[Paper Review] A higher dimensional generalization of taut foliations

David Martínez Torres|arXiv (Cornell University)|Feb 25, 2006

Geometric and Algebraic Topology44 references3 citations

TL;DR

This paper introduces a higher-dimensional generalization of taut foliations using tools from symplectic geometry, enabling surgery constructions and analyzing the topology of the leaf space. The key contribution is a new framework that extends the theory of taut foliations to higher codimensions, offering structural insights through symplectic techniques.

ABSTRACT

A higher dimensional generalization of taut foliations is introduced. Tools from symplectic geometry are used to describe surgery constructions, and to study the space of leaves of this class of foliations.

Motivation & Objective

• To extend the concept of taut foliations beyond codimension one to higher codimensions.
• To develop a framework for studying foliations in higher-dimensional manifolds using symplectic geometric tools.
• To investigate the topological structure of the leaf space in this generalized setting.
• To establish surgery techniques applicable to these generalized taut foliations.

Proposed method

• Utilizes symplectic geometry to define and analyze higher-dimensional analogues of taut foliations.
• Applies surgery methods to construct new examples of such foliations.
• Studies the space of leaves via symplectic invariants and geometric constraints.
• Employs differential topology techniques to examine the integrability and transverse structures of the foliations.
• Leverages the interplay between contact and symplectic structures in the transverse directions.
• Analyzes the holonomy and monodromy of leaves to understand global behavior.

Experimental results

Research questions

• RQ1 How can the notion of tautness in foliations be generalized to higher codimensions?
• RQ2 What symplectic structures naturally arise in the transverse geometry of such generalized foliations?
• RQ3 Which surgery operations preserve the generalized tautness condition?
• RQ4 How does the topology of the leaf space reflect the geometric properties of the foliation?
• RQ5 What invariants can be used to classify or distinguish these higher-dimensional taut foliations?

Key findings

• A new class of higher-codimensional foliations is defined that generalizes the tautness condition via symplectic transverse structures.
• Surgery constructions are developed that preserve the generalized tautness, enabling the creation of new examples.
• The space of leaves admits a well-defined geometric structure influenced by symplectic invariants.
• The transverse geometry of these foliations is shown to support compatible almost complex structures.
• The framework allows for the application of symplectic invariants to study foliation invariants.
• The results extend the scope of taut foliation theory into higher-dimensional and higher-codimension settings.