Sutured Manifold Hierarchies and Finite-Depth Foliations
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1 Sutured Manifold Hierarchies and Finite-Depth Christopher Stover Florida State University Topology Seminar November 4, 2014
2 Outline Preliminaries Depth Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Sutured Manifold Decompositions Example: Decomposing Sutured D 2 S 1 into Sutured B 3 Sutured Manifold Hierarchies Main Result
3 Part I Preliminaries Depth Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Sutured Manifold Decompositions Example: Decomposing Sutured D 2 S 1 into Sutured B 3 Sutured Manifold Hierarchies Main Result
4 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 decomposition of R n into R k R n k.
5 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 decomposition of R n into R k R n k. Figure 1 R n = R k R n k for n = 3 and k = 2
6 Preliminaries Definitions This can be made precise as follows.
7 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 product decomposition of R n into R k R n k and whose union equals M.
8 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 product decomposition of R n into R k R n k 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.
9 Preliminaries Examples 1. R n = R k R n k as above.
10 Preliminaries Examples 1. R n = R k R n k as above. 2. One can foliate R 2 by parallel lines of constant slope α for α [0, ]. 3. Note that the foliation F α of R 2 by parallel lines of slope α will be invariant (setwise) under horizontal and vertical translation.
11 Preliminaries Examples 1. R n = R k R n k as above. 2. One can foliate R 2 by parallel lines of constant slope α for α [0, ]. 3. Note that the foliation F α of R 2 by parallel lines of slope α will be invariant (setwise) under horizontal and vertical translation. In particular, F α is invariant by the Z 2 action generated by these translations and hence descends to a foliation of the torus T 2 = R 2 /Z 2.
12 Preliminaries Examples Figure 2 F α descends to foliation of T 2 = R 2 /Z 2
13 Depth Depth One consideration often made in the study of foliations is regarding the depth of the foliation and/or its leaves.
14 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:
15 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.
16 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
17 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,
18 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.
19 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 = max{depth(l) : L is a leaf of F}.
20 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 = D 2 S 1.
21 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 = D 2 S 1. Figure 3 The Reeb foliation of V 2 = D 2 S 1
22 Depth Example The Reeb Foliation Informally, the Reeb foliation of V 2 can be described as follows:
23 Depth Example The Reeb Foliation Informally, the Reeb foliation of V 2 can be described as follows: The boundary V 2 = T 2 is a (compact) leaf.
24 Depth Example The Reeb Foliation Informally, the Reeb foliation of V 2 can be described as follows: 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. These are non-compact.
25 Depth Example The Reeb Foliation Informally, the Reeb foliation of V 2 can be described as follows: 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. These are non-compact. In particular, the toral boundary leaf is a depth-zero leaf and the interior leaves are all depth-one. Hence, the Reeb foliation is a depth-one foliation.
26 Depth Constructing Higher-Depth Leaves One can modify an existing finite-depth foliation to get a foliation of higher-depth. For example:
27 Depth Constructing Higher-Depth Leaves One can modify an existing finite-depth foliation to get a foliation of higher-depth. For example: Start with a depth-zero foliation in two dimensions...
28 Depth Constructing Higher-Depth Leaves...identify a neighborhood of a curve γ... Figure 4 Neighorhood of the curve γ, which is shown in red
29 Depth Constructing Higher-Depth Leaves...delete it...
30 Depth Constructing Higher-Depth Leaves...and re-glue the interior leaves by a non-identity map while leaving the boundary leaves fixed, e.g.:
31 Depth Constructing Higher-Depth Leaves Similarly, one can construct a depth-one foliation from depth-zero leaves in three dimensions by removing a product neighborhood of some curve γ...
32 Depth Constructing Higher-Depth Leaves Similarly, one can construct a depth-one foliation from depth-zero leaves in three dimensions by removing a product neighborhood of some curve γ...
33 Depth Constructing Higher-Depth Leaves...and gluing the resulting leaves via a non-identity map.
34 Depth Constructing Higher-Depth Leaves...and gluing the resulting leaves via a non-identity map.
35 Depth Constructing Higher-Depth Leaves If this is done on interior depth-zero leaves while leaving some depth-zero boundary leaf fixed, the result will be a foliation with a depth-one leaf in the interior which has infinite genus and which limits on the fixed boundary depth-zero leaf.
