The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(\Delta_n^m)$ 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 $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.
@article{10_37236_6212,
author = {Benjamin Braun and Wesley K. Hough},
title = {Matching and independence complexes related to small grids},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {4},
doi = {10.37236/6212},
zbl = {1373.05137},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6212/}
}
TY - JOUR
AU - Benjamin Braun
AU - Wesley K. Hough
TI - Matching and independence complexes related to small grids
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/6212/
DO - 10.37236/6212
ID - 10_37236_6212
ER -
%0 Journal Article
%A Benjamin Braun
%A Wesley K. Hough
%T Matching and independence complexes related to small grids
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/6212/
%R 10.37236/6212
%F 10_37236_6212
Benjamin Braun; Wesley K. Hough. Matching and independence complexes related to small grids. The electronic journal of combinatorics, Tome 24 (2017) no. 4. doi: 10.37236/6212
