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Witten-Morse functions and Morse inequalities on digraphs

Yong Lin Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China Chong Wang School of Mathematics, Renmin University of China, Beijing 100872, China; School of Mathematics and Statistics, Cangzhou Normal University, 061000 China

Combinatorics Algebraic Topology and General Topology mathscidoc:2207.06006

arXiv, 2021.8
In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of paths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.
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@inproceedings{yong2021witten-morse,
  title={Witten-Morse functions and Morse inequalities on digraphs},
  author={Yong Lin, and Chong Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707161428607763584},
  booktitle={arXiv},
  year={2021},
}
Yong Lin, and Chong Wang. Witten-Morse functions and Morse inequalities on digraphs. 2021. In arXiv. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707161428607763584.
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