Sutured Manifold Hierarchies and Finite-Depth Foliations Christopher Stover Florida State University Topology Seminar November 4, 2014 Foliations & Hierarchies Outline Foliations Preliminaries Depth Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Sutured Manifold Decompositions Example: Decomposing Sutured D2 × S1 into Sutured B3 Sutured Manifold Hierarchies Main Result Foliations & Hierarchies Foliations Part I Foliations Preliminaries Depth Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Sutured Manifold Decompositions Example: Decomposing Sutured D2 × S1 into Sutured B3 Sutured Manifold Hierarchies Main Result Figure 1 n k n−k R = R × R for n = 3 and k = 2 Foliations & Hierarchies Foliations Preliminaries Definitions Informally, a dimension-k foliation of a manifold M = M n is a decomposition of M into disjoint, connected submanifolds of dimension k < n which, on a small scale, looks like the n k n−k decomposition of R into R × R . Foliations & Hierarchies Foliations Preliminaries Definitions Informally, a dimension-k foliation of a manifold M = M n is a decomposition of M into disjoint, connected submanifolds of dimension k < n which, on a small scale, looks like the n k n−k decomposition of R into R × R . Figure 1 n k n−k R = R × R for n = 3 and k = 2 Definition. A k-dimensional foliation of a manifold M = M n is a disjoint union F of connected, properly embedded dimension-k submanifolds of M which is locally homeomorphic to the direct n k n−k product decomposition of R into R × R and whose union equals M. Here, k is the dimension of F, n − k is its codimension, the submanifolds which comprise F are called its leaves, and the collection of all leaves is known as the leaf space of F. Foliations & Hierarchies Foliations Preliminaries Definitions This can be made precise as follows. Here, k is the dimension of F, n − k is its codimension, the submanifolds which comprise F are called its leaves, and the collection of all leaves is known as the leaf space of F. Foliations & Hierarchies Foliations Preliminaries Definitions This can be made precise as follows. Definition. A k-dimensional foliation of a manifold M = M n is a disjoint union F of connected, properly embedded dimension-k submanifolds of M which is locally homeomorphic to the direct n k n−k product decomposition of R into R × R and whose union equals M. Foliations & Hierarchies Foliations Preliminaries Definitions This can be made precise as follows. Definition. A k-dimensional foliation of a manifold M = M n is a disjoint union F of connected, properly embedded dimension-k submanifolds of M which is locally homeomorphic to the direct n k n−k product decomposition of R into R × R and whose union equals M. Here, k is the dimension of F, n − k is its codimension, the submanifolds which comprise F are called its leaves, and the collection of all leaves is known as the leaf space of F. 2 2. One can foliate R by parallel lines of constant slope α for α 2 [0; 1]. 2 3. Note that the foliation Fα of R by parallel lines of slope α will be invariant (setwise) under horizontal and vertical 2 translation. In particular, Fα is invariant by the Z action generated by these translations and hence descends to a 2 2 2 foliation of the torus T = R =Z . Foliations & Hierarchies Foliations Preliminaries Examples n k n−k 1. R = R × R as above. 2 In particular, Fα is invariant by the Z action generated by these translations and hence descends to a 2 2 2 foliation of the torus T = R =Z . Foliations & Hierarchies Foliations Preliminaries Examples n k n−k 1. R = R × R as above. 2 2. One can foliate R by parallel lines of constant slope α for α 2 [0; 1]. 2 3. Note that the foliation Fα of R by parallel lines of slope α will be invariant (setwise) under horizontal and vertical translation. Foliations & Hierarchies Foliations Preliminaries Examples n k n−k 1. R = R × R as above. 2 2. One can foliate R by parallel lines of constant slope α for α 2 [0; 1]. 