Geodesic
Multipath cohomology of directed graphs
Caputi, Luigi1 ; Collari, Carlo2 ; Di Trani, Sabino3
1 Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom
2 New York University Abu Dhabi, Abu Dhabi, United Arab Emirates
3 Dipartimento di Matematica “G Castelnuovo”, Sapienza Università di Roma, Rome, Italy
Algebraic and Geometric Topology, Tome 24 (2024) no. 8, pp. 4373-4421

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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.

DOI : 10.2140/agt.2024.24.4373
Keywords: graph homology, chromatic homology, Hochschild homology

Caputi, Luigi 1 ; Collari, Carlo 2 ; Di Trani, Sabino 3

1 Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom
2 New York University Abu Dhabi, Abu Dhabi, United Arab Emirates
3 Dipartimento di Matematica “G Castelnuovo”, Sapienza Università di Roma, Rome, Italy 
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Caputi, Luigi; Collari, Carlo; Di Trani, Sabino. Multipath cohomology of directed graphs. Algebraic and Geometric Topology, Tome 24 (2024) no. 8, pp. 4373-4421. doi: 10.2140/agt.2024.24.4373

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