surfaces in seifert fibered spaces
TRANSCRIPT
Surfaces in Seifert fibered spaces
Jennifer Schultens
University of California, Davis
June 26, 2019
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 in3-dimensional space.
I will discuss the totality of surfaces in a particular class of3-manifolds.
Rather than giving you a formal definition of manifold, in particular3-manifold, I will describe some of my favorite examples.
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 in3-dimensional space.
I will discuss the totality of surfaces in a particular class of3-manifolds.
Rather than giving you a formal definition of manifold, in particular3-manifold, I will describe some of my favorite examples.
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 in3-dimensional space.
I will discuss the totality of surfaces in a particular class of3-manifolds.
Rather than giving you a formal definition of manifold, in particular3-manifold, I will describe some of my favorite examples.
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 in3-dimensional space.
I will discuss the totality of surfaces in a particular class of3-manifolds.
Rather than giving you a formal definition of manifold, in particular3-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
C (K ) = S3 − 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-’
4KC sot’? kftco V
(Jattant
i2
KcV•
Figure: Satellite knots
Jennifer Schultens Surfaces in Seifert fibered spaces
Thurston’s theorem
Theorem
(Trichotomy for knots) Let K be a knot in S3. Then exactly one ofthe 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 anddiscrete groups”.
Jennifer Schultens Surfaces in Seifert fibered spaces
Thurston’s theorem
Theorem
(Trichotomy for knots) Let K be a knot in S3. Then exactly one ofthe 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 anddiscrete 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 aknot has two points for every point in the knot complement andone 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 ofinvariants. The structure of these manifolds allows us to concretelydescribe essential surfaces embedded in them.
We are interested in the collection of all surfaces in a given Seifertfibered space. However, we must restrict our attention to essentialsurfaces in Seifert fibered spaces. The Kakimizu complex of a knotprovides an example of how to encode surfaces. We discus theanalogous concept for Seifert fibered spaces.
We will always assume that curves in surfaces are simple and thatsurfaces 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
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 cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2
about its center point x by 2πνµ .
Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.
All fibers except the one determined by x × I traverse the torus µtimes. 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, thenthe central fiber is also called a regular fiber.
Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.
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 cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2
about its center point x by 2πνµ .
Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.
All fibers except the one determined by x × I traverse the torus µtimes. 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, thenthe central fiber is also called a regular fiber.
Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.
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 cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2
about its center point x by 2πνµ .
Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.
All fibers except the one determined by x × I traverse the torus µtimes. 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, thenthe central fiber is also called a regular fiber.
Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.
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 cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2
about its center point x by 2πνµ .
Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.
All fibers except the one determined by x × I traverse the torus µtimes. 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, thenthe central fiber is also called a regular fiber.
Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.
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 cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2
about its center point x by 2πνµ .
Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.
All fibers except the one determined by x × I traverse the torus µtimes. 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, thenthe central fiber is also called a regular fiber.
Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.
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 cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2
about its center point x by 2πνµ .
Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.
All fibers except the one determined by x × I traverse the torus µtimes. 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, thenthe central fiber is also called a regular fiber.
Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.
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 cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2
about its center point x by 2πνµ .
Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.
All fibers except the one determined by x × I traverse the torus µtimes. 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, thenthe central fiber is also called a regular fiber.
Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.
Jennifer Schultens Surfaces in Seifert fibered spaces
Fibered solid tori
Every fiber in a fibered solid torus except the singular onerepresents the element µ ∈ π1(V (ν, µ)) = Z which is generated bythe class of the central fiber.
Hence µ is an invariant of V (ν, µ) and ν is an invariant up to signand mod µ.
In other words, V (ν, µ) is isomorphic to V (ν ′, µ′) via a fiberpreserving homeomorphism if and only if µ′ = µ andν ′ = ±ν ′(modµ).
If we keep track of orientations we can normalize the invariant ν sothat 0 ≤ ν < µ.
Jennifer Schultens Surfaces in Seifert fibered spaces
Fibered solid tori
Every fiber in a fibered solid torus except the singular onerepresents the element µ ∈ π1(V (ν, µ)) = Z which is generated bythe class of the central fiber.
Hence µ is an invariant of V (ν, µ) and ν is an invariant up to signand mod µ.
In other words, V (ν, µ) is isomorphic to V (ν ′, µ′) via a fiberpreserving homeomorphism if and only if µ′ = µ andν ′ = ±ν ′(modµ).
If we keep track of orientations we can normalize the invariant ν sothat 0 ≤ ν < µ.
Jennifer Schultens Surfaces in Seifert fibered spaces
Fibered solid tori
Every fiber in a fibered solid torus except the singular onerepresents the element µ ∈ π1(V (ν, µ)) = Z which is generated bythe class of the central fiber.
Hence µ is an invariant of V (ν, µ) and ν is an invariant up to signand mod µ.
In other words, V (ν, µ) is isomorphic to V (ν ′, µ′) via a fiberpreserving homeomorphism if and only if µ′ = µ andν ′ = ±ν ′(modµ).
