Construction of Lefschetz fibrations via Luttinger surgery

Construction of Lefschetz fibrations via

Luttinger surgery

Anar Akhmedov

University of Minnesota, Twin Cities

August 2, 2013

Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 1 / 43

Outline

1 Symplectic 4-manifolds and Lefschetz fibrationsSymplectic 4-manifolds

Lefschetz fibrations

Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas

2 Surgery on symplectic 4-manifoldsLuttinger Surgery

Symplectic Connected Sum

3 Construction of Symplectic 4-Manifolds

Luttinger surgeries on product 4-manifolds Σn × Σ2 and Σn × T2

Construction of Lefschetz fibration via Luttinger SurgeryExotic Stein fillings


Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds

Symplectic manifolds




Definition

A (compact) symplectic 2n-manifold (X , ω) is a smooth 2n-manifold with a

symplectic form ω ∈ Ω2(X) (i.e., ω is closed (dω = 0) and non-degenerate(ωn = ω ∧ · · · ∧ ω > 0 everywhere) 2-form.




Definition


symplectic form ω ∈ Ω2(X) (i.e., ω is closed (dω = 0) and non-degenerate(ωn = ω ∧ · · · ∧ ω > 0 everywhere) 2-form. A diffeomorphism

f : (X1, ω1) → (X2, ω2) is a symplectomorphism if ω1 = f ∗(ω2).




Definition




Examples




Definition




Examples

X = R2n with linear coordinates x1, · · · , xn, y1, · · · , yn and with the 2-form

ω0 =∑n

i=1 dxi ∧ dyi .




Definition




Examples


ω0 =∑n

i=1 dxi ∧ dyi .

If (X1, ω1) and (X2, ω2) are symplectic manifolds, then π1∗ω1 +π2

∗ω2 gives

a symplectic structure on X1 × X2.




Definition




Examples


ω0 =∑n

i=1 dxi ∧ dyi .


∗ω2 gives

a symplectic structure on X1 × X2. Σn × Σm are symplectic 4-manifolds




Definition




Examples


ω0 =∑n

i=1 dxi ∧ dyi .


∗ω2 gives


Every Kähler manifold is also a symplectic manifold.




Definition




Examples


ω0 =∑n

i=1 dxi ∧ dyi .


∗ω2 gives


Every Kähler manifold is also a symplectic manifold.

A closed complex surface S is Kähler iff the first Betti number b1(S) is

even.



Kodaira-Thurston manifold




Example




Example

Consider R4 with the 2-form ω0 = dx1 ∧ dy1 + dx2 ∧ dy2.




Example


Let Γ be the discrete group generated by the following symplectomorphisms:




Example



γ1 : (x1, x2, y1, y2) −→ (x1, x2 + 1, y1, y2)γ2 : (x1, x2, y1, y2) −→ (x1, x2, y1, y2 + 1)

γ3 : (x1, x2, y1, y2) −→ (x1 + 1, x2, y1, y2)

γ4 : (x1, x2, y1, y2) −→ (x1, x2 + y2, y1 + 1, y2)




Example



γ1 : (x1, x2, y1, y2) −→ (x1, x2 + 1, y1, y2)γ2 : (x1, x2, y1, y2) −→ (x1, x2, y1, y2 + 1)

γ3 : (x1, x2, y1, y2) −→ (x1 + 1, x2, y1, y2)

γ4 : (x1, x2, y1, y2) −→ (x1, x2 + y2, y1 + 1, y2)

M = R4/Γ admits both symplectic structure and complex structures, but

non-Kähler.



Kodaira-Thurston manifold continued

Example




Example

Let φ = Db denote the right-handed Dehn twist on T2 = a × b along the curve

b.




Example


b.

φ∗(a) = a + b

φ∗(b) = b




Example


b.

φ∗(a) = a + b

φ∗(b) = b

Let Zφ denote the mapping torus of φ.




Example


b.

