Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces

作     者：Lipiński, Michał Kubica, Jacek Mrozek, Marian Wanner, Thomas 

作者机构：Division of Computational Mathematics Faculty of Mathematics and Computer Science Jagiellonian University ul. St. Łojasiewicza 6 Kraków 30-348 Poland Department of Mathematical Sciences George Mason University Fairfax 22030 VA United States 

出 版 物：《Journal of Applied and Computational Topology》 (J Appl. and Comput. Topology)

年 卷 期：2023年第7卷第2期

页      面：139-184页

基　　金：Center for Selective C-H Functionalization, National Science Foundation Center for Hierarchical Manufacturing, National Science Foundation, CHM, NSF Narodowe Centrum Nauki, NCN, (Maestro 2014/14/A/ST1/00453, Preludium 2018/29/N/ST1/00449, 2014/14/A/ST1/00453, 2018/29/N/ST1/00449) Narodowe Centrum Nauki, NCN Simons Foundation, SF, (581334, Award 581334) Simons Foundation, SF National Science Foundation, NSF, (1407087, 1407087) National Science Foundation, NSF 

主　　题：Combinatorial vector field Conley theory Discrete Morse theory Finite topological space Isolated invariant set 

摘      要：We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in Mrozek (Found Comput Math 17(6):1585–1633, 2017). The generalization is threefold. First, we drop the restraining assumption in Mrozek (Found Comput Math 17(6):1585–1633, 2017) that every multivector must have a unique maximal element. Second, we define the dynamical system induced by the multivector field in a less restrictive way. Finally, we also change the setting from Lefschetz complexes to finite topological spaces. Formally, the new setting is more general, because every Lefschetz complex is a finite topological space, but the main reason for switching to finite topologcial spaces is because the latter better explain some peculiarities of combinatorial topological dynamics. We define isolated invariant sets, isolating neighborhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities. © 2022, The Author(s).