Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 2

Künneth formulas for path homology

Pages: 697 – 712

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n2.a10


Fang Li (School of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Bin Yu (Department of Mathematics, China Jiliang University, Hangzhou, China)


We study the path homology groups with coefficients in a general ring $R$, and show that such groups are always finitely generated. We further prove two versions of Eilenberg–Zilber theorem for the Cartesian product and the join of two regular path complexes over a commutative ring $R$. Hence Künneth formulas are derived for the two cases over a PID. Note that this generalizes the related results proved for regular path complexes over a field $K$ in [7], whose proofs can not be carried over here parallelly.


directed graphs, path complexes, path (co)homology, cross product, Künneth formula

2010 Mathematics Subject Classification

Primary 05C25, 13D03, 55U25. Secondary 13D07, 55N35.

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The first-named author was supported by the National Natural Science Foundation of China (No.12071422, No.12131015).

The second-named author was supported by the Zhejiang Provincial Natural Science Foundation (No. LQ20A010008).

Received 5 October 2021

Received revised 17 November 2022

Accepted 11 February 2023

Published 7 April 2023