φ∗(a) = a + b

φ∗(b) = b

Let Zφ denote the mapping torus of φ. W = Zφ × S1 admits T2 bundle

structure over T2.




Example


b.

φ∗(a) = a + b

φ∗(b) = b

Let Zφ denote the mapping torus of φ. W = Zφ × S1 admits T2 bundle

structure over T2. Presentation for the fundamental group of W :

π1(W ) = 〈g1, g2, g3, x ; |

[g1, g2] = 1, [g2, g3] = 1, [g1, g3] = g2, [x , gi ] = 1〉.



Symplectic 4-manifolds via surface bundles over surfaces




Theorem (W. Thurston (1976))

Assume that Σg, Σh are closed, oriented, 2-dimensional surfaces. Iff : X → Σh is a bundle with fiber Σg and the homology class of the fiber is

nonzero in H2(X ;R), then X admits a symplectic structure.




Theorem (W. Thurston (1976))

Assume that Σg, Σh are closed, oriented, 2-dimensional surfaces. Iff : X → Σh is a bundle with fiber Σg and the homology class of the fiber is

nonzero in H2(X ;R), then X admits a symplectic structure.

Theorem (J. Bryan - R. Donagi)

For any integers n ≥ 2, there exisit smooth algebraic surface Xn that havesignature σ(Xn) = 8/3n(n − 1)(n + 1) and admit two smooth fibrations

Xn −→ B and Xn −→ B′ such that the base and fiber genus are(3, 3n3 − n2 + 1) and (2n2 + 1, 3n) respectively.


Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations

Lefschetz fibrations

Definition

Let X be a compact, connected, oriented, smooth 4-manifold. A Lefschetz

fibration on X is a smooth map f : X −→ Σh, where Σh is a compact, oriented,smooth 2-manifold of genus h, such that f is surjective and each critical point

of f has an orientation preserving chart on which f : C2 −→ C is given by

f (z1, z2) = z12 + z2

2.



Vanishing cycle α

S2

X

f

Figure: Lefschetz fibration on X over S2



The genus of the regular fiber of f is defined to be the genus of the Lefschetzfibration.



The genus of the regular fiber of f is defined to be the genus of the Lefschetzfibration. Let p1, · · · , ps denote the critical points of Lefschetz fibration

f : X −→ Σh.

e(X) = e(Σh)e(Σg) + s

σ(X) well understood for fibrations over S2

(Y .Matsumoto,H.Endo,B.Ozbagci, I.Smith, ...)



Monodromy of Lefschetz fibration

A singular fiber of the genus g Lefschetz fibration can be described by itsmonodromy, i.e., an element of the mapping class group Mg . This element is

a right-handed (or a positive) Dehn twist along a simple closed curve on Σg ,

called the vanishing cycle.

Vanishing cycle α

S2

X

f

Figure: Vanishing cycle of Lefschetz fibrationAnar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 10 / 43


For a genus g Lefschetz fibration over S2, the product of right handed Dehn

twists tαialong the vanishing cycles αi , for i = 1, · · · , s, determines the global

monodromy of the Lefschetz fibration, the relation tα1· tα2

· · · · · tαs= 1 in Mg .

Conversely, such a relation in Mg determines a genus g Lefschetz fibration

over S2 with the vanishing cycles α1, · · · , αs.

Vanishing cycle α

S2

X

f

Figure: Lefschetz fibration on X over S2



Example (Genus one Lefschetz fibrations)

M1 be the mapping class group of the torus T2 = a × b.




M1 be the mapping class group of the torus T2 = a × b. M1 = SL(2,Z) isgenerated by

ta =

(

1 1

0 1

)

tb =

(

1 0−1 1

)

Subject to the relations

tatbta = tbtatb(tatb)

6 = 1





ta =

(

1 1

0 1

)

tb =

(

1 0−1 1

)



6 = 1

(tatb)6n = 1 in M1.





ta =

(

1 1

0 1

)

tb =

(

1 0−1 1

)



6 = 1

(tatb)6n = 1 in M1. The total space of this fibration is the elliptic surface E(n).

