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arXiv · 1606.01204

Matching and Independence Complexes Related to Small Grids

Benjamin Braun·Wesley K. Hough
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Abstract

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 $\mathrm{Ind}(Δ_n^m)$ 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 $\mathrm{Ind}(Δ_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(Δ_n^m)$ and prove that several homology groups of $\mathrm{Ind}(Δ_n^m)$ are non-zero.

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Benjamin Braun, Wesley K. Hough. 2016-06-03. Matching and Independence Complexes Related to Small Grids. https://arxiv.org/abs/1606.01204

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