Surfaces in Seifert fibered spaces

Jennifer Schultens

University of California, Davis

June 26, 2019

Jennifer Schultens Surfaces in Seifert fibered spaces We also study submanifolds. E.g., 1-dimensional knots in 3-dimensional space.

I will discuss the totality of surfaces in a particular class of 3-manifolds.

Rather than giving you a formal definition of manifold, in particular 3-manifold, I will describe some of my favorite examples.

General comments

Low-dimensional topology concerns the topology of manifolds in 1, 2, 3, and 4 dimensions.

Jennifer Schultens Surfaces in Seifert fibered spaces I will discuss the totality of surfaces in a particular class of 3-manifolds.

Rather than giving you a formal definition of manifold, in particular 3-manifold, I will describe some of my favorite examples.

General comments

Low-dimensional topology concerns the topology of manifolds in 1, 2, 3, and 4 dimensions.

We also study submanifolds. E.g., 1-dimensional knots in 3-dimensional space.

Jennifer Schultens Surfaces in Seifert fibered spaces Rather than giving you a formal definition of manifold, in particular 3-manifold, I will describe some of my favorite examples.

General comments

Low-dimensional topology concerns the topology of manifolds in 1, 2, 3, and 4 dimensions.

We also study submanifolds. E.g., 1-dimensional knots in 3-dimensional space.

I will discuss the totality of surfaces in a particular class of 3-manifolds.

Jennifer Schultens Surfaces in Seifert fibered spaces General comments

Low-dimensional topology concerns the topology of manifolds in 1, 2, 3, and 4 dimensions.

We also study submanifolds. E.g., 1-dimensional knots in 3-dimensional space.

I will discuss the totality of surfaces in a particular class of 3-manifolds.

Rather than giving you a formal definition of manifold, in particular 3-manifold, I will describe some of my favorite examples.

Jennifer Schultens Surfaces in Seifert fibered spaces Examples of manifolds: Not knot

Definition Let K be a knot. The complement of K is

3 C(K) = S − η(K)

where η(K) is a regular neighborhood of K.

Jennifer Schultens Surfaces in Seifert fibered spaces Not knot

Figure: The complement of the unknot

Jennifer Schultens Surfaces in Seifert fibered spaces Not knot

Figure: The complement of a knot

Jennifer Schultens Surfaces in Seifert fibered spaces Torus knots

Figure: T(2, 3), also known as the trefoill

Jennifer Schultens Surfaces in Seifert fibered spaces Satellite knots

I-’ 4 KC sot’? kftco V

i2

(Jattant

KcV •

Figure: Satellite knots

Jennifer Schultens Surfaces in Seifert fibered spaces A proof can be found in M. Kapovich’s “Hyperbolic manifolds and discrete groups”.

Thurston’s theorem

Theorem 3 (Trichotomy for knots) Let K be a knot in S . Then exactly one of the following holds: K is a torus knot; K is a satellite knot; K is hyperbolic.

Jennifer Schultens Surfaces in Seifert fibered spaces Thurston’s theorem

Theorem 3 (Trichotomy for knots) Let K be a knot in S . Then exactly one of the following holds: K is a torus knot; K is a satellite knot; K is hyperbolic.

A proof can be found in M. Kapovich’s “Hyperbolic manifolds and discrete groups”.

Jennifer Schultens Surfaces in Seifert fibered spaces Examples of manifolds: 2-fold branched cover

2-fold branched cover of a knot: The 2-fold branched cover of a knot has two points for every point in the knot complement and one point for every point on the knot.

Jennifer Schultens Surfaces in Seifert fibered spaces Examples of manifolds: 2-fold branched cover

Figure: A Seifert surface

Theorem (Seifert) Every knot admits a Seifert surface.

Jennifer Schultens Surfaces in Seifert fibered spaces Examples of manifolds: Seifert fibered spaces

Seifert fibered spaces are a family of 3-dimensional spaces (3-manifolds) whose members are classified by a finite set of invariants. The structure of these manifolds allows us to concretely describe essential surfaces embedded in them.

