Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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L. Caputi - C. Collari - S. Di Trani

Multipath cohomology of directed graphs

created by collari on 10 Aug 2022
modified on 15 Nov 2023


Accepted Paper

Inserted: 10 aug 2022
Last Updated: 15 nov 2023

Journal: Algebraic & Geometric Topology
Year: 2021

ArXiv: 2108.02690 PDF


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 -- i.e., 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 the multipath cohomology does not vanish on trees, and 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.

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