Topological 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.

Alternative to Morse–Novikov theory for a closed 1-form. I

“…To understand the relations between this paper and the previous works, [2], [3] and [1], the following observations are useful:…”

“…the map f ω is never proper when degree of irrationality k is greater than 1, so the homology vector spaces of the levels sets are not of finite dimension in general, 2. the set of critical values of f ω is not discrete when k > 1 but the opposite, always dense if not empty; the approach of Zig-Zag persistence based on graph representations, cf. [2], is apparently not applicable.…”

“…The support of δ ω r and γ ω r are real numbers of the form c ′ − c ′′ with c ′ and c ′′ critical values of any lift f of the TC1-form ω (cf. section (2) for definitions). Item 1. explains in what sense δ ω r refines the Novikov-Betti numbers.…”

Sketches of a platypus: a survey of persistent homology and its algebraic foundations

“…Drawing on results by Donovan and Freislich [36] and by Nazarova [57], demonstrating that these quivers have indecomposables classified by barcode spirals coupled with Jordan cells, Burghelea and Dey [4] produces algorithms and methods to both compute these indecomposable descriptions, and to solve numerous Betti number computation problems with the spiral and Jordan cell description of a circular persistent homology module.…”

“…Circular persistence. In a sequence of preprints, Burghelea and Dey [4], Burghelea, Dey, and Dong [5] study what they call persistence for circle valued maps. This treats the question of how to adapt the methods of persistent homology in order to deal with studying maps f : X → S 1 instead of f : X → R. Such maps appear naturally when studying cohomology, a fact also underlying the work by Morozov, de Silva, and Vejdemo-Johansson [56] that we described in Section 4.2.3.…”