We are interested in the collection of all surfaces in a given Seifert fibered space. However, we must restrict our attention to essential surfaces in Seifert fibered spaces. The Kakimizu complex of a knot provides an example of how to encode surfaces. We discus the analogous concept for Seifert fibered spaces.

We will always assume that curves in surfaces are simple and that surfaces in 3-manifolds are properly embedded.

Jennifer Schultens Surfaces in Seifert fibered spaces Fibered solid tori

identify after twist

Figure: A fibered solid torus

Jennifer Schultens Surfaces in Seifert fibered spaces The fibered solid torus V (ν, µ) is obtained from the solid cylinder D2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0, 1] for y ∈ D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism.

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g.c.d.(ν, µ) = 1.

Jennifer Schultens Surfaces in Seifert fibered spaces 2 Intervals of the form y × [0, 1] for y ∈ D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism.

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g.c.d.(ν, µ) = 1. The fibered solid torus V (ν, µ) is obtained from the solid cylinder D2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2 2πν about its center point x by µ .

Jennifer Schultens Surfaces in Seifert fibered spaces All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism.

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g.c.d.(ν, µ) = 1. The fibered solid torus V (ν, µ) is obtained from the solid cylinder D2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0, 1] for y ∈ D match up to form simple closed curves called fibers.

Jennifer Schultens Surfaces in Seifert fibered spaces If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism.

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g.c.d.(ν, µ) = 1. The fibered solid torus V (ν, µ) is obtained from the solid cylinder D2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0, 1] for y ∈ D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber.

Jennifer Schultens Surfaces in Seifert fibered spaces The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism.

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g.c.d.(ν, µ) = 1. The fibered solid torus V (ν, µ) is obtained from the solid cylinder D2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0, 1] for y ∈ D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber.

Jennifer Schultens Surfaces in Seifert fibered spaces Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism.

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g.c.d.(ν, µ) = 1. The fibered solid torus V (ν, µ) is obtained from the solid cylinder D2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0, 1] for y ∈ D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber.

Jennifer Schultens Surfaces in Seifert fibered spaces Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g.c.d.(ν, µ) = 1. The fibered solid torus V (ν, µ) is obtained from the solid cylinder D2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0, 1] for y ∈ D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism.

Jennifer Schultens Surfaces in Seifert fibered spaces Hence µ is an invariant of V (ν, µ) and ν is an invariant up to sign and mod µ. In other words, V (ν, µ) is isomorphic to V (ν0, µ0) via a fiber preserving homeomorphism if and only if µ0 = µ and ν0 = ±ν0(modµ). If we keep track of orientations we can normalize the invariant ν so that 0 ≤ ν < µ.

Fibered solid tori

Every fiber in a fibered solid torus except the singular one represents the element µ ∈ π1(V (ν, µ)) = Z which is generated by the class of the central fiber.

Jennifer Schultens Surfaces in Seifert fibered spaces In other words, V (ν, µ) is isomorphic to V (ν0, µ0) via a fiber preserving homeomorphism if and only if µ0 = µ and ν0 = ±ν0(modµ). If we keep track of orientations we can normalize the invariant ν so that 0 ≤ ν < µ.

Fibered solid tori

Every fiber in a fibered solid torus except the singular one represents the element µ ∈ π1(V (ν, µ)) = Z which is generated by the class of the central fiber. Hence µ is an invariant of V (ν, µ) and ν is an invariant up to sign and mod µ.

Jennifer Schultens Surfaces in Seifert fibered spaces If we keep track of orientations we can normalize the invariant ν so that 0 ≤ ν < µ.

Fibered solid tori

Every fiber in a fibered solid torus except the singular one represents the element µ ∈ π1(V (ν, µ)) = Z which is generated by the class of the central fiber. Hence µ is an invariant of V (ν, µ) and ν is an invariant up to sign and mod µ. In other words, V (ν, µ) is isomorphic to V (ν0, µ0) via a fiber preserving homeomorphism if and only if µ0 = µ and ν0 = ±ν0(modµ).