If we keep track of orientations we can normalize the invariant ν sothat 0 ≤ ν < µ.
Jennifer Schultens Surfaces in Seifert fibered spaces
Fibered solid tori
Every fiber in a fibered solid torus except the singular onerepresents the element µ ∈ π1(V (ν, µ)) = Z which is generated bythe class of the central fiber.
Hence µ is an invariant of V (ν, µ) and ν is an invariant up to signand mod µ.
In other words, V (ν, µ) is isomorphic to V (ν ′, µ′) via a fiberpreserving homeomorphism if and only if µ′ = µ andν ′ = ±ν ′(modµ).
If we keep track of orientations we can normalize the invariant ν sothat 0 ≤ ν < µ.
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 thatis a union of disjoint circles called fibers such that each fiber has aneighborhood that is homeomorphic to a fibered solid torus.
Definition
Fibers are called regular (or singular) if they are regular (orsingular) fibers in the fibered solid torus containing them.
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 thatis a union of disjoint circles called fibers such that each fiber has aneighborhood that is homeomorphic to a fibered solid torus.
Definition
Fibers are called regular (or singular) if they are regular (orsingular) fibers in the fibered solid torus containing them.
Jennifer Schultens Surfaces in Seifert fibered spaces
Seifert fibered spaces
Remark 1: Fibers of a Seifert fibered space are either regular orsingular. (Not both.) Moreover, a fiber of a Seifert fibered spaceuniquely determines normalized ν, µ.
Remark 2: Because it is compact, a Seifert fibered space is coveredby finitely many fibered solid tori.
Remark 3: A Seifert fibered space will have only a finite number ofsingular fibers.
Jennifer Schultens Surfaces in Seifert fibered spaces
Seifert fibered spaces
Remark 1: Fibers of a Seifert fibered space are either regular orsingular. (Not both.) Moreover, a fiber of a Seifert fibered spaceuniquely determines normalized ν, µ.
Remark 2: Because it is compact, a Seifert fibered space is coveredby finitely many fibered solid tori.
Remark 3: A Seifert fibered space will have only a finite number ofsingular fibers.
Jennifer Schultens Surfaces in Seifert fibered spaces
Seifert fibered spaces
Remark 1: Fibers of a Seifert fibered space are either regular orsingular. (Not both.) Moreover, a fiber of a Seifert fibered spaceuniquely determines normalized ν, µ.
Remark 2: Because it is compact, a Seifert fibered space is coveredby finitely many fibered solid tori.
Remark 3: A Seifert fibered space will have only a finite number ofsingular fibers.
Jennifer Schultens Surfaces in Seifert fibered spaces
Seifert fibered spaces
Given a Seifert fibered space, we identify each fiber to a point toobtain the base orbifold, a compact surface with a finite number ofmarked 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 completelydetermined 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
Given a Seifert fibered space, we identify each fiber to a point toobtain the base orbifold, a compact surface with a finite number ofmarked 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 completelydetermined 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
Given a Seifert fibered space, we identify each fiber to a point toobtain the base orbifold, a compact surface with a finite number ofmarked 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 completelydetermined 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
Given a Seifert fibered space, we identify each fiber to a point toobtain the base orbifold, a compact surface with a finite number ofmarked 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 completelydetermined 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 boundsa disk. If it is not inessential, then it is essential.
Definition
The surface F in the 3-manifold M is compressible if there is anessential simple closed curve in F that bounds a disk in M. If F isnot 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 Seifertfibered 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 isnecessarily incompressible.
Remark 2: A Seifert fibered space admits horizontal surfaces if andonly 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 surfaceF 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
πidentify after twist through
Figure: Only a vertical Mobius band in a fibered solid Klein bottle wouldproject 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 becompressible or incompressible. If it is compressible, then it boundsa 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 orientablebase orbifold. If F is a connected, two-sided, incompressiblesurface 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 asa surface bundle over the circle (in particular, F is horizontal);
(iii) F is an annulus or a torus and, after isotopy, F consists offibers, 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 spanningsurfaces for knots.
Definition
Given an knot K in S3, a compact connected orientable surfacethat represents a generator of H2(S3,K ) is called a Seifert surface.More concretely, a Seifert surface is an embedded orientablesurface 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
(Eisner) Many knots K in S3 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
The vertices of the Kakimizu complex Kak(K ) of a knot K in S3
are given by the isotopy classes of minimal genus Seifert surfacesfor K .
The n-simplices of the Kakimizu complex of K , for n > 1, are givenby n-tuples of vertices that admit pairwise disjoint representatives.
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
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 isconnected.
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 locallyfinite.
Theorem
(Banks) Approximately: The Kakimizu complex of a sum of twoknots 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 isquasi-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
What’s special about Seifert surfaces of knots?
Key Property: They represent the generator ofH2(S3,K ) = H2(S3 − η(K ), ∂S3 − η(K )).
