New invariants for a real valued and angle valued map (an Alternative to Morse- Novikov theory)

TL;DR

This paper introduces new invariants for real and angle-valued maps as an alternative to Morse-Novikov theory, providing refined topological information that is computationally accessible and relevant for flow dynamics.

Contribution

It defines and explores invariants like point configurations, vector space collections, and Jordan cells, offering a novel framework as an alternative to Morse-Novikov theory.

Findings

Invariants refine Betti numbers, homology, and monodromy.

Invariants are computable via algorithms for simplicial complexes.

Relevance demonstrated for flow dynamics with Lyapunov maps.

Abstract

This paper but section 6 is essentially my lecture at The Eighth Congress of Romanian Mathematicians, June 26 - July 1, 2015, Iasi, Romania. The paper summarizes the definitions and the properties of the invariants associated to a real or an angle valued map in the framework of what we call an Alternative to Morse-Novikov theory. These invariants are configurations of points in the complex plane, configurations of vector spaces or modules indexed by complex numbers and collections of Jordan cells. The first are refinements of Betti numbers, the second of homology and the third of monodromy. Although not discussed in this paper but discussed in works this report is based on, these invariants are computer friendly (i.e. can be calculated by computer implementable algorithms when the source of the map is a simplicial complex and the map is simplicial) and are of relevance for the dynamics of flows which admit Lyapunov real or angle valued map.