# Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes

@article{Mrozek2017ConleyMorseFormanTF, title={Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes}, author={Marian Mrozek}, journal={Foundations of Computational Mathematics}, year={2017}, volume={17}, pages={1585-1633} }

We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through…

## Figures and Topics from this paper

## 22 Citations

Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions

- Mathematics, Computer ScienceFound. Comput. Math.
- 2020

We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric…

Persistence of the Conley Index in Combinatorial Dynamical Systems

- Mathematics, Computer ScienceSoCG
- 2020

It is shown how one can use zigzag persistence to summarize changes to the Conley index, and techniques to capture such changes in the presence of noise are developed.

Persistent Homology of Morse Decompositions in Combinatorial Dynamics

- Computer Science, MathematicsSIAM J. Appl. Dyn. Syst.
- 2019

The homological persistence of {\em Morse decompositions} of combinatorial dynamical systems on simplicial complexes, an important descriptor of the dynamics, is studied as a tool for validating the reconstruction.

Morse theory for loop-free categories

- Mathematics
- 2021

We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen’s Theorem A in this context and introduce the notion…

Approaches to Smooth and Discrete Dynamics

- 2017

Motivation to revisit the Conley index theory for discrete multivalued dynamical systems [T. Kaczynski and M. Mrozek, Topology Appl., 65(1995), pp. 83–96] stems from the needs of broader real…

Persistence of Conley-Morse Graphs in Combinatorial Dynamical Systems

- Computer Science, MathematicsArXiv
- 2021

This paper proposes a method to summarize changes in combinatorial dynamical systems by capturing changes in the socalled Conley-Morse graphs, which summarizes the changing structure of a sequence of dynamical Systems at a finer granularity than previous approaches.

Abstract elements and Morse hyper-graphs of topological spaces and decompositions

- Mathematics
- 2021

We introduce topological invariants for topological spaces and decompositions, which are analogous to abstract (weak) orbit spaces and Morse graphs for flows. To achieve these, we define analogous…

Topological realizations of groups in Alexandroff spaces

- Mathematics
- 2019

Given a group $G$, we provide a constructive method to get infinitely many (non-homotopy-equivalent) Alexandroff spaces, such that the group of autohomeomorphisms, the group of homotopy classes of…

Combinatorial vs. classical dynamics: Recurrence

- MathematicsCommunications in Nonlinear Science and Numerical Simulation
- 2022

Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On…

A T ] 1 3 Ju l 2 02 1 MORSE THEORY FOR LOOP-FREE CATEGORIES

- 2021

We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen’s Theorem A in this context and introduce the notion…

## References

SHOWING 1-10 OF 42 REFERENCES

Some remarks on Morse theory for posets, homological Morse theory and finite manifolds

- Mathematics
- 2010

We introduce a version of discrete Morse theory for posets. This theory studies the topology of the order complexes K(X) of h-regular posets X from the critical points of admissible matchings on X.…

Leray functor and cohomological Conley index for discrete dynamical systems

- Mathematics
- 1990

We introduce the Leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological Conley index of an isolated invariant…

A Morse equation in Conley's index theory for semiflows on metric spaces

- Mathematics
- 1985

Given a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition ( M 1 , …, M n ) of S , there is a generalized Morse equation, proved by Conley and Zehnder, which relates the…

Equivariant discrete Morse theory

- Computer Science, MathematicsDiscret. Math.
- 2009

This paper generalizes the notion of a Morse matching, and obtains a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex.

Morse Theory for Cell Complexes

- Mathematics
- 1998

In this paper we will present a very simple discrete Morse theory for CW complexes. In addition to proving analogues of the main theorems of Morse theory, we also present discrete analogues of such…

Combinatorial vector fields and dynamical systems

- Mathematics
- 1998

Abstract. In this paper we introduce the notion of a combinatorial dynamical system on any CW complex. Earlier in [Fo3] and [Fo4], we presented the idea of a combinatorial vector field (see also…

Discrete Morse Theory for free chain complexes

- Mathematics, Computer ScienceArXiv
- 2005

Abstract We extend the combinatorial Morse complex construction to arbitrary free chain complexes, and give a short, self-contained, and elementary proof of the quasi-isomorphism between the original…

Towards a formal tie between combinatorial and classical vector field dynamics

- Medicine
- 2016

A flow-like upper semi-continuous acyclic-valued mapping on the underlying topological space whose dynamics is equivalent to the dynamics of Forman's combinatorial vector field on the level of isolated invariant sets and isolating blocks is constructed.

Morse theory from an algebraic viewpoint

- Mathematics
- 2005

Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the…

Resolution of the residue class field via algebraic discrete Morse theory

- Mathematics
- 2005

Forman's Discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoeldberg we show that this theory can be extended to chain complexes of free modules…