E(1) = CP2#9CP2, the complex projective plane blown up at 9 points, and

E(2) is K 3 surface. E(n) also admits a genus n − 1 Lefschetz fibration overS2.



Example (Hyperelliptic Lefschetz fibrations)

Let α1, α2, .... , α2g , α2g+1 denote the collection of simple closed curves given

in Figure, and ci denote the right handed Dehn twists tαialong the curve αi .

α1

α2

α3

α4 α2g−2

α2g−1

α2g

α2g+1

Figure: Vanishing cycles of the genus g Lefschetz fibration given by hyperelliptic

involution

The following relations hold in the mapping class group Mg:

Γ1(g) = (c1c2 · · · c2g−1c2gc2g+12c2gc2g−1 · · · c2c1)

2 = 1.Γ2(g) = (c1c2 · · · c2g−1c2gc2g+1)

2g+2 = 1.Γ3(g) = (c1c2 · · · c2g−1c2g)

2(2g+1) = 1.(1)



The monodromy relation Γ1(g) = 1, the corresponding genus g Lefschetz

fibrations over S2 has total space X(g, 1) = CP2#(4g + 5)CP2, the complexprojective plane blown up at 4g + 5 points.

It is known that for g ≥ 2, the above fibration on X(g, 1) admits 4g + 4 disjoint(−1)-sphere sections (proof of this fact using a mapping class group

argument is due to S. Tanaka).

The fiber class is of the form (g + 2)h − ge1 − e2 − · · · − e4g+5, where ei

denotes the homology class of the exceptional sphere of the i-th blow up andh denotes the pullback of the hyperplane class of CP2. The exceptional

spheres represented by the homology classes e2, e3, . . . , e4g+5 are sections of

the Lefschetz fibration X(g, 1) → S2.



Theorem (S. Donaldson)

For any symplectic 4-manifold X, there exists a non-negative integer n such

that the n-fold blowup X#nCP2 of X admits a Lefschetz fibrationf : X#nCP2 → S2.

Theorem (R. Gompf)

Assume that the closed 4-manifold X admits a genus g Lefschetz fibration

f : X → Σh, and let [F ] denote the homology class of the fiber. Then X admitsa symplectic structure with symplectic fibers iff [F ] 6= 0 in H2(X ;R). If e1, · · · ,en is a finite set of sections of the Lefschetz fibration, the symplectic form ωcan be chosen in such a way that all these sections are symplectic.

Theorem (S. Donaldson and R. Gompf, J. Amoros - F. Bogomolov - L.

Katzarkov - T. Pantev)

For any finitely presented group G, there exist a Lefschetz fibration X(G) over

S2 with π1(X(G)) = G.

b2+(X(G)) is very large, and depends from the presentation of G.



Main Theorems

Theorem (A. Akhmedov - B. Ozbagci, 2012)

For any finitely presented group G, there exist a closed symplectic 4-manifoldXn(G) with π1(X(G)) = G, which admits a genus 2g + n − 1 Lefschetz

fibration over S2 that has al least 4n + 4 pairwise disjoint sphere sections of

self intersection −2. Moreover, Xn(G) contains a homologically essentialembedded torus of square zero disjoint from these sections which intersects

each fiber of the Lefschetz fibrations twice.


There exist an infinite family of non-holomorphic Lefschetz fibrations Xn(G,Ki)over S2 with π1(Xn(G,Ki)) = G that can be obtained from Xn(G) via knot

surgery along Ki , where Ki are inf. family of genus g ≥ 2 fibered knots with

distinct Alexander polynomials.


Symplectic 4-manifolds and Lefschetz fibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas

Lefschetz fibrations by Y. Matsumoto and M. Korkmaz

Let assume g = 2k .

