Path homology of digraphs without multisquares and its comparison with homology of spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917731750641664 |
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| author | Fu, Xin Ivanov, Sergei O. |
| author_facet | Fu, Xin Ivanov, Sergei O. |
| contents | For a digraph $G$ without multisquares and a field $\mathbb{F}$, we construct a basis of the vector space of path $n$-chains $Ω_n(G;\mathbb{F})$ for $n\geq 0$, generalising the basis of $Ω_3(G;\mathbb{F})$ constructed by Grigory'an. For a field $\mathbb{F},$ we consider the $\mathbb{F}$-path Euler characteristic $χ^\mathbb{F}(G)$ of a digraph $G$ defined as the alternating sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If $Ω_\bullet(G;\mathbb{F})$ is a bounded chain complex, the constructed bases can be applied to compute $χ^\mathbb{F}(G)$. We provide an explicit example of a digraph $\mathcal{G}$ whose $\mathbb{F}$-path Euler characteristic depends on whether the characteristic of $\mathbb{F}$ is two, revealing the differences between GLMY theory and the homology theory of spaces. This allows us to prove that there is no topological space $X$ whose homology is isomorphic to path homology of the digraph $H_*(X;\mathbb{K})\cong {\rm PH}_*(\mathcal{G};\mathbb{K})$ simultaneously for $\mathbb{K}=\mathbb{Z}$ and $\mathbb{K}=\mathbb{Z}/2\mathbb{Z}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17001 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Path homology of digraphs without multisquares and its comparison with homology of spaces Fu, Xin Ivanov, Sergei O. Algebraic Topology K-Theory and Homology For a digraph $G$ without multisquares and a field $\mathbb{F}$, we construct a basis of the vector space of path $n$-chains $Ω_n(G;\mathbb{F})$ for $n\geq 0$, generalising the basis of $Ω_3(G;\mathbb{F})$ constructed by Grigory'an. For a field $\mathbb{F},$ we consider the $\mathbb{F}$-path Euler characteristic $χ^\mathbb{F}(G)$ of a digraph $G$ defined as the alternating sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If $Ω_\bullet(G;\mathbb{F})$ is a bounded chain complex, the constructed bases can be applied to compute $χ^\mathbb{F}(G)$. We provide an explicit example of a digraph $\mathcal{G}$ whose $\mathbb{F}$-path Euler characteristic depends on whether the characteristic of $\mathbb{F}$ is two, revealing the differences between GLMY theory and the homology theory of spaces. This allows us to prove that there is no topological space $X$ whose homology is isomorphic to path homology of the digraph $H_*(X;\mathbb{K})\cong {\rm PH}_*(\mathcal{G};\mathbb{K})$ simultaneously for $\mathbb{K}=\mathbb{Z}$ and $\mathbb{K}=\mathbb{Z}/2\mathbb{Z}.$ |
| title | Path homology of digraphs without multisquares and its comparison with homology of spaces |
| topic | Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2407.17001 |