Discrete Morse theory, simplicial nonpositive curvature, and simplicial collapsibility
Ioana-Claudia Laz · 2012
We flnd combinatorial conditions which ensure the collapsibil- ity of a flnite simplicial complex of dimension 3 or less. Our main result states that any flnite systolic standard piecewise Euclidean simplicial com- plex of dimension 3 and 2, satisfying the property that any 2-simplex is a face of at most two 3-simplices in the complex, simplicially collapses to a point. A simplicial complex is systolic if it is simply connected, connected and locally 6-large. Our proof relies on the fact that any cycle in a systolic complex has a van Kampen diagram of minimal area whose disk is itself systolic. Our proof follows by applying discrete Morse theory.