Fact: This group is dual toH1(S3 − η(K )) = Hom(H1(S3 − η(K )) = Hom(Z) = Z
Consequence: There is a canonical infinite cyclic cover of(S3,K ) in which we can compare lifts of distinct Seifertsurfaces.
Jennifer Schultens Surfaces in Seifert fibered spaces
What’s special about Seifert surfaces of knots?
Key Property: They represent the generator ofH2(S3,K ) = H2(S3 − η(K ), ∂S3 − η(K )).
Fact: This group is dual toH1(S3 − η(K )) = Hom(H1(S3 − η(K )) = Hom(Z) = Z
Consequence: There is a canonical infinite cyclic cover of(S3,K ) in which we can compare lifts of distinct Seifertsurfaces.
Jennifer Schultens Surfaces in Seifert fibered spaces
What’s special about Seifert surfaces of knots?
Key Property: They represent the generator ofH2(S3,K ) = H2(S3 − η(K ), ∂S3 − η(K )).
Fact: This group is dual toH1(S3 − η(K )) = Hom(H1(S3 − η(K )) = Hom(Z) = Z
Consequence: There is a canonical infinite cyclic cover of(S3,K ) in which we can compare lifts of distinct Seifertsurfaces.
Jennifer Schultens Surfaces in Seifert fibered spaces
The covering space associated with α
S’0
m
m−2
S
m−1S
S
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-manifoldM3, choose a primitive element α of H2(M3, ∂M3). Or, moregenerally, choose a primitive element α of Hn−1(Mn, ∂Mn).
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, denotedKak(M, α): Vertices are given by weighted multi-surfaces thatrepresent α.
The multi-surface is required to be Thurston norm minimizing, tohave 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 ′ span an edge e = (v , v ′) if and only ifrepresentatives of v , v ′ can be chosen so that the complement of alift of one respresentative to the covering space associated with αintersects exactly two lifts of the complement of the otherrepresentative.
(This condition implies that the representatives are disjoint, butnot vice versa.)
Kak(M, α) is the flag complex with the vertices and edgesdescribed 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) EveryKakimizu complex of a 2- or 3-manifold is connected.
Theorem
(Przytycki-S, S) Every Kakimizu complex of a 2- or 3-manifold iscontractible.
Theorem
(S) A Kakimizu complex of a 2- or 3-manifold need not be quasiEuclidean.
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 baseorbifold. Let α ∈ H2(M, ∂M,Z) be a primitive relative secondhomology class that is represented by a horizontal surface. ThenKak(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 bya vertical surface. Then Kak(M, α) is isomorphic to thecorresponding Kakimizu complex of the surface obtained from thebase orbifold of M by removing neighborhoods of the singularpoints.
Jennifer Schultens Surfaces in Seifert fibered spaces
Vertical Kakimizu complexes of Seifert fibered spaces
−ξ
~
b~
A
b’~
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 in2010, Brie Finegold studied torus complexes in all dimensions froman algebraic point of view. In dimension 2, where vertices aresimple closed curves in the 2-torus, Finegold’s torus complexcoincides with the curve complex of the torus, i.e., the Farey graph.
In dimension 3, where the vertices are isotopy classes of 2-toriembedded in the 3-torus, there are no disjoint non isotopicessential surfaces, and edges are defined as pairs of vertices withrepresentatives 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 sequenceof complexes {Si (M)}, and the surface complex, S(M), of M asfollows:
Vertices in {Si (M)} and S(M) correspond to isotopy classesof 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) ifand 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). GivenSi (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 + 1components.
For all i , Si (M) is a flag complex.
The surface complex of M, denoted S(M), is defined to beSi0(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 Eulernumber, then S(M) is isomorphic to the curve complex of Q.
Theorem
If M is a totally orientable Seifert fibered space with base orbifoldof genus 0 and Euler number 0, then S(M) contains a subcomplexisomorphic to the curve complex of the surface obtained from Q.Moreover, S(M) is contained in the cone on this subcomplex.
Corollary
(Special case) If M is a totally orientable Seifert fibered space withbase orbifold of genus 0, Euler number 0, and either 4 or 5exceptional fibers with identical invariants, then S(M) isisomorphic to the cone on the curve complex of Q.
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 number0 and base orbifold of positive genus, then S(M) contains asubcomplex isomorphic to the curve complex of the surfaceobtained from Q. Moreover, Sd(M) is connected, for d the leastcommon 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 |
h−bΠg1 [ai , bi ]Π
k1xi , [a1, h], [b1, h], . . . , [ag , h], [bg , h],
[x1, h], . . . , [xk , h], xα11 hβ1 , . . . , xαk
k hβk >
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 αixi + βih 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 proceduresinvolving Nielsen equivalence and the Euclidean algorithm. Nielsenequivalence oftenprovides a method for reducing the number ofgenerators.
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 invitingme to your conference!
Jennifer Schultens Surfaces in Seifert fibered spaces