Jennifer Schultens Surfaces in Seifert fibered spaces Fibered solid tori

Every fiber in a fibered solid torus except the singular one represents the element µ ∈ π1(V (ν, µ)) = Z which is generated by the class of the central fiber. Hence µ is an invariant of V (ν, µ) and ν is an invariant up to sign and mod µ. In other words, V (ν, µ) is isomorphic to V (ν0, µ0) via a fiber preserving homeomorphism if and only if µ0 = µ and ν0 = ±ν0(modµ). If we keep track of orientations we can normalize the invariant ν so that 0 ≤ ν < µ.

Jennifer Schultens Surfaces in Seifert fibered spaces Definition Fibers are called regular (or singular) if they are regular (or singular) fibers in the fibered solid torus containing them.

Seifert fibered spaces

For simplicity, we consider only orientable manifolds.

Definition A Seifert Fibered Space is a compact connected 3-manifold M that is a union of disjoint circles called fibers such that each fiber has a neighborhood that is homeomorphic to a fibered solid torus.

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert fibered spaces

For simplicity, we consider only orientable manifolds.

Definition A Seifert Fibered Space is a compact connected 3-manifold M that is a union of disjoint circles called fibers such that each fiber has a neighborhood that is homeomorphic to a fibered solid torus.

Definition Fibers are called regular (or singular) if they are regular (or singular) fibers in the fibered solid torus containing them.

Jennifer Schultens Surfaces in Seifert fibered spaces Remark 2: Because it is compact, a Seifert fibered space is covered by finitely many fibered solid tori.

Remark 3: A Seifert fibered space will have only a finite number of singular fibers.

Seifert fibered spaces

Remark 1: Fibers of a Seifert fibered space are either regular or singular. (Not both.) Moreover, a fiber of a Seifert fibered space uniquely determines normalized ν, µ.

Jennifer Schultens Surfaces in Seifert fibered spaces Remark 3: A Seifert fibered space will have only a finite number of singular fibers.

Seifert fibered spaces

Remark 1: Fibers of a Seifert fibered space are either regular or singular. (Not both.) Moreover, a fiber of a Seifert fibered space uniquely determines normalized ν, µ.

Remark 2: Because it is compact, a Seifert fibered space is covered by finitely many fibered solid tori.

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert fibered spaces

Remark 1: Fibers of a Seifert fibered space are either regular or singular. (Not both.) Moreover, a fiber of a Seifert fibered space uniquely determines normalized ν, µ.

Remark 2: Because it is compact, a Seifert fibered space is covered by finitely many fibered solid tori.

Remark 3: A Seifert fibered space will have only a finite number of singular fibers.

Jennifer Schultens Surfaces in Seifert fibered spaces (We will assume that this surface is orientable.) The marked points are called singular points. Remark: A closed oriented Seifert fibered space M is completely determined by a set of invariants called a signature:

{g, b, e | (α1, β1),..., (αr , βr )}

(More on the Euler number e in a moment.)

Seifert fibered spaces

Given a Seifert fibered space, we identify each fiber to a point to obtain the base orbifold, a compact surface with a finite number of marked points.

Jennifer Schultens Surfaces in Seifert fibered spaces The marked points are called singular points. Remark: A closed oriented Seifert fibered space M is completely determined by a set of invariants called a signature:

{g, b, e | (α1, β1),..., (αr , βr )}

(More on the Euler number e in a moment.)

Seifert fibered spaces

Given a Seifert fibered space, we identify each fiber to a point to obtain the base orbifold, a compact surface with a finite number of marked points. (We will assume that this surface is orientable.)

Jennifer Schultens Surfaces in Seifert fibered spaces Remark: A closed oriented Seifert fibered space M is completely determined by a set of invariants called a signature:

{g, b, e | (α1, β1),..., (αr , βr )}

(More on the Euler number e in a moment.)

