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.

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