Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups
type: Item
instance of: scholarly article
title: Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups
author: Wacław Marzantowicz
author: Łukasz Patryk Michalak
published in: Fundamenta Mathematicae
publication date: 2024-10-07T00:00:00Z
review text: The \textit{Reeb graph} \(\mathcal{R}(f)\) of a function \(f:M\to\mathbf{R}\) is a topological space obtained by contracting connected components of its level sets to points, endowed with the quotient topology. For a Morse function on a compact manifold, \(\mathcal{R}(f)\) is a finite graph, so its fundamental group is a free group, \(\pi_1(\mathcal{R}(f))=F_r\). The quotient map \(q_f:M\to\mathcal{R}(f)\) induces the epimorphism of the fundamental groups \(q_{f\sharp}:\pi_1(M)\to F_r\), called the \textit{Reeb epimorphism} of \(f\).\N\NThe authors prove that conversely, it is possible to represent any epimorphism \(\varphi:\pi_1(M)\to F_r\) as the Reeb epimorphism of a Morse function on \(M\). To construct the Morse function, the authors study a correspondence between epimorphisms \(\varphi:\pi_1(M)\to F_r\) and systems of \(r\) framed non-separating hypersurfaces in \(M\), and show that this correspondence induces a bijection onto framed cobordism classes of such systems. The hypersurfaces can be considered as connected components of some level sets of the constructed Morse function. This correspondence also provides a geometric tool for studying the general problem of classifying epimorphisms of groups \(G\to F_r\).\N\NSome applications to the Reeb graph theory are also presented, including consideration of manifolds with boundary.
reviewed by: Irina Gelbukh
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57R90
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57R70
zbMATH DE Number: 7922309
zbMATH Keywords: Reeb graph
zbMATH Keywords: 2-sided submanifold
zbMATH Keywords: corank of a group
zbMATH Keywords: framed cobordism
zbMATH Keywords: equivalence of epimorphisms onto a free group
MaRDI profile type: Publication
cites work: Notes on Sela's work: Limit groups and Makanin-Razborov diagrams
cites work: Introduction to group theory. Translated from the Russian. With a new chapter.
cites work: Loops in Reeb graphs of 2-manifolds
cites work: Q4278192
cites work: Q3469916
cites work: The edit distance for Reeb graphs of surfaces
cites work: The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations
cites work: Loops in Reeb graphs of \(n\)-manifolds
cites work: Approximation of metric spaces by Reeb graphs: Cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifolds
cites work: Classification of epimorphisms of fundamental groups of surfaces onto free groups
cites work: Q4278191
cites work: Q4016464
cites work: Q4114528
cites work: Heegaard Splittings and Splitting Homomorphisms
cites work: Geometric realizations for free quotients
cites work: On representation of the Reeb graph as a sub-complex of manifold
cites work: On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds with finitely many singular values
cites work: Q4939881
cites work: Q3552800
cites work: EQUATIONS IN A FREE GROUP
cites work: Realization of a graph as the Reeb graph of a Morse function on a manifold
cites work: Combinatorial modifications of Reeb graphs and the realization problem
cites work: Q5543303
cites work: Q5510566
cites work: ON SYSTEMS OF EQUATIONS IN A FREE GROUP
cites work: Q5845505
cites work: Morse functions with sphere fibers
cites work: Reeb Spaces of Smooth Functions on Manifolds
cites work: Diophantine geometry over groups. I: Makanin-Razborov diagrams
cites work: Smooth and Topological Equivalence of Functions on Surfaces
cites work: Q3440673
cites work: Q4866269
DOI: 10.4064/FM263-1-2024
description: scientific article; zbMATH DE number 7922309