Seifert fibered spaces

Given a Seifert fibered space, we identify each fiber to a point to obtain the base orbifold, a compact surface with a finite number of marked points. (We will assume that this surface is orientable.) The marked points are called singular points.

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert fibered spaces

Given a Seifert fibered space, we identify each fiber to a point to obtain the base orbifold, a compact surface with a finite number of marked points. (We will assume that this surface is orientable.) The marked points are called singular points. Remark: A closed oriented Seifert fibered space M is completely determined by a set of invariants called a signature:

{g, b, e | (α1, β1),..., (αr , βr )}

(More on the Euler number e in a moment.)

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert fibered spaces

Figure: A base orbifold for a Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert fibered spaces

Figure: A base orbifold for a Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces Compressible/incompressible surfaces

Definition The simple closed curve α in a surface F is inessential if it bounds a disk. If it is not inessential, then it is essential.

Definition The surface F in the 3-manifold M is compressible if there is an essential simple closed curve in F that bounds a disk in M. If F is not compressible, then it is incompressible.

Jennifer Schultens Surfaces in Seifert fibered spaces Compressible/incompressible surfaces

D

Figure: A compressible surface

Jennifer Schultens Surfaces in Seifert fibered spaces Incompressible surfaces in Seifert fibered spaces

Definition If the surface F is everywhere transverse to the fibers of the Seifert fibered space M, then F is said to be horizontal.

1

x [0, 1] identify

Figure: A surface bundle over the circle

Jennifer Schultens Surfaces in Seifert fibered spaces Incompressible surfaces in Seifert fibered spaces

Remark 1: A horizontal surface in a Seifert fibered space is necessarily incompressible.

Remark 2: A Seifert fibered space admits horizontal surfaces if and only if its Euler number is 0.

Jennifer Schultens Surfaces in Seifert fibered spaces Incompressible surfaces in Seifert fibered spaces

Definition If every fiber of the Seifert fibered space M that meets the surface F is entirely contained in F , then F is said to be vertical.

Figure: A vertical surface in a Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces Incompressible surfaces in Seifert fibered spaces

CAUTION:

Figure: Not a vertical surface in an orientable Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces Incompressible surfaces in Seifert fibered spaces

identify after twist through π

Figure: Only a vertical M¨obiusband in a fibered solid Klein bottle would project to an arc

Jennifer Schultens Surfaces in Seifert fibered spaces Incompressible surfaces in Seifert fibered spaces

Remark: A vertical surface in a Seifert fibered space can be compressible or incompressible. If it is compressible, then it bounds a fibered solid torus. (Dehn’s lemma + Loop theorem)

Jennifer Schultens Surfaces in Seifert fibered spaces A theorem of Jaco

Theorem (Jaco) Let M be an orientable Seifert fibered space with orientable base orbifold. If F is a connected, two-sided, incompressible surface in M, then one of the following alternatives holds: (i) F is a disk or an annulus and F is parallel into ∂M; (ii) F does not separate M and F is a fiber in a fiberation of M as a surface bundle over the circle (in particular, F is horizontal); (iii) F is an annulus or a torus and, after isotopy, F consists of fibers, in some Seifert fiberation of M.

Jennifer Schultens Surfaces in Seifert fibered spaces The Kakimizu complex

Introduction to the Kakimizu complex

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert surfaces

The Kakimizu complex evolved from the study of spanning surfaces for knots.

Definition 3 Given an knot K in S , a compact connected orientable surface 3 that represents a generator of H2(S , K) is called a Seifert surface. More concretely, a Seifert surface is an embedded orientable surface S such that ∂S = K.

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert Surfaces

Figure: A Seifert surface

Theorem (Seifert) Every knot admits a Seifert surface.

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert Surfaces

Figure: Two knots

Theorem 3 (Eisner) Many knots K in S admit non-isotopic Seifert surfaces.

