Discrete Morse theory and the topology of matching complexes of complete graphs
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arXiv
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2023
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| author | Mondal, Anupam Mukherjee, Sajal Saha, Kuldeep |
| author_facet | Mondal, Anupam Mukherjee, Sajal Saha, Kuldeep |
| contents | We denote the matching complex of the complete graph with $n$ vertices by $M_n$. Bouc first studied the topological properties of $M_n$ in connection with the Quillen complex. Later Björner, Lovász, Vrećica, and Živaljević showed that $M_n$ is homotopically $(ν_n-1)$-connected, where $ν_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, but in general the topology of $M_n$ 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 $M_n$. We show that $M_n$ is \emph{geometrically} $(ν_n-1)$-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 $M_8$ is simply connected, but not 2-connected. The technique developed here helped us determine that $M_8$ is in fact homotopy equivalent to a wedge of 132 spheres of dimension 2. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_07054 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Discrete Morse theory and the topology of matching complexes of complete graphs Mondal, Anupam Mukherjee, Sajal Saha, Kuldeep Combinatorics Algebraic Topology Geometric Topology 18G85, 57Q70, 05C70, 05E45 We denote the matching complex of the complete graph with $n$ vertices by $M_n$. Bouc first studied the topological properties of $M_n$ in connection with the Quillen complex. Later Björner, Lovász, Vrećica, and Živaljević showed that $M_n$ is homotopically $(ν_n-1)$-connected, where $ν_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, but in general the topology of $M_n$ 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 $M_n$. We show that $M_n$ is \emph{geometrically} $(ν_n-1)$-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 $M_8$ is simply connected, but not 2-connected. The technique developed here helped us determine that $M_8$ is in fact homotopy equivalent to a wedge of 132 spheres of dimension 2. |
| title | Discrete Morse theory and the topology of matching complexes of complete graphs |
| topic | Combinatorics Algebraic Topology Geometric Topology 18G85, 57Q70, 05C70, 05E45 |
| url | https://arxiv.org/abs/2303.07054 |