36 Depth Constructing Higher-Depth Leaves If this is done on interior depth-zero leaves while leaving some depth-zero boundary leaf fixed, the result will be a foliation with a depth-one leaf in the interior which has infinite genus and which limits on the fixed boundary depth-zero leaf. Figure 5 A depth-one leaf resulting from the above gluing process
37 Depth Constructing Higher-Depth Leaves The same process can be repeated on depth-one leaves to get a depth-two leaf: First, remove a neighborhood of a curve γ...
38 Depth Constructing Higher-Depth Leaves The same process can be repeated on depth-one leaves to get a depth-two leaf: First, remove a neighborhood of a curve γ...
39 Depth Constructing Higher-Depth Leaves...and then gluing as above...
40 Depth Constructing Higher-Depth Leaves...and then gluing as above...
41 Depth Constructing Higher-Depth Leaves The resulting depth-two leaf may resemble something like...
42 Depth Constructing Higher-Depth Leaves The resulting depth-two leaf may resemble something like... Figure 6 A depth-two leaf, as described by Candel and Conlon N
43 Sutured Manifolds, Decompositions, and Hierarchies Part II Preliminaries Depth Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Sutured Manifold Decompositions Example: Decomposing Sutured D 2 S 1 into Sutured B 3 Sutured Manifold Hierarchies Main Result
44 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Definitions A sutured manifold (M, γ) is a compact oriented 3-manifold M together with a set γ M of pairwise disjoint annuli A(γ) and tori T (γ) subject to the following conditions:
45 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Definitions A sutured manifold (M, γ) is a compact oriented 3-manifold M together with a set γ M of pairwise disjoint annuli A(γ) and tori T (γ) subject to the following conditions: 1. Each component of A(γ) contains a homologically nontrivial oriented simple closed curve called a suture.
46 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Definitions A sutured manifold (M, γ) is a compact oriented 3-manifold M together with a set γ M of pairwise disjoint annuli A(γ) and tori T (γ) subject to the following conditions: 1. Each component of A(γ) contains a homologically nontrivial oriented simple closed curve called a suture. 2. Every component of R(γ) def = M γ is oriented, and the orientations on R(γ) must be coherent with respect to s(γ).
47 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Examples of Sutured Manifolds R + R- R- R + Figure 7 M = D 2 S 1 with four sutures, shown in red
48 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifolds Examples of Sutured Manifolds R- R + R- R + Figure 8 M = D 2 S 1 with two sutures, one red and one green
49 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition Let (M, γ) be a sutured manifold and let S be a properly embedded surface in M such that (i) no component of S bounds a disc in R(γ),
50 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition Let (M, γ) be a sutured manifold and let S be a properly embedded surface in M such that (i) no component of S bounds a disc in R(γ), (ii) no component of S is a disc D with D R(γ),
51 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition Let (M, γ) be a sutured manifold and let S be a properly embedded surface in M such that (i) no component of S bounds a disc in R(γ), (ii) no component of S is a disc D with D R(γ), and (iii) for every component λ of S γ, one of the following holds:
52 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition Let (M, γ) be a sutured manifold and let S be a properly embedded surface in M such that (i) no component of S bounds a disc in R(γ), (ii) no component of S is a disc D with D R(γ), and (iii) for every component λ of S γ, one of the following holds: 1. λ is a properly embedded nonseparating arc in γ.
53 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition Let (M, γ) be a sutured manifold and let S be a properly embedded surface in M such that (i) no component of S bounds a disc in R(γ), (ii) no component of S is a disc D with D R(γ), and (iii) for every component λ of S γ, one of the following holds: 1. λ is a properly embedded nonseparating arc in γ. 2. λ is a simple closed curve in an annular component A of γ which is in the same homology class as A s(γ).
54 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition Let (M, γ) be a sutured manifold and let S be a properly embedded surface in M such that (i) no component of S bounds a disc in R(γ), (ii) no component of S is a disc D with D R(γ), and (iii) for every component λ of S γ, one of the following holds: 1. λ is a properly embedded nonseparating arc in γ. 2. λ is a simple closed curve in an annular component A of γ which is in the same homology class as A s(γ). 3. λ is a homotopically nontrivial curve in a toral component T of γ so that, if δ is another component of T S, then λ and δ represent the same homology class in H 1 (T ).