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert Surfaces

Figure: Connect sum of knots with swallow-follow torus

Jennifer Schultens Surfaces in Seifert fibered spaces Seifert Surfaces

Figure: Schematic for spinning Seifert surface around swallow-follow torus

Jennifer Schultens Surfaces in Seifert fibered spaces Kakimizu complex

Definition 3 The vertices of the Kakimizu complex Kak(K) of a knot K in S are given by the isotopy classes of minimal genus Seifert surfaces for K. The n-simplices of the Kakimizu complex of K, for n > 1, are given by n-tuples of vertices that admit pairwise disjoint representatives.

Jennifer Schultens Surfaces in Seifert fibered spaces Example II: Hyperbolic knots have finite Kakimizu complexes.

Examples of Kakimizu complexes

Example I: Fibered knots have trivial Kakimizu complexes.

Jennifer Schultens Surfaces in Seifert fibered spaces Examples of Kakimizu complexes

Example I: Fibered knots have trivial Kakimizu complexes.

Example II: Hyperbolic knots have finite Kakimizu complexes.

Jennifer Schultens Surfaces in Seifert fibered spaces Topology of the Kakimizu complex

Theorem (Scharlemann-Thompson, Kakimizu) The Kakimizu complex is connected.

Theorem (S) The Kakimizu complex of a knot K is a flag complex.

Jennifer Schultens Surfaces in Seifert fibered spaces Topology of the Kakimizu complex

Theorem (Banks) The Kakimizu complex of a knot is not necessarily locally finite.

Theorem (Banks) Approximately: The Kakimizu complex of a sum of two knots is the product of their Kakimizu complexes with Z.

Jennifer Schultens Surfaces in Seifert fibered spaces Geometry of the Kakimizu complex

Theorem (Przytycki-S) The Kakimizu complex of a knot is contractible.

Theorem (Johnson-Pelayo-Wilson) The Kakimizu complex of a knot is quasi-Euclidean.

Jennifer Schultens Surfaces in Seifert fibered spaces Just for fun: Find a Seifert surface

Figure: Whitehead double of the figure eight knot

Jennifer Schultens Surfaces in Seifert fibered spaces What’s special about Seifert surfaces of knots?

Figure: A Seifert surface

Jennifer Schultens Surfaces in Seifert fibered spaces Fact: This group is dual to 1 3 3 H (S − η(K)) = Hom(H1(S − η(K)) = Hom(Z) = Z Consequence: There is a canonical infinite cyclic cover of 3 (S , K) in which we can compare lifts of distinct Seifert surfaces.

What’s special about Seifert surfaces of knots?

Key Property: They represent the generator of 3 3 3 H2(S , K) = H2(S − η(K), ∂S − η(K)).

Jennifer Schultens Surfaces in Seifert fibered spaces Consequence: There is a canonical infinite cyclic cover of 3 (S , K) in which we can compare lifts of distinct Seifert surfaces.

What’s special about Seifert surfaces of knots?

Key Property: They represent the generator of 3 3 3 H2(S , K) = H2(S − η(K), ∂S − η(K)). Fact: This group is dual to 1 3 3 H (S − η(K)) = Hom(H1(S − η(K)) = Hom(Z) = Z

Jennifer Schultens Surfaces in Seifert fibered spaces What’s special about Seifert surfaces of knots?

Key Property: They represent the generator of 3 3 3 H2(S , K) = H2(S − η(K), ∂S − η(K)). Fact: This group is dual to 1 3 3 H (S − η(K)) = Hom(H1(S − η(K)) = Hom(Z) = Z Consequence: There is a canonical infinite cyclic cover of 3 (S , K) in which we can compare lifts of distinct Seifert surfaces.

Jennifer Schultens Surfaces in Seifert fibered spaces The covering space associated with α

Sm

Sm−1

Sm−2

S’0

Figure: No edge of a Kakimizu complex

Jennifer Schultens Surfaces in Seifert fibered spaces What’s special about Seifert surfaces?

