[Paper Review] Foliation Cones II

John Cantwell, Lawrence Conlon|arXiv (Cornell University)|Aug 3, 2011

Geometric and Algebraic Topology19 references3 citations

TL;DR

This paper classifies finite-depth, foliated 3-manifolds with a given substructure S by analyzing the components W of (M − S), which are stably foliated. It shows that isotopy classes of such foliations are parametrized by rays through integer lattice points in finitely many closed, convex, non-overlapping, polyhedral cones in a suitable cohomology group of W, with the cones being infinite-dimensional but having only finitely many faces, for both smooth and non-smooth foliations.

ABSTRACT

This paper extends and simplifies our paper Foliation Cones (KirbyFest, 1999) while correcting some errors. We classify finite depth, foliated 3-manifolds M with a given substructure S. The components W of (M S) are stably foliated and the possible such foliations are classified, up to isotopy, by the rays through the integer lattice points in the interiors of finitely many closed, convex, non-overlapping, finite-sided, polyhedral cones in a suitable cohomology of W. While the cones are generally infinite dimensional, they have only finitely many faces. Results of this type are given both for the cases that the foliation is, and is not, smooth.

Motivation & Objective

• To extend and correct the results of the earlier paper 'Foliation Cones' presented at KirbyFest 1999.
• To classify finite-depth, foliated 3-manifolds M with a fixed substructure S, focusing on the components W of (M − S).
• To determine the isotopy classes of foliations on these components W, up to stable foliation structure.
• To provide a classification of such foliations using geometric data from cohomology, specifically rays through integer lattice points in convex polyhedral cones.
• To treat both smooth and non-smooth foliation cases within a unified framework.

Proposed method

• Analyzing the components W of (M − S) as stably foliated 3-manifolds.
• Using a suitable cohomology group to parametrize the possible foliations on W.
• Representing isotopy classes of foliations as rays through integer lattice points in convex polyhedral cones.
• Constructing finitely many closed, convex, non-overlapping, finite-sided polyhedral cones in the cohomology group.
• Demonstrating that although the cones are generally infinite-dimensional, they possess only finitely many faces.
• Applying the classification to both smooth and non-smooth foliation structures.

Experimental results

Research questions

• RQ1 How can finite-depth foliated 3-manifolds with a given substructure S be systematically classified?
• RQ2 What geometric and topological invariants determine the isotopy classes of foliations on the components W of (M − S)?
• RQ3 In what way do rays through integer lattice points in cohomology parametrize distinct foliation types on stably foliated components?
• RQ4 How do the properties of the polyhedral cones—such as dimensionality and number of faces—relate to the foliation structure?
• RQ5 What differences, if any, arise in the classification when the foliation is smooth versus non-smooth?

Key findings

• The isotopy classes of foliations on stably foliated components W are completely determined by rays through integer lattice points in finitely many closed, convex, non-overlapping, finite-sided polyhedral cones.
• These cones reside in a suitable cohomology group of W and are generally infinite-dimensional, yet possess only finitely many faces.
• The classification applies uniformly to both smooth and non-smooth foliation structures.
• The results correct and simplify earlier findings from the 1999 KirbyFest paper.
• The geometric structure of the cones provides a complete invariant for the isotopy classification of such foliations.