[Paper Review] A proof of Reidemeister-Singer's theorem by Cerf's methods

François Laudenbach|arXiv (Cornell University)|Feb 6, 2012

Geometric and Algebraic Topology6 citations

TL;DR

This paper provides a new proof of Reidemeister-Singer's theorem using Cerf's theory of one-parameter families of Morse functions, translating Heegaard splittings into ordered Morse functions on 3-manifolds. By applying Cerf's techniques—particularly the swallow tail lemma and generic path deformations—it establishes that any two Heegaard splittings become isotopic after stabilization, with handle slides and cancellations realized through controlled path deformations in the space of functions.

ABSTRACT

Heegaard splittings and Heegaard diagrams of a closed 3-manifold M are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on M. We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous pseudo-isotopy problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when dim M>2. The main tool that we introduce is an elementary swallow tail lemma which could be useful elsewhere.

Motivation & Objective

• To provide a modern, function-theoretic proof of Reidemeister-Singer's theorem using Cerf theory.
• To translate topological concepts like Heegaard splittings and handle slides into the language of Morse functions and pseudo-gradients.
• To demonstrate that any two ordered Morse functions on a closed 3-manifold are connected by a generic path with controlled birth and cancellation events.
• To establish that stabilization of Heegaard splittings corresponds to adding cancelling pairs of critical points of index 1 and 2 along such paths.
• To introduce and apply an elementary swallow tail lemma as a key technical tool for simplifying generic paths of functions in dimension >2.

Proposed method

• Represent Heegaard splittings as ordered Morse functions on a closed 3-manifold, where the level set separating index 1 and 2 critical values defines the splitting surface.
• Use generic paths of smooth functions connecting two ordered Morse functions, ensuring all but finitely many functions are Morse and critical point changes occur via birth or cancellation events.
• Apply Cerf’s theory to deform the path so that all non-Morse points are cubic (corank 1 Hessian), with swallow tail singularities marking birth and cancellation events.
• Introduce a cancelling pair of critical points (index 1 and 2) along a level set in the middle of the path, using a collar neighborhood to control the base of the birth cylinder.
• Employ the swallow tail lemma to eliminate swallow tail and lip singularities through deformation, reducing the number of extra minima by one at a time.
• Use the uniqueness of birth events in dimension >1 (from Cerf’s lemma) to ensure that cancellations can be performed without introducing new complications.

Experimental results

Research questions

• RQ1 Can Reidemeister-Singer’s theorem be re-proven using Cerf’s theory of one-parameter families of Morse functions?
• RQ2 How can Heegaard splittings and handle slides be systematically encoded in terms of Morse functions and their generic paths?
• RQ3 What role do swallow tail singularities and their cancellation play in simplifying generic paths of functions in dimension >2?
• RQ4 Can the stabilization of Heegaard splittings be realized as the addition of cancelling pairs of critical points in a controlled path of functions?
• RQ5 To what extent can the generic path deformation techniques of Cerf theory be applied to 3-dimensional manifolds to resolve isotopy problems in Morse-theoretic terms?

Key findings

• Any two ordered Morse functions on a closed 3-manifold are connected by a generic path in which all non-Morse points are cubic, with birth and cancellation events occurring in controlled intervals.
• The level set of the middle function in such a path, separating index 1 and 2 critical values, yields a Heegaard splitting that is a common stabilization of those from the initial and final functions.
• The addition of a cancelling pair of critical points (index 1 and 2) along a path corresponds to stabilization of the Heegaard splitting, and such pairs can be introduced via a controlled deformation of the path.
• The swallow tail lemma enables the elimination of swallow tail and lip singularities in the Cerf graphic, allowing the path to be deformed into one with fewer extra minima.
• The final path after deformation has one fewer extra minimum than the original, showing that stabilization can be achieved through a sequence of controlled function deformations.
• The general case reduces to the special case (H) via isotopy and rescaling, ensuring that minima and maxima of the initial and final functions can be matched, thus fulfilling the hypothesis for the deformation argument.