Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 5

Discrete Morse theory on digraphs

Pages: 1711 – 1737

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n5.a4


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

Chong Wang (Department of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, Hebei, China; and School of Mathematics, Renmin University of China, Beijing, China)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


In this paper, we give a necessary and sufficient condition that discrete Morse functions on a digraph can be extended to be Morse functions on its transitive closure, from this we can extend the Morse theory to digraphs by using quasi-isomorphism between path complex and discrete Morse complex, we also prove a general sufficient condition for digraphs that the Morse functions satisfying this necessary and sufficient condition.


discrete Morse theory, quasi-isomorphism, path homology

2010 Mathematics Subject Classification

55N35, 55U15

Yong Lin was partially supported by National Natural Science Foundation of China, Grant No. 12071245.

Chong Wang was partially supported by Science and Technology Project of Hebei Education Department (QN2019333), by the Natural Fund of Cangzhou Science and Technology Bureau (No. 197000002), and by a Project of Cangzhou Normal University (No. xnjjl1902).

Shing-Tung Yau was partially supported by DMS-1737873.

Received 29 January 2021

Accepted 25 May 2021

Published 26 January 2022