The complex of discrete Morse matchings of the $n$-simplex: homotopy types and structural results
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2026
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| _version_ | 1866914491644510208 |
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| author | Scoville, Nicholas A. |
| author_facet | Scoville, Nicholas A. |
| contents | The complex of discrete Morse matchings $\M(K)$, introduced by Chari and Joswig, is a simplicial complex whose simplices are the acyclic matchings on the Hasse diagram of $K$. Its homotopy type is known in only a handful of cases. In this paper, we compute the homotopy types of $\M(Δ^3)$ and $\M(\partialΔ^3)$, the corresponding pure complexes $\M_{P}(Δ^3) \simeq \M_{P}(\partialΔ^3)$, and the generalized complex of discrete Morse matchings $\GM(Δ^3) \simeq \GM(\partialΔ^3)$. For general $n$ we prove the identity $f(n) = (n+1) \cdot |\text{top-dimensional facets of } \M(Δ^n_{(n-2)})|$, reducing the enumeration of optimal matchings on $Δ^n$ to an enumeration on its $(n-2)$-skeleton, and we show that the inclusion $\M(K) \hookrightarrow \M(CK)$ is null-homotopic for any cone. We also compute the $f$-vector of $\M(Δ^4)$, whose top entry $f(4) = 380{,}125$ is the number of optimal discrete Morse matchings on $Δ^4$. We conclude with two conjectures extending the $\M_{P}$ and $\GM$ equivalences to all $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_18172 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The complex of discrete Morse matchings of the $n$-simplex: homotopy types and structural results Scoville, Nicholas A. Algebraic Topology Combinatorics Primary 05E45, 57Q70, Secondary 55P10, 05C70 The complex of discrete Morse matchings $\M(K)$, introduced by Chari and Joswig, is a simplicial complex whose simplices are the acyclic matchings on the Hasse diagram of $K$. Its homotopy type is known in only a handful of cases. In this paper, we compute the homotopy types of $\M(Δ^3)$ and $\M(\partialΔ^3)$, the corresponding pure complexes $\M_{P}(Δ^3) \simeq \M_{P}(\partialΔ^3)$, and the generalized complex of discrete Morse matchings $\GM(Δ^3) \simeq \GM(\partialΔ^3)$. For general $n$ we prove the identity $f(n) = (n+1) \cdot |\text{top-dimensional facets of } \M(Δ^n_{(n-2)})|$, reducing the enumeration of optimal matchings on $Δ^n$ to an enumeration on its $(n-2)$-skeleton, and we show that the inclusion $\M(K) \hookrightarrow \M(CK)$ is null-homotopic for any cone. We also compute the $f$-vector of $\M(Δ^4)$, whose top entry $f(4) = 380{,}125$ is the number of optimal discrete Morse matchings on $Δ^4$. We conclude with two conjectures extending the $\M_{P}$ and $\GM$ equivalences to all $n$. |
| title | The complex of discrete Morse matchings of the $n$-simplex: homotopy types and structural results |
| topic | Algebraic Topology Combinatorics Primary 05E45, 57Q70, Secondary 55P10, 05C70 |
| url | https://arxiv.org/abs/2604.18172 |