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Path homology of digraphs without multisquares and its comparison with homology of spaces

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arxiv 2407.17001 v1 pith:VNFK326I submitted 2024-07-24 math.AT math.KT

Xin Fu , Sergei O. Ivanov This is my paper · ORCID
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keywords mathbbhomologypathdigraphcharacteristicomegabasisconstructed
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abstract

For a digraph $G$ without multisquares and a field $\mathbb{F}$, we construct a basis of the vector space of path $n$-chains $\Omega_n(G;\mathbb{F})$ for $n\geq 0$, generalising the basis of $\Omega_3(G;\mathbb{F})$ constructed by Grigory'an. For a field $\mathbb{F},$ we consider the $\mathbb{F}$-path Euler characteristic $\chi^\mathbb{F}(G)$ of a digraph $G$ defined as the alternating sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If $\Omega_\bullet(G;\mathbb{F})$ is a bounded chain complex, the constructed bases can be applied to compute $\chi^\mathbb{F}(G)$. 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 $\mathbb{K}=\mathbb{Z}/2\mathbb{Z}.$

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Inductive construction of path homology chains

    math.AT 2024-11 conditional novelty 8.0 of 10

    New inductive elements, built from face multihypergraphs, generate the path homology chain modules over finite fields and produce digraphs whose path Euler characteristic changes with the coefficient field.

  2. Inductive construction of path homology chains and the structure of $\Omega_3(G;R)$

    math.AT 2026-07 conditional novelty 7.0 of 10

    Path homology chains of any digraph are generated by inductive extensions over face multigraphs: completely in characteristic 2 and in dimensions 0–3 in characteristic 0.

  3. Primitive path homology

    math.AT 2024-11 conditional novelty 6.0 of 10

    Primitive path homology is a new digraph invariant that coincides with GLMY path homology on asymmetric digraphs and differs on symmetric ones.

  4. Stability of persistent path homology of path complexes

    math.AT 2026-07 reject novelty 5.0 of 10

    The claimed stability of persistent path homology for arbitrary path complexes is unsupported because the key homotopy argument assumes path complexes are closed under concatenation.

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