Persistence for Circle Valued Maps

Abstract

We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle valued map on an input simplicial complex.

A complete algorithm to compute barcodes and Jordan cells is provided in this version. The paper is accepted in in the journal Discrete & Computational Geometry. arXiv admin note: text overlap with arXiv:1210.3092 by other authors

References in corpus (3)

• Dynamics, Laplace transform and Spectral geometry
• Defining and Computing Topological Persistence for 1-cocycles
• Graph Representations and Topology of Real and Angle Valued Maps

Cited by in corpus (14)

• Sheaves, Cosheaves and Applications
• Topology of angle valued maps, bar codes and Jordan blocks
• Defining and Computing Topological Persistence for 1-cocycles
• Linear relations, monodromy and Jordan cells of a circle valued map
• A refinement of Betti numbers and homology in the presence of a continuous function II (the case of an angle valued map)
• Graph Representations and Topology of Real and Angle Valued Maps
• Topology Of Real And Angle Valued Maps And Graph Representations (A Brief Survey)
• Refinement of Novikov - Betti numbers and of Novikov homology provided by an angle valued map
• Barcodes in level and sub level persistence and Morse-Novikov theory
• Barcodes for closed one form - an alternative to Novikov theory
• Alternative to Morse-Novikov Theory for closed 1-form (I)
• Contributions to Persistence Theory
• Comparison between "Parametrized homology via zigzag persistence" and "Refined homology in the presence of a real-valued continuous function"
• On Level persistence (Relevant level persistence numbers)