Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 2

Analytic and Reidemeister torsions of digraphs and path complexes

Pages: 703 – 755

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n2.a3

Authors

Alexander Grigor’yan (Department of Mathematics, University of Bielefeld, Germany)

Yong Lin (Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, China)

Shing-Tung Yau (Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, China)

Abstract

We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in [ $\href{https://arxiv.org/abs/1207.2834}{7}$, $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3324763}{8}$, $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3431683}{9}$, $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3845076}{11}$]. We prove the identity of the two notions of torsions as well as obtain formulas for torsions of Cartesian products and joins of digraphs.

Keywords

analytic torsion, Reidemeister torsion, digraphs, path homology

2010 Mathematics Subject Classification

Primary 05C20, 05C38. Secondary 55U25.

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Dedicated to our dear friend Peter Li on the occasion of his 70th birthday

A.G. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - SFB 1283/2 2021 - 317210226.

Y.L. is supported by the National Science Foundation of China (Grant No. 12071245 and 11761131002).

Received 21 February 2022

Received revised 26 April 2023

Accepted 14 May 2023

Published 3 April 2024