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

2016/05/24 by Dan Burghelea, Burghelea, Dan
Computer Science · Mathematics · #55 N35 #55 N99 #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1605.07688

openalex publication_date 2016/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Linear relations, monodromy and Jordan cells of a circle valued map
• Topology Of Real And Angle Valued Maps And Graph Representations (A Brief Survey)

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• Graph Representations and Topology of Real and Angle Valued Maps
• Alternative to Morse-Novikov Theory for closed 1-form (I)
• Refinement of Novikov - Betti numbers and of Novikov homology provided by an angle valued map