Physical Sciences and Mathematics Commons^™

Matching And Independence Complexes Related To Small Grids, Benjamin Braun, Wesley K. Hough

Mathematics Faculty Publications

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(Δ^m_n) 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(Δ^m_n). Furthermore, we determine the Euler characteristic of Ind(Δ^m_n) and prove that several homology groups of Ind(Δ^m_n) are non-zero.