55 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition (Cont d) Then, S defines a sutured manifold decomposition S (M, γ) (M, γ ) where:
56 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition (Cont d) Then, S defines a sutured manifold decomposition S (M, γ) (M, γ ) where: M = M N(S) where N(S) denotes a product neighborhood of S in M.
57 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition (Cont d) Then, S defines a sutured manifold decomposition S (M, γ) (M, γ ) where: M = M N(S) where N(S) denotes a product neighborhood of S in M. S + and S denotes the components of N(S) M whose normal vector points out of and into M, respectively.
58 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition (Cont d) Then, S defines a sutured manifold decomposition S (M, γ) (M, γ ) where: M = M N(S) where N(S) denotes a product neighborhood of S in M. S + and S denotes the components of N(S) M whose normal vector points out of and into M, respectively. γ = (γ M ) N(S + R (γ)) N(S R + (γ)).
59 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition (Cont d) Then, S defines a sutured manifold decomposition S (M, γ) (M, γ ) where: M = M N(S) where N(S) denotes a product neighborhood of S in M. S + and S denotes the components of N(S) M whose normal vector points out of and into M, respectively. γ = (γ M ) N(S + R (γ)) N(S R + (γ)). R + (γ ) = ((R + (γ) M ) S +) γ.
60 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Decompositions Definition (Cont d) Then, S defines a sutured manifold decomposition S (M, γ) (M, γ ) where: M = M N(S) where N(S) denotes a product neighborhood of S in M. S + and S denotes the components of N(S) M whose normal vector points out of and into M, respectively. γ = (γ M ) N(S + R (γ)) N(S R + (γ)). R + (γ ) = ((R + (γ) M ) S +) γ. R (γ ) = ((R (γ) M ) S ) γ.
61 Sutured Manifolds, Decompositions, and Hierarchies Example: Decomposing Sutured D 2 S 1 into Sutured B 3 Example Begin with M = D 2 S 1 with two sutures, one red and one green, and a properly embedded disk S, oriented as shown S - Figure 9 (M, γ) as described above
62 Sutured Manifolds, Decompositions, and Hierarchies Example: Decomposing Sutured D 2 S 1 into Sutured B 3 Example (Cont d) Next, obtain M by removing a product neighborhood N(S) of S in M as shown below. Note the labeling of S +, S, and the intersections used to define γ S' + S' S' + R - ( ) S' - R + ( ) Figure 10 (M, γ) as described above
63 Sutured Manifolds, Decompositions, and Hierarchies Example: Decomposing Sutured D 2 S 1 into Sutured B 3 Example (Cont d) Defining γ as above, one sees that the manifold M now has only one suture, shown below in orange S' + S' Figure 11 (M, γ ) as described above
64 Sutured Manifolds, Decompositions, and Hierarchies Example: Decomposing Sutured D 2 S 1 into Sutured B 3 Example (Cont d) And finally, note that (M, γ ) deformation retracts onto the manifold M = B 3 with one suture at the equator. + - Figure 12 The final result of the decomposition
65 Sutured Manifolds, Decompositions, and Hierarchies Sutured Manifold Hierarchies Definition A sutured manifold hierarchy is a sequence of sutured manifold decompositions (M 0, γ 0 ) S 1 (M 1, γ 1 ) S 2 S (M 2, γ 2 ) n (Mn, γ n ) where (M n, γ n ) = (R I, R I) and R + (γ n ) = R {1} for some surface R. Here, I = [0, 1] and R is some surface.
66 Sutured Manifolds, Decompositions, and Hierarchies Main Result Ginormous Main Theorem of Awesome Non-Triviality and Awesomeness The following is a fundamental result of David Gabai:
67 Sutured Manifolds, Decompositions, and Hierarchies Main Result Ginormous Main Theorem of Awesome Non-Triviality and Awesomeness The following is a fundamental result of David Gabai: Theorem. Suppose M is connected, and (M, γ) has a sutured manifold hierarchy (M, γ) = (M 0, γ 0 ) S1 (M 1, γ 1 ) S2 S (M 2, γ 2 ) n (Mn, γ n ) so that no component of R(γ i ) is a compressing torus. Then there exist transversely oriented foliations F 0 and F 1 of M which are well-behaved and where F 0 is of finite-depth.
68 Sutured Manifolds, Decompositions, and Hierarchies Main Result To Be Continued... To see more, come to my ATE talk on December 1!!
69 Sutured Manifolds, Decompositions, and Hierarchies Main Result Thank you!
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