The 4-manifold Y (1, k) = Σk × S2#4CP2 is the total space of the genus g

Lefschetz fibration over S2 with 2g + 4 singular fibers. This was shown byYukio Matsumoto for k = 1, and in the case k ≥ 2 by Mustafa Korkmaz, by

factorizing the vertical involution θ of the genus 2k surface.

θ

1

2

k

k + 1

2k − 1

2k

Figure: The involution θ of the genus 2k surface



Theorem (Y. Matsumoto, M. Korkmaz)

Let θ denote the vertical involution of the genus g surface with 2 fixed points.

In the mapping class group Mg , the follwing relations between right handed

Dehn twists hold:a) (tB0

tB1tB2

· · · tBgtc)

2 = θ2 = 1 if g is even,

b) (tB0tB1

tB2· · · tBg

(ta)2(tb)

2)2 = θ2 = 1 if g is odd.

Bk , a, b, c are the simple closed curves defined as in Figure.

B0

B1 B2

Bg

c

a

b

B0

B1B2

Bg

Figure: The vanishing cycles



Lefschetz fibrations by Y. Gurtas

Yusuf Gurtas generalized the constructions of Matsumoto and Korkmaz even

further. He presented the positive Dehn twist expression for a new set ofinvolutions in the mapping class group M2k+n−1 of a compact, closed, oriented

2-dimensional surface Σ2k+n−1. The total space of these genus

g = 2k + n − 1 Lefschetz fibration over S2 is Y (n, k) = Σk × S2#4nCP2.



k

1

2

e2n−2

2k

k + 2

k + 1

e4e2

e1 e3 e2n−1e2n−3

e2n−4

e2n−5

e2n−6

e5

e6 θ

Figure: The involution θ of the surface Σ2k+n−1



The branched-cover description for Σk × S2#4nCP2

A generic horizontal fiber is the double cover of S2, branched over two points.Thus, we have a sphere fibration on Y (n, k) = Σk × S2#4nCP2. A generic

fiber of the vertical fibration is the double cover of Σk , branched over 2n

points. Thus, a generic fiber of the vertical fibration has genus n + 2k − 1.

Σg Σg

S2

S2

S2

S2

S2

S2

S2

S2

2n copies

Figure: The branch locus for Σk × S2#4nCP2



Theorem (Y. Gurtas)

The positive Dehn twist expression for the involution θ is given by

θ = e2i+2 · · ·e2n−2e2n−1e2i · · · e1B0e2n−1 · · · e2i+2e1 · · · e2iB1B2 · · ·B4k−1B4k e2i+1.



k

1

2

e2n−2

2k

k + 2

k + 1

e4e2

e1 e3 e2n−1e2n−3

e2n−4

e2n−5

e2n−6

e5

e6 θ

Figure: The involution θ of the surface Σ2k+n−1


Surgery on symplectic 4-manifolds

Construction Tools


Surgery on symplectic 4-manifolds

Construction Tools

Symplectic Connected Sum (1995) (M. Gromov, R. Gompf, J. McCarthy-

J. Wolfson)

Luttinger Surgery (1995) (K. Luttinger, D. Auroux- S. Donaldson- L.Katzarkov)

Knot Surgery (1998) (R. Fintushel- R. Stern)


Surgery on symplectic 4-manifolds Luttinger Surgery

Luttinger surgery

Definition

Let X be a symplectic 4-manifold with a symplectic form ω, and the torus Λ be

a Lagrangian submanifold of X with self-intersection 0. Given a simple loop λon Λ, let λ′ be a simple loop on ∂(νΛ) that is parallel to λ under the

Lagrangian framing. For any integer m, the (Λ, λ, 1/m) Luttinger surgery on Xwill be XΛ,λ(1/m) = (X − ν(Λ)) ∪φ (S1 × S1 × D2), the 1/m surgery on Λ with

respect to λ under the Lagrangian framing. Here

φ : S1 × S1 × ∂D2 → ∂(X − ν(Λ)) denotes a gluing map satisfyingφ([∂D2]) = m[λ′] + [µΛ] in H1(∂(X − ν(Λ)), where µΛ is a meridian of Λ.

