Suspension Foliations: Interesting Examples of Topology and Geometry

We give examples of foliations on suspensions and comment on their topological and geometric properties 1. Idea of foliation by suspension Here is the simplest example of foliation by suspension. Let X be a manifold of dimension q, and let f : X → X be a bijection. Then we define the suspension M = S ×f X as the quotient of [0, 1]×X by the equivalence relation (1, x) ∼ (0, f (x)). M = S ×f X = [0, 1]×X ∼ Then automatically M carries two foliations: F2 consisting of sets of the form F2,t = {(t, x)∼ : x ∈ X} and F1 consisting of sets of the form F2,x0 = {(t, x) : t ∈ [0, 1] , x ∈ Ox0}, where the orbit Ox0 is defined as Ox0 = { ..., f−2 (x0) , f −1 (x0) , x0, f (x0) , f 2 (x0) , ... } , where the exponent refers to the number of times the function f is composed with itself. Note that Ox0 = Of(x0) = Of−2(x0), etc., so the same is true for F1,x0 . Understanding the foliation F1 is equivalent to understanding the dynamics of the map f . If the manifold X is already foliated, you can use the construction to increase the codimension of the foliation, as long as f maps leaves to leaves. The first set of examples concerns foliations of a map from the circle to itself. Example A: Let X = S, let α be a fixed real number, and let f : S → S be defined by f (z) = ez. The S ×f S is topologically the 2-torus. It is a cylinder with the two ends identified with a twist. Note that if α is a rational multiple of 2π, then all of the leaves are closed. If α is irrational, then all of the leaves are dense. This is called a Kronecker foliation. Note that all leaves have no holonomy. Example B: Let X = S, let f : S → S be defined by f (z) = z. The S ×f S is topologically the Klein bottle. It is a cylinder with the two ends identified with a reflection. Observe that all leaves are closed — two of them have z2 holonomy, and the others have trivial holonomy. The next example is a codimension-2 foliation on a 3-manifold. Example C: (This one is from [8] and [9].) Consider the one-dimensional foliation obtained by suspending an irrational rotation on the standard unit sphere S. On S we use the cylindrical coordinates (z, θ), related to the standard rectangular coordinates by x′ = √ (1− z2) cos θ, y′ = √ (1− z2) sin θ, z′ = z. Let α be an irrational multiple Date: August, 2009. 1991 Mathematics Subject Classification. 53C12, 58G11, 58G18, 58G25. 1