taut foliations, positive 3-braids, and the l-space conjecture

The L-Space ConjectureInvestigating the LSC

Proof Strategy

Taut Foliations, Positive 3-Braids, and theL-Space Conjecture

Siddhi Krishna

Boston College

January 17, 2019

Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors

Proof Strategy

Taut FoliationsL-Spaces

The L-Space Conjecture

The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:

Y admits a taut foliation

Y is a non-L-space

π1(Y ) is not left orderable

Proof Strategy

Y admits a taut foliation

Y is a non-L-space

π1(Y ) is not left orderable

Proof Strategy

Y admits a taut foliation “geometry”

Y is a non-L-space “Heegaard-Floer homology”

π1(Y ) is not left orderable “algebra”

Proof Strategy

• Y is a non-L-space “Heegaard-Floer homology”

Proof Strategy

Taut Foliations

Definition

A taut foliation is

• a decomposition of a manifold into codim-1 submanifolds, calledleaves, such that

• there exists a simple closed curve meeting each leaf transversely

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

a compact, connected, oriented surface F

ϕ : F → F , a diffeomorphism of F

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

a compact, connected, orientable surface F

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

An Example

Fibered 3-Manifolds

Start with

This determines:

Mϕ = F × I/ϕ = F × I

/((x , 0) ∼ (ϕ(x), 1))

F × {0}

F × {1}

Proof Strategy

Taut Foliations in the Literature

Theorem (Thurston ’86):Compact leaves of taut foliations minimize genus in their homologyclass.

Theorem (Gabai ’87):The “Property R” Conjecture is true, i.e.

If K ⊂ S3, and S30 (K ) ≈ S1 × S2, then K ≈ U.

Proof Strategy

Taut Foliations in the Literature

Theorem (Thurston ’86):Compact leaves of taut foliations minimize genus in their homologyclass.

Theorem (Gabai ’87):The “Property R” Conjecture is true, i.e.

If K ⊂ S3, and S30 (K ) ≈ S1 × S2, then K ≈ U.

Proof Strategy

L-Spaces

“L-spaces are simple from the perspective ofHeegaard Floer homology”

Definition

A closed, irreducible 3-manifold Y is an L-space if

rk(HF (Y ;Z/2Z)) = |H1(Y ;Z)| <∞

Remark: rk(HF (Y ;Z/2Z)) ≥ |H1(Y ;Z)| holds for QHS3

Examples: Lens Spaces ⊂ Manifolds with Elliptic Geometry

Proof Strategy

L-Spaces

Definition

rk(HF (Y ;Z/2Z)) = |H1(Y ;Z)| <∞

Proof Strategy

L-Spaces

Definition

rk(HF (Y ;Z/2Z)) = |H1(Y ;Z)| <∞

Proof Strategy

L-Spaces

Definition

rk(HF (Y ;Z/2Z)) = |H1(Y ;Z)| <∞

Proof Strategy

The L-Space Conjecture Revisited

These manifolds are “extra”.

Proof Strategy

The L-Space Conjecture Revisited

These manifolds are “extra”.

Proof Strategy

EvidenceProducing Non-L-SpacesA New Theorem

Evidence for the LSC

Theorem (Ozsvath and Szabo):If Y admits a taut foliation, then Y is a non-L-space.

Theorem: The L-Space Conjecture is true for graph manifolds.Proof: Input by many!

Eisenbud, Hirsch, Neumann, Naimi, Calegari, Walker, Boyer,Gordon, Clay, Watson, Lisca, Stipsicz, J. & S. Rasmussen,Hanselman, and many more!

Proof Strategy

Theorem: The L-Space Conjecture is true for graph manifolds.

Proof: Input by many!

Proof Strategy

Theorem: The L-Space Conjecture is true for graph manifolds.Proof: Input by many!

Proof Strategy

Focusing on Foliations

For Y is a closed, irreducible 3-manifold:

Y admits a taut foliation Y is a non-L-space

Ozsvath-Szabo

Questions: How do we(1) identify non-L-spaces, and(2) build taut foliations in them?

Proof Strategy

Ozsvath-Szabo

Questions: How do we(1) identify non-L-spaces?(2) build taut foliations in them?

Proof Strategy

Ozsvath-Szabo

Questions: How do we(1) identify non-L-spaces?

(2) build taut foliations in them?