XΛ,λ(1/m) possesses a symplectic form that restricts to the original

symplectic form ω on X \ νΛ.

Luttinger’s surgery has been very effective tool recently for constructing exotic

smooth structures.



Example



Example

Let T4 = a × b × c × d ∼= (c × d)× (a × b). Let Kn be an n-twist knot.



Example

Let T4 = a × b × c × d ∼= (c × d)× (a × b). Let Kn be an n-twist knot. Let MKn

denote the result of performing 0 Dehn surgery on S3 along Kn. S1 × MKnis

obtained from T4 = (c × d)× (a × b) = c × (d × a × b) = S1 × T3 by first

performing a Luttinger surgery (c × a, a,−1) followed by a surgery(c × b, b,−n). The tori c × a and c × b are Lagrangian and the second tilde

circle factors in T3 are as pictured. Use the Lagrangian framing to trivialize

their tubular neighborhoods. When n = 1 the second surgery is also aLuttinger surgery.

b

d

b~

a a~

Figure: The 3-torus d × a × bAnar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 27 / 43

Surgery on symplectic 4-manifolds Symplectic Connected Sum

Symplectic Connected Sum

Definition

Let X1 and X2 are symplectic 4-manifolds, and Fi ⊂ Xi are 2-dimensional,

smooth, closed, connected symplectic submanifolds in them. Suppose that

[F1]2 + [F2]

2 = 0 and the genera of F1 and F2 are equal. Take anorientation-preserving diffemorphism ψ : F1 −→ F2 and lift it to an

orientation-reversing diffemorphism Ψ : ∂νF1 −→ ∂νF2 between theboundaries of the tubular neighborhoods of νFi . Using Ψ, we glue X1 \ νF1

and X2 \ νF2 along the boundary. The 4-manifold X1#ΨX2 is called the

(symplectic) connected sum of X1 and X2 along F1 and F2, determined by Ψ.

e(X1#ΨX2) = e(X1) + e(X2) + 4(g − 1),

σ(X1#ΨX2) = σ(X1) + σ(X2),


Surgery on symplectic 4-manifolds Symplectic Connected Sum

The vanishing cycles of twisted fiber sum of Lefschetz fibration

Lemma

Let f : X → S2 be a genus g Lefschetz fibration with global monodromy given

by the relation tα1· tα2

· · · · · tαs= 1. Let X#ψX denote the fiber sum of X with

itself by a self-diffemorphism ψ of the generic fiber Σ. Then X#ψX has the

vanishing cycles α1, α2, · · · , αs, ψ(α1), ψ(α2), · · · , ψ(αs).


Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn × Σ2 and Σn × T2

Luttinger surgeries on product manifolds Σn × Σ2 and Σn × T2

Fix integers n ≥ 2, pi ≥ 0 and qi ≥ 0 , where 1 ≤ i ≤ n. Let

Yn(1/p1, 1/q1, · · · , 1/pn, 1/qn) denote symplectic 4-manifold obtained byperforming the following 2n + 4 Luttinger surgeries on Σn × Σ2.

These 2n + 4 surgeries comprise of the following 8 surgeries

(a′

1 × c′

1, a′

1,−1), (b′

1 × c′′

1 , b′

1,−1),

(a′

2 × c′

2, a′

2,−1), (b′

2 × c′′

2 , b′

2,−1),

(a′

2 × c′

1, c′

1,+1/p1), (a′′

2 × d ′

1, d′

1,+1/q1),

(a′

1 × c′

2, c′

2,+1/p2), (a′′

1 × d ′

2, d′

2,+1/q2),

together with the following 2(n − 2) additional Luttinger surgeries

(b′

1 × c′

3, c′

3,−1/p3), (b′

2 × d ′

3, d′

3,−1/q3),

· · · , · · · ,

(b′

1 × c′

n, c′

n,−1/pn), (b′

2 × d ′

n, d′

n,−1/qn).