Generalization of Key Property: In an orientable 3-manifold 3 3 3 M , choose a primitive element α of H2(M , ∂M ). Or, more n n generally, choose a primitive element α of Hn−1(M , ∂M ).

Jennifer Schultens Surfaces in Seifert fibered spaces Kakimizu complex of a 3-manifold

Let α be a primitive element of H2(M, ∂M, Z).

Definition We define the Kakimizu complex of M with respect to α, denoted Kak(M, α): Vertices are given by weighted multi-surfaces that represent α. The multi-surface is required to be Thurston norm minimizing, to have connected complement, and is considered up to isotopy.

Jennifer Schultens Surfaces in Seifert fibered spaces Kakimizu complex of a 3-manifold

Definition The vertices v, v 0 span an edge e = (v, v 0) if and only if representatives of v, v 0 can be chosen so that the complement of a lift of one respresentative to the covering space associated with α intersects exactly two lifts of the complement of the other representative. (This condition implies that the representatives are disjoint, but not vice versa.) Kak(M, α) is the flag complex with the vertices and edges described above. (I.e., add simplicies whenever possible.)

Jennifer Schultens Surfaces in Seifert fibered spaces The topology and geometry of Kakimizu complexes of 2- and 3-manifolds

Theorem (Scharlemann-Thompson, Kakimizu, Przytycki-S, S) Every Kakimizu complex of a 2- or 3-manifold is connected.

Theorem (Przytycki-S, S) Every Kakimizu complex of a 2- or 3-manifold is contractible.

Theorem (S) A Kakimizu complex of a 2- or 3-manifold need not be quasi Euclidean.

Jennifer Schultens Surfaces in Seifert fibered spaces The Kakimizu complexes of Seifert fibered spaces

The Kakimizu complexes of Seifert fibered spaces

Jennifer Schultens Surfaces in Seifert fibered spaces Horizontal Kakimizu complexes of Seifert fibered spaces

Theorem Let M be an orientable Seifert fibered space with orientable base orbifold. Let α ∈ H2(M, ∂M, Z) be a primitive relative second homology class that is represented by a horizontal surface. Then Kak(M, α) is trivial, i.e., consists of a single vertex.

This result also follows from work of Jaco. Question: What if α is represented by a vertical surface?

Jennifer Schultens Surfaces in Seifert fibered spaces Vertical Kakimizu complexes of Seifert fibered spaces

Theorem (S) Let M be a Seifert fibered space with orientable base space. Let α ∈ H2(M, ∂M, Z) be a homology class that is represented by a vertical surface. Then Kak(M, α) is isomorphic to the corresponding Kakimizu complex of the surface obtained from the base orbifold of M by removing neighborhoods of the singular points.

Jennifer Schultens Surfaces in Seifert fibered spaces Vertical Kakimizu complexes of Seifert fibered spaces

ξ

~ b’

~ ~ b A

−ξ

Figure: A lune

Jennifer Schultens Surfaces in Seifert fibered spaces The surface complex of a Seifert fibered space

The surface complex of a Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces Finegold’s torus complex

In her dissertation, written under the direction of Daryl Cooper in 2010, Brie Finegold studied torus complexes in all dimensions from an algebraic point of view. In dimension 2, where vertices are simple closed curves in the 2-torus, Finegold’s torus complex coincides with the curve complex of the torus, i.e., the Farey graph.

In dimension 3, where the vertices are isotopy classes of 2-tori embedded in the 3-torus, there are no disjoint non isotopic essential surfaces, and edges are defined as pairs of vertices with representatives meeting in a single simple closed curve.

Jennifer Schultens Surfaces in Seifert fibered spaces Finegold’s torus complex

Theorem (Finegold) The torus complex in dimension 3 is connected.

Theorem (Finegold) The torus complex in dimension 3 is simply-connected.

Theorem (Finegold) The torus complex in dimension 3 has diameter 2.

