimal surface in $R3$ has nonpositive Gauss curvature (see theorem 28). We also investigate another combinatorial question related to curvature. We prove a combinatorial isoperimetric inequality by finding an exact answer for the largest possible number of interior vertices in a triangulated n-gon satisfying the property that every interior vertex has degree at least six.