2 3. Note that the foliation Fα of R by parallel lines of slope α will be invariant (setwise) under horizontal and vertical 2 translation. In particular, Fα is invariant by the Z action generated by these translations and hence descends to a 2 2 2 foliation of the torus T = R =Z . Foliations & Hierarchies Foliations Preliminaries Examples Figure 2 2 2 2 Fα descends to foliation of T = R =Z Foliations & Hierarchies Foliations Depth Depth One consideration often made in the study of foliations is regarding the depth of the foliation and/or its leaves. 1. A leaf L of F is depth zero if L is compact. 2. Having defined depth j ≤ k, a leaf L of F is said to be at depth k + 1 provided that (i) L − L is a union of depth j ≤ k leaves, and (ii) L − L contains at least one leaf of depth k. Definition. Under the same assumptions as above, F is said to be depth k if k = maxfdepth(L): L is a leaf of Fg: Foliations & Hierarchies Foliations Depth Definitions Let M = M 3 be compact and orientable and let F be a codimension-1 foliation on M. The depth of a leaf L of F is defined inductively as follows: 2. Having defined depth j ≤ k, a leaf L of F is said to be at depth k + 1 provided that (i) L − L is a union of depth j ≤ k leaves, and (ii) L − L contains at least one leaf of depth k. Definition. Under the same assumptions as above, F is said to be depth k if k = maxfdepth(L): L is a leaf of Fg: Foliations & Hierarchies Foliations Depth Definitions Let M = M 3 be compact and orientable and let F be a codimension-1 foliation on M. The depth of a leaf L of F is defined inductively as follows: 1. A leaf L of F is depth zero if L is compact. (i) L − L is a union of depth j ≤ k leaves, and (ii) L − L contains at least one leaf of depth k. Definition. Under the same assumptions as above, F is said to be depth k if k = maxfdepth(L): L is a leaf of Fg: Foliations & Hierarchies Foliations Depth Definitions Let M = M 3 be compact and orientable and let F be a codimension-1 foliation on M. The depth of a leaf L of F is defined inductively as follows: 1. A leaf L of F is depth zero if L is compact. 2. Having defined depth j ≤ k, a leaf L of F is said to be at depth k + 1 provided that and (ii) L − L contains at least one leaf of depth k. Definition. Under the same assumptions as above, F is said to be depth k if k = maxfdepth(L): L is a leaf of Fg: Foliations & Hierarchies Foliations Depth Definitions Let M = M 3 be compact and orientable and let F be a codimension-1 foliation on M. The depth of a leaf L of F is defined inductively as follows: 1. A leaf L of F is depth zero if L is compact. 2. Having defined depth j ≤ k, a leaf L of F is said to be at depth k + 1 provided that (i) L − L is a union of depth j ≤ k leaves, Definition. Under the same assumptions as above, F is said to be depth k if k = maxfdepth(L): L is a leaf of Fg: Foliations & Hierarchies Foliations Depth Definitions Let M = M 3 be compact and orientable and let F be a codimension-1 foliation on M. The depth of a leaf L of F is defined inductively as follows: 1. A leaf L of F is depth zero if L is compact. 2. Having defined depth j ≤ k, a leaf L of F is said to be at depth k + 1 provided that (i) L − L is a union of depth j ≤ k leaves, and (ii) L − L contains at least one leaf of depth k. Foliations & Hierarchies Foliations Depth Definitions Let M = M 3 be compact and orientable and let F be a codimension-1 foliation on M. The depth of a leaf L of F is defined inductively as follows: 1. A leaf L of F is depth zero if L is compact. 2. Having defined depth j ≤ k, a leaf L of F is said to be at depth k + 1 provided that (i) L − L is a union of depth j ≤ k leaves, and (ii) L − L contains at least one leaf of depth k. Definition. Under the same assumptions as above, F is said to be depth k if k = maxfdepth(L): L is a leaf of Fg: Figure 3 The Reeb foliation of V 2 = D2 × S1 Foliations & Hierarchies Foliations Depth Example{The Reeb Foliation One of the most commonly-encountered examples of a finite-depth foliation is the Reeb foliation of the solid torus V 2 = D2 × S1. Foliations & Hierarchies Foliations Depth Example{The Reeb Foliation One of the most commonly-encountered examples of a finite-depth foliation is the Reeb foliation of the solid torus V 2 = D2 × S1. Figure 3 The Reeb foliation of V 2 = D2 × S1 • The boundary @V 2 = T 2 is a (compact) leaf. • All other leaves are \interior leaves," all of which are topological planes which spiral towards the boundary torus.