On diagonal digraphs, Koszul algebras and triangulations of homology spheres
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916500129972224 |
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| author | Ivanov, Sergei O. Mukoseev, Lev |
| author_facet | Ivanov, Sergei O. Mukoseev, Lev |
| contents | The article is devoted to the magnitude homology of digraphs, with a primary focus on diagonal digraphs, i.e., digraphs whose magnitude homology is concentrated on the diagonal. For any digraph $G$, we provide a complete description of the second magnitude homology ${\rm MH}_{2,k}(G)$. This allows us to define a combinatorial condition, denoted by $(\mathcal{V}_\ell)$, which is equivalent to the vanishing of ${\rm MH}_{2,k}(G, \mathbb{Z})$ for all $k > \ell$. In particular, diagonal digraphs satisfy $(\mathcal{V}_2)$. As a corollary, we obtain that the 2-dimensional CW-complex obtained from a diagonal undirected graph by attaching 2-cells to all squares and triangles of the graph is simply connected. We also give an interpretation of diagonality in terms of Koszul algebras: a digraph $G$ is diagonal if and only if the distance algebra $σG$ is Koszul for any ground field, and if and only if $G$ satisfies $(\mathcal{V}_2)$ and the path cochain algebra $Ω^\bullet(G)$ is Koszul for any ground field. To provide a source of examples of digraphs, we study the extended Hasse diagram $\hat G_K$ of a pure simplicial complex $K$. For a triangulation $K$ of a topological manifold $M$, we express the non-diagonal part of the magnitude homology of $\hat G_K$ in terms of the homology of $M$. As a corollary, we obtain that if $K$ is a triangulation of a closed manifold $M$, then $\hat G_K$ is diagonal if and only if $M$ is a homology sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On diagonal digraphs, Koszul algebras and triangulations of homology spheres Ivanov, Sergei O. Mukoseev, Lev K-Theory and Homology Algebraic Topology Combinatorics Representation Theory The article is devoted to the magnitude homology of digraphs, with a primary focus on diagonal digraphs, i.e., digraphs whose magnitude homology is concentrated on the diagonal. For any digraph $G$, we provide a complete description of the second magnitude homology ${\rm MH}_{2,k}(G)$. This allows us to define a combinatorial condition, denoted by $(\mathcal{V}_\ell)$, which is equivalent to the vanishing of ${\rm MH}_{2,k}(G, \mathbb{Z})$ for all $k > \ell$. In particular, diagonal digraphs satisfy $(\mathcal{V}_2)$. As a corollary, we obtain that the 2-dimensional CW-complex obtained from a diagonal undirected graph by attaching 2-cells to all squares and triangles of the graph is simply connected. We also give an interpretation of diagonality in terms of Koszul algebras: a digraph $G$ is diagonal if and only if the distance algebra $σG$ is Koszul for any ground field, and if and only if $G$ satisfies $(\mathcal{V}_2)$ and the path cochain algebra $Ω^\bullet(G)$ is Koszul for any ground field. To provide a source of examples of digraphs, we study the extended Hasse diagram $\hat G_K$ of a pure simplicial complex $K$. For a triangulation $K$ of a topological manifold $M$, we express the non-diagonal part of the magnitude homology of $\hat G_K$ in terms of the homology of $M$. As a corollary, we obtain that if $K$ is a triangulation of a closed manifold $M$, then $\hat G_K$ is diagonal if and only if $M$ is a homology sphere. |
| title | On diagonal digraphs, Koszul algebras and triangulations of homology spheres |
| topic | K-Theory and Homology Algebraic Topology Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2405.04748 |