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Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

Wacław Marzantowicz, Łukasz Patryk Michalak

TL;DR

The paper answers affirmatively whether every epimorphism $\pi_1(M)\to F_r$ can be realized as the Reeb epimorphism of a Morse function by establishing a geometric correspondence between epimorphisms and regular independent framed systems of hypersurfaces via an extended Pontryagin–Thom construction. It shows that epimorphisms are classified by framed cobordism classes of such systems and, under suitable diffeomorphism actions, by equivalence relations analogous to Nielsen transformations. It then integrates Reeb graph theory, proving that any epimorphism factoring through the boundary quotient can be realized as a Reeb epimorphism of a Morse function with a prescribed Reeb graph, and explores corank, Reeb number, extendability, and topological conjugacy of Morse functions within this framework. The results yield a purely geometric/topological proof of the known surface-case classification and provide a robust toolkit for realizing and comparing epimorphisms through Reeb graphs and hypersurface systems, with potential applications to higher-dimensional and bordered manifolds.

Abstract

We construct a correspondence between epimorphisms $\varphi \colon π_1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which induces a bijection onto framed cobordism classes of such systems. In consequence, for closed manifolds any such $\varphi$ can be represented by the Reeb epimorphism of a Morse function $f\colon M \to \mathbb{R}$, i.e. by the epimorphism induced by the quotient map $M \to \mathcal{R}(f)$ onto the Reeb graph of $f$. Applying this construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups, providing a new purely geometrical-topological proof of the solution of this problem for surface groups.

Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

TL;DR

The paper answers affirmatively whether every epimorphism can be realized as the Reeb epimorphism of a Morse function by establishing a geometric correspondence between epimorphisms and regular independent framed systems of hypersurfaces via an extended Pontryagin–Thom construction. It shows that epimorphisms are classified by framed cobordism classes of such systems and, under suitable diffeomorphism actions, by equivalence relations analogous to Nielsen transformations. It then integrates Reeb graph theory, proving that any epimorphism factoring through the boundary quotient can be realized as a Reeb epimorphism of a Morse function with a prescribed Reeb graph, and explores corank, Reeb number, extendability, and topological conjugacy of Morse functions within this framework. The results yield a purely geometric/topological proof of the known surface-case classification and provide a robust toolkit for realizing and comparing epimorphisms through Reeb graphs and hypersurface systems, with potential applications to higher-dimensional and bordered manifolds.

Abstract

We construct a correspondence between epimorphisms from the fundamental group of a compact manifold onto the free group of rank , and systems of framed non-separating hypersurfaces in , which induces a bijection onto framed cobordism classes of such systems. In consequence, for closed manifolds any such can be represented by the Reeb epimorphism of a Morse function , i.e. by the epimorphism induced by the quotient map onto the Reeb graph of . Applying this construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups, providing a new purely geometrical-topological proof of the solution of this problem for surface groups.

Paper Structure

This paper contains 15 sections, 43 theorems, 33 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Systems of hypersurfaces and induced homomorphisms
  3. Systems of hypersurfaces
  4. Epimorphisms and independency of inducing systems
  5. Systems of hypersurfaces in the classification of epimorphisms
  6. Systems of hypersurfaces up to framed cobordism and diffeomorphism
  7. Analogue of Nielsen transformations for systems of hypersurfaces
  8. Reeb graphs and Reeb epimorphisms
  9. The initial graph and factorization by Reeb epimorphism
  10. Reeb number of manifolds with boundary
  11. Realization of system of hypersurfaces as components of level sets of function
  12. Realization of epimorphisms onto free groups as Reeb epimorphisms
  13. Extendability of independent systems of hypersurfaces
  14. Topological conjugacy of Morse functions
  15. Final remarks

Key Result

Proposition 2.2

Any homomorphism $\varphi \colon \pi_1(W) \to F_r$ is induced by a system of hypersurfaces. If a system $\mathcal{N}$ is independent, then $\varphi_\mathcal{N}$ is an epimorphism.

Figures (8)

  • Figure 1: Example of connected sum operation of submanifolds $X$ and $Y$ along a curve $\gamma$ which joins them from the same side. The arrows indicate normal vectors from the framing.
  • Figure 2: Equivalence (a) and strong equivalence (b) of epimorphisms onto free groups.
  • Figure 3: Operation (H3) which transforms $\mathcal{N}=(N_j,N_i)$ into $\mathcal{N}^{ij} =(N_j\#_\gamma P_1(N_i),N_i)$.
  • Figure 4: (a) the initial graph with distinguished tree and ordered edges outside it; (b) the initial graph admissible for $(W,W_-,W_+)$.
  • Figure 5: Cutting along edge.
  • ...and 3 more figures

Theorems & Definitions (93)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 83 more