Topological classification of circle-valued simple Morse-Bott functions. (English) Zbl 1407.58013
In this interesting paper, the authors investigate the classification of Morse-Bott functions from \(S^2\) to \(S^1\) up to topological conjugacy. They give a complete topological invariant of simple Morse-Bott functions \(f : S^2 \rightarrow S^1\) based on the generalized Reeb graph associated to \(f\). Moreover, a realization theorem is obtained. The paper is organized into four sections as follows : introduction, Morse-Bott functions, circle-valued Morse-Bott functions, the MB-Reeb graph of a simple circle-valued \(\mathcal{MB}\) function, realization theorem. Other first two authors papers directly connected to this topic are [Proc. Edinb. Math. Soc., II. Ser. 60, No. 2, 319–348 (2017; Zbl 1376.58016); Bull. Braz. Math. Soc. (N.S.) 49, No. 2, 369–394 (2018; Zbl 1415.58022)].
Reviewer: Dorin Andrica (Riyadh)
MSC:
| 58K15 | Topological properties of mappings on manifolds |
| 58K65 | Topological invariants on manifolds |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 57R70 | Critical points and critical submanifolds in differential topology |
Keywords:
topological invariant; Morse-Bott functions; Reeb graph; topological equivalence; realization theoremReferences:
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