Jennifer Schultens Surfaces in Seifert fibered spaces The surface complex

Let M be a compact orientable 3-manifold. We define a sequence of complexes {Si (M)}, and the surface complex, S(M), of M as follows:

Vertices in {Si (M)} and S(M) correspond to isotopy classes of compact connected orientable essential (incompressible, boundary incompressible and not boundary parallel) surfaces (properly embedded) in M.

Jennifer Schultens Surfaces in Seifert fibered spaces The surface complex

A pair of distinct vertices (v1, v2) spans an edge in S0(M) if and only if v1 and v2 admit disjoint representatives. Inductively, we construct a sequence of complexes, {Si (M)}, whose vertices coincide, for all i, with those of S0(M). Given Si (M), the pair of vertices {v1, v2} spans an edge in Si+1(M) if and only if v1 and v2 lie in distinct components of Si (M) and admit representatives whose intersection has i + 1 components.

For all i, Si (M) is a flag complex. The surface complex of M, denoted S(M), is defined to be

Si0 (M), where i0 is the smallest natural number such that Si0 (M) is connected.

Jennifer Schultens Surfaces in Seifert fibered spaces Surface complexes of Seifert fibered spaces

Theorem If M is a totally orientable Seifert fibered space with nonzero Euler number, then S(M) is isomorphic to the curve complex of Qˆ.

Theorem If M is a totally orientable Seifert fibered space with base orbifold of genus 0 and Euler number 0, then S(M) contains a subcomplex isomorphic to the curve complex of the surface obtained from Qˆ. Moreover, S(M) is contained in the cone on this subcomplex.

Corollary (Special case) If M is a totally orientable Seifert fibered space with base orbifold of genus 0, Euler number 0, and either 4 or 5 exceptional fibers with identical invariants, then S(M) is isomorphic to the cone on the curve complex of Qˆ.

Jennifer Schultens Surfaces in Seifert fibered spaces Surface complexes of Seifert fibered spaces

Theorem If M is a totally orientable Seifert fibered space with Euler number 0 and base orbifold of positive genus, then S(M) contains a subcomplex isomorphic to the curve complex of the surface obtained from Qˆ. Moreover, Sd (M) is connected, for d the least common multiple of α1, . . . , αk . In particular, S(M) = Sd (M).

Jennifer Schultens Surfaces in Seifert fibered spaces Computations

π1(M) =< a1, b1,..., ag , bg , x1,..., xk , h | −b g k h Π1 [ai , bi ]Π1 xi , [a1, h], [b1, h],..., [ag , h], [bg , h],

α1 β1 αk βk [x1, h],..., [xk , h], x1 h ,..., xk h >

Jennifer Schultens Surfaces in Seifert fibered spaces Computations

H1(M) =< a1, b1,..., ag , bg , x1,..., xn, h | x1 + ··· + xn,

α1x1 + β1h, . . . , αnxn + βnh >

Jennifer Schultens Surfaces in Seifert fibered spaces Computations

Relations of the form αi xi + βi h yield relations between the xi s. E.g.: α1 = 3, β1 = 2, α2 = 5, β2 = 3

9x1 + 6h = 10x2 + 6h

9(x1 − x2) = x2 So:

< x1, x2 | 9x1 = 10x2 >=< x1 − x2, x2 | 9(x1 − x2) = x2 >

=< x1 − x2 >= Z

Jennifer Schultens Surfaces in Seifert fibered spaces Computations

Substitution of this type are examples of standard procedures involving Nielsen equivalence and the Euclidean algorithm. Nielsen equivalence oftenprovides a method for reducing the number of generators.

This allows us to compute H1 explicitly:

H1(M) =< a1, b1,..., ag , bg , η >

Jennifer Schultens Surfaces in Seifert fibered spaces Gratitude

Thank you for listening and thanks to the organizers for inviting me to your conference!

Jennifer Schultens Surfaces in Seifert fibered spaces