Multipath cohomology of directed graphs

Caputi, Luigi; Collari, Carlo; Di Trani, Sabino

doi:10.2140/agt.2024.24.4373

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Multipath cohomology of directed graphs

Luigi Caputi, Carlo Collari and Sabino Di Trani

Algebraic & Geometric Topology 24 (2024) 4373–4421

DOI: 10.2140/agt.2024.24.4373

Abstract

This work is part of a series of papers focusing on multipath cohomology of directed graphs. Multipath cohomology is defined as the (poset) homology of the path poset — ie the poset of disjoint simple paths in a graph — with respect to a certain functor. This construction is essentially equivalent, albeit more computable, to taking the higher limits of said functor on (a certain modification of) the path poset. We investigate the functorial properties of multipath cohomology. We provide a number of sample computations, show that multipath cohomology does not vanish on trees, and show that, when evaluated at the coherently oriented polygon, it recovers Hochschild homology. Finally, we use the same techniques employed to study the functoriality to investigate the connection with the chromatic homology of (undirected) graphs introduced by L Helme-Guizon and Y Rong.

Keywords

graph homology, chromatic homology, Hochschild homology

Mathematical Subject Classification

Primary: 05C20, 13D03, 18G85

References

Publication

Received: 30 May 2022

Revised: 23 July 2023

Accepted: 10 August 2023

Published: 17 December 2024

Authors

Luigi Caputi
Institute of Mathematics University of Aberdeen Aberdeen United Kingdom

Carlo Collari
New York University Abu Dhabi Abu Dhabi United Arab Emirates

Sabino Di Trani
Dipartimento di Matematica “G Castelnuovo” Sapienza Università di Roma Rome Italy

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Volume 24, issue 8 (2024)