The topology of the matching complex for the 2 x n grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes Ind(Δmn) that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain Ind(Δmn). Furthermore, we determine the Euler characteristic of Ind(Δmn) and prove that several homology groups of Ind(Δmn) are non-zero.
Benjamin Braun partially supported by grant H98230-16-1-0045 from the U.S. National Security Agency.
Braun, Benjamin and Hough, Wesley K., "Matching and Independence Complexes Related to Small Grids" (2017). Mathematics Faculty Publications. 26.