Simplicial collapsibility, 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 finite simplicial complex of dimension three or less 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 result 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, a combinatorial analog of the classical smooth theory developed in the 1920s. A key ingredient in our proof is a combinatorial analog of the fact that a minimal surface in $${{\mathbb R}^{3}}$$ has nonpositive Gauss curvature.