Proof Strategy

Ozsvath-Szabo

Proof Strategy

Ozsvath-Szabo

Proof Strategy

Producing Non-L-Spaces via Dehn Surgery

Definition

A non-trivial knot K ⊂ S3 is an L-Space knot if K admits anon-trivial surgery to an L-space.

Examples: Torus knots

Proof Strategy

Definition

A non-trivial knot K ⊂ S3 is an L-Space knot if K admits anon-trivial surgery to an L-space.

Examples: Torus knots

Proof Strategy

Suppose K ⊂ S3.

Then either:

(1) K isn’t an L-space knot:For all r ∈ Q, S3

r (K ) is a non-L-space.

(2) K is an L-space knot:

Theorem: (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):

For r ∈ Q, S3r (K ) =

{non-L-space r < 2g(K )− 1

L-space r ≥ 2g(K )− 1

If r < 2g(K )− 1, then S3r (K ) is always a non-L-space.

Therefore, the LSC predicts it admits a taut foliation.

Proof Strategy

Suppose K ⊂ S3. Then either:

For r ∈ Q, S3r (K ) =

Proof Strategy

For r ∈ Q, S3r (K ) =

Proof Strategy

For r ∈ Q, S3r (K ) =

Therefore, the LSC predicts a taut foliation.

Proof Strategy

Theorem (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):

For r ∈ Q, S3r (K ) =

Proof Strategy

(1) K isn’t an L-space knotFor all r ∈ Q, S3

(2) K is an L-space knot

Theorem (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):

For r ∈ Q, S3r (K ) =

Proof Strategy

A New Theorem

Theorem (K.)

Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3

r (K ) admits a taut foliation.

Example: The P(−2, 3, 7) (Fintushel-Stern) Pretzel Knot

Proof Strategy

A New Theorem

Theorem (K.)

Proof Strategy

A New Theorem

Theorem (K.)

Proof Strategy

A New Theorem

Theorem (K.)

Remark: Applying (Lidman-Moore ’16), we get the first example of

Y is a non-L-space ⇐⇒ Y admits a taut foliation

for every non-L-space obtained by Dehn surgery along an infinitefamily of hyperbolic L-space knots.

Proof Strategy

A New Theorem

Theorem (K.)

Remark: Applying (Lidman-Moore ’16), we get the first example of

Y is a non-L-space ⇐⇒ Y admits a taut foliation

for every non-L-space obtained by Dehn surgery along an infinitefamily of hyperbolic L-space knots.

Proof Strategy

Branched SurfacesProof Sketch

What’s Known, Revisited

Ozsvath-Szabo

Proof Strategy

Ozsvath-Szabo

Questions: How do we(1) identify non-L-spaces? Dehn Surgery(2) build taut foliations in them?

Proof Strategy

Ozsvath-Szabo

Questions: How do we(1) identify non-L-spaces? Dehn Surgery(2) build taut foliations in them? Branched Surfaces

Proof Strategy

Branched Surfaces

Definition

A branched surface is a “co-oriented 2-complex”, locallymodelled by:

Branched surfaces can encode data about(1) surfaces in 3-manifolds (Oertel; Floyd-Oertel)(2) laminations in 3-manifolds (Li)(3) foliations of 3-manifolds (Roberts)

Proof Strategy

Branched Surfaces

Definition

Proof Strategy

Branched Surfaces

Definition

Proof Strategy

Branched Surfaces

Definition

Proof Strategy

Branched Surfaces and Taut Foliations

Theorem (K.)

Proof Sketch:(1) Identify a fiber surface for K in

XK = S3 − ◦ν(K ) = F × I

(2) Use the fibration to build a branched surface B in XK

(3) Use B to build a taut foliation in XK

(4) Understand how F meets ∂XK

(5) Dehn fill to produce a taut foliation in S3r (K )

Branched surfaces give a recipe for building taut foliations.

Proof Strategy

Theorem (K.)

XK = S3 − ◦ν(K ) = F × I

Proof Strategy

Theorem (K.)

XK = S3 − ◦ν(K ) = F × I

Proof Strategy

Theorem (K.)

XK = S3 − ◦ν(K ) = F × I

Proof Strategy

Theorem (K.)

XK = S3 − ◦ν(K ) = F × I

Proof Strategy

Theorem (K.)

XK = S3 − ◦ν(K ) = F × I

Proof Strategy

Theorem (K.)

XK = S3 − ◦ν(K ) = F × I

taut foliations, positive 3-braids, and the l-space conjecture

Documents