Here, ai , bi (i = 1, 2) and cj , dj (j = 1, . . . , n) are the standard loops thatgenerate π1(Σ2) and π1(Σn).

x

x

y

x

x

y

y y

x

y

aia′

i

ai

bi

d′

ja′′

idj

bi

cj

c′

j

cj

dj

Figure: Lagrangian tori a′

i × c′

j and a′′

i × d ′

j



The Euler characteristic of Yn(1/p1, 1/q1, · · · , 1/pn, 1/qn) is 4n − 4 and itssignature is 0. The fundamental group π1(Yn(1/p1, 1/q1, · · · , 1/pn, 1/qn)) is

generated by ai , bi , cj , dj (i = 1, 2 and j = 1, . . . , n) and the following relationshold in π1(Yn(1/p1, 1/q1, · · · , 1/pn, 1/qn)):

[b−11 , d−1

1 ] = a1, [a−11 , d1] = b1, [b−1

2 , d−12 ] = a2, [a−1

2 , d2] = b2, (2)

[d−11 , b−1

2 ] = cp1

1 , [c−11 , b2] = d

q1

1 , [d−12 , b−1

1 ] = cp2

2 , [c−12 , b1] = d

q2

2 ,

[a1, c1] = 1, [a1, c2] = 1, [a1, d2] = 1, [b1, c1] = 1,

[a2, c1] = 1, [a2, c2] = 1, [a2, d1] = 1, [b2, c2] = 1,

[a1, b1][a2, b2] = 1,

n∏

j=1

[cj , dj ] = 1,

[a−11 , d−1

3 ] = cp3

3 , [a−12 , c−1

3 ] = dq3

3 , . . . , [a−11 , d−1

n ] = cpnn , [a−1

2 , c−1n ] = d

qnn ,

[b1, c3] = 1, [b2, d3] = 1, . . . , [b1, cn] = 1, [b2, dn] = 1.



The surfaces Σ2 × pt and pt × Σn in Σ2 × Σn descend to surfaces inYn(1/p1, 1/q1, · · · , 1/pn, 1/qn). They are symplectic submanifolds in

Yn(1/p1, 1/q1, · · · , 1/pn, 1/qn). Denote their images by Σ2 and Σn. Note that

[Σ2]2 = [Σn]

2 = 0 and [Σ2] · [Σn] = 1.



Let pi , qi ≥ 0 : 1 ≤ i ≤ g be a set of nonnegative integers and let

p = (p1, . . . , pg) and q = (q1, . . . , qg).



Let pi , qi ≥ 0 : 1 ≤ i ≤ g be a set of nonnegative integers and let

p = (p1, . . . , pg) and q = (q1, . . . , qg). Denote by Mg(p, q) the symplectic

4-manifold obtained by performing the following 2g Luttinger surgeries on thesymplectic 4-manifold Σg × T2:

(a′

1 × c′, a′

1,−1/p1), (b′

1 × c′′, b′

1,−1/q1), (3)

(a′

2 × c′, a′

2,−1/p2), (b′

2 × c′′, b′

2,−1/q2),

· · · · · ·

(a′

g−1 × c′, a′

g−1,−1/pg−1), (b′

g−1 × c′′, b′

g−1,−1/qg−1),

(a′

g × c′, a′

g,−1/pg), (b′

g × c′′, bg,−1/qg).

Here, ai , bi (i = 1, 2, · · · , g) and c, d denote the standard generators of π1(Σg)and π1(T

2), respectively.



The fundamental group of Mg(p, q) is generated by ai , bi (i = 1, 2, 3 · · · , g)

and c, d , and the following relations hold in Mg(p, q):

[b−11 , d−1] = a1

p1 , [a−11 , d ] = b1

q1 , [b−12 , d−1] = a2

p2 , [a−12 , d ] = b2

q2 , (4)

· · · , · · · , · · · ,

[b−1g , d−1] = ag

pg , [a−1g , d ] = bg

qg , [a1, c] = 1, [b1, c] = 1, [a2, c] = 1, [b2, c] =

[a3, c] = 1, [b3, c] = 1,

[ag , c] = 1, [bg , c] = 1,

[a1, b1][a2, b2] · · · [ag, bg ] = 1, [c, d ] = 1.

Let Σg ⊂ Mg(p, q) and T be a genus g and genus 1 surfaces that desendfrom the surfaces Σg × pt and pt × T2 in Σg × T2.


Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery

Mg(p, q) is a locally trivial genus g bundle over T2 where T is a section.

The (a′

i × c′, a′

i ,−pi) or (b′

i × c′′, b′

i ,−qi) Luttinger surgery in the trivialbundle Σg × T2 preserves the fibration structure over T2 introducing a

monodromy of the fiber Σg along the curve c in the base. Depending on

the type of the surgery the monodromy is either (tai)pi or (tbi

)qi , where tdenotes a Dehn twist.

E(n) can be obtained as a desingularization of the branched doublecover of S2 × S2 with the branching set being 4 copies of pt × S2 and 2n

copies of S2 × pt.



Mg(p, q) is a locally trivial genus g bundle over T2 where T is a section.

The (a′

i × c′, a′

i ,−pi) or (b′

i × c′′, b′

i ,−qi) Luttinger surgery in the trivialbundle Σg × T2 preserves the fibration structure over T2 introducing a

monodromy of the fiber Σg along the curve c in the base. Depending on

the type of the surgery the monodromy is either (tai)pi or (tbi

)qi , where tdenotes a Dehn twist.

E(n) can be obtained as a desingularization of the branched doublecover of S2 × S2 with the branching set being 4 copies of pt × S2 and 2n

copies of S2 × pt. E(n) admits a genus n − 1 fibration over S2 and an

elliptic fibration over S2. A regular fiber of the elliptic fibration on E(n)intersects every genus n − 1 fiber of the other Lefschetz fibration twice.



Construction of Lefschetz fibrations over S2

Let Xg,n(p, q) denote the symplectic sum of Mg(p, q) along the torus

T = c × d with the elliptic surface E(n) along a regular elliptic fiber.

The symplectic 4-manifold Xg,n(p, q) admits a genus 2g + n − 1

Lefschetz fibration over S2 with at least 4n + 4 pairwise disjoint spheresections of self intersection −2. Moreover, Xg,n(p, q) contains a

homologically essential embedded torus of square zero disjoint from

these sections which intersects each fiber of the Lefschetz fibration twice.

The symplectic 4-manifold Xg,n(p, q) can also be constructed as the

twisted fiber sum of two copies of a genus 2g + n − 1 Lefschetz fibrationon Σg × S2 #4nCP2. This follows from the fact that the symplectic sum of

E(n) along a regular elliptic fiber with Σg ×T2 along a natural square zero

torus is diffeomorphic to the untwisted fiber sum of two copies of thegenus 2g + n − 1 fibration on Σg × S2 #4nCP2. The gluing φdiffeomorphism can be described explicitly using the curves along which

we perform our Luttinger surgeries.



The fundamental group of the symplectic 4-manifold Xg,n(p, q) is

generated by the set ai , bi : 1 ≤ i ≤ g subject to the relations:a

pi

i = 1, bqi

i = 1, for all 1 ≤ i ≤ g, and

Πgj=1[aj , bj ] = 1.

By setting pi = 1 and qi = 0, for all 1 ≤ i ≤ g, we see that the

fundamental group of Xg,n((1, 1, . . . , 1), (0, 0, . . . , 0)) is a free group of

rank g. The gluing diffeomorphism: φ = ta1· · · tag

.

By setting pi = 1 and qi = 1, for all 1 ≤ i ≤ g, we see that the

fundamental group of Xg,n((1, 1, . . . , 