Discrete Morse theory and the topology of matching complexes of complete graphs

Abstract

We denote the matching complex of the complete graph with n vertices by Mn. Bouc first studied the topological properties of Mn in connection with the Quillen complex. Later Björner, Lovász, Vrećica, and Živaljević showed that Mn is homotopically (νn−1)-connected, where νn=⌊n+13⌋−1, but in general the topology of Mn is not very well-understood even for smaller natural numbers. Forman developed discrete Morse theory, which has various applications in diverse fields of studies. In this article, we develop a discrete Morse theoretic technique to capture deeper structural topological properties of Mn. We show that Mn is \emph{geometrically} (νn−1)-connected, where the notion of geometrical k-connectedness as defined in this article, is stronger than that of homotopical k-connectedness. Previously, Björner et al. showed that M8 is simply connected, but not 2-connected. The technique developed here helped us determine that M8 is in fact homotopy equivalent to a wedge of 132 spheres of dimension 2.