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Taut Foliations and Heegard Floer L-Spaces

Antonio Alfieri

University of Pisa

ECSTATIC, June 2015

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Foliation Theory in Low Dimensional Topology

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Foliation Theory in Low Dimensional Topology

• in 1923, Alexander proved that a two-sphere in R3 bounds a

three-ball mainly using a ”foliation argument”;

• during the eighties, Gabai used taut foliations in order to

compute the genus of arborescent links;

• more recently, Ozsváth and Szabó used a mixture of foliation

theory and contact geometry to prove that Heegaard Floer homology detects the Thurston norm of a three-manifold and the minimal Seifert genus of a knot.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

What is a Foliation?

M oriented n dimensional manifold. A foliation F of M is a decomposition of M M =

• α

Σα in a disjoint union of smoothly immersed hypersurfaces - the leaves

• f F- locally modelled on
• R3, horizontal planes
• .

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Examples

S =

• −π

2 , π 2

• × R

Define a foliation F on S setting F =

• Ct
• t ∈ R
• ∪
• ± π

2 × R

• where Ct : y = sec(x) + t.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Examples

Rotating the leaves of F around the z-axis we obtain a foliation on D2 × R. The foliated manifold

• D2 × R, F
• is called the Reeb

tube.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

F induce a foliation on the solid torus D2 × S1 = D2 × R/z→z+1 called Reeb foliation.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Examples

Take a genus one Heegaard splitting L(p, q) = U0 ∪Σ U1 Filling U0 and U1 with the interior of the Reeb foliation and declaring Σ as leaf we obtain a foliation on the whole L(p, q). Theorem Every closed oriented three-manifold has a foliation.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Reebless foliations

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Reebless foliations

Definition Y 3 connected orientable smooth 3-manifold, F foliation of Y . A Reeb component of F is a smoothly embedded solid torus T ⊆ Y such that

• ∂T is a compact leaf of F,
• F|T coincides with the Reeb foliation.

A foliation without Reeb components is said Reebless. Theorem (Palmeira) If a closed connected and orientable three-manifold Y , with Y = S2 × S1, has a Reebless foliation then the universal cover of Y is homeomorphic to R3. Consequently: π1(Y ) is torsion free, πk(Y ) = 0 for k ≥ 2 and Y is irreducible.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

An example of Reebless foliation

K = (−2, 3, 7) pretzel knot.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

An example of Reebless foliation

Proposition Y = S3

+ 37

2 (K) contains a Reebless foliation.

Sketch of proof: Y = X1 ∪ψ X2 where X1 = S3 N(31) X2 = S3 N(3m

1 )

φi : Xi → S1 fibration, set Fi =

• φ−1

i

(θ) ⊂ Xi

• θ ∈ S1

Spinning around the boundary both F1 and F2 we obtain two foliations Fo

1 and Fo 2 filling ˚

X1 and ˚ X2 respectively. A Reebless foliation on Y is now given by F = Fo

1 ∪ Fo 2 ∪ separating torus of Y .

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Taut foliations

Definition A foliation F is called taut if: for each leaf Σ there exist a simple closed curve γ ⊆ Y intersecting Σ and γ ⋔ F. Example: Y = S2 × S1, F =

• S2 × θ|θ ∈ S1

. A transverse curve intersecting all the leaves is given by γ = pt. × S1. Theorem Taut foliations are Reebless.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

What about existence of taut foliations?

Theorem (Gabai) A prime, compact, connected and orientable three-manifold with b1 > 0 has a taut foliation. Furthermore, if Y is such a manifold and Σ ⊂ Y is a properly embedded surface minimizing the Thurston norm of its homology class, then Y has a taut foliation having Σ as leaf. Question When does an irreducible rational homology sphere have a taut foliation?

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Heegard Floer Homology

Ozsváth and Szabó introduced a package of invariants HF +, HF −, HF ∞, HF called Heegaard Floer homology.

• HF : Y rational homology 3-sphere f.d. F2 vector space
• HF(Y ) =
• s∈Spinc
• HF(Y , s)

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Heegaard Floer Lens Spaces

Proposition If Y is a rational homology sphere dim HF(Y , s) ≥ 1 for all Spinc structures. If the equality hold for all Spinc structures we say that Y is an Heegaard Floer lens space, or an L-space for short. dim HF(Y ) = #Spinc(Y ) = |H1(Y , Z)|.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Examples of L-Spaces

Exaples of L-Spaces are:

• S3
• Lens spaces (whence the name)
• All manifolds with finite fundamental group
• Branched double covers of quasi-alternating links

Question Is there a topological characterization (i.e. that does not refer to Heegard Floer homology) of L-spaces?

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Taut foliations and L-Spaces

Theorem (Ozsváth & Szabó) A rational homology three-sphere that has a taut foliation is not an L-space.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

An application

K = (−2, 3, 7) pretzel knot.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

An application

Proposition Then Y = S3

+ 37

2 (K) does not contains a taut foliation.

Sketch of proof: The +18 surgery on K produces the lens space L(18, 5). HF surgery exact sequence ⇒ if S3

r (K) is an L-space for some

r > 0 then so is S3

s (K) for all s > r.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces

Conjectural characterizations L-spaces

Conjecture (L-space Conjecture) An irreducible QH sphere is an L-space if and only if has not a taut foliation. Conjecture (Hedden & Levine) If Y is an irreducible ZH sphere that is an L-space, then Y is homeomorphic to either S3 or the Poincaré homology sphere.

Antonio Alfieri Taut Foliations and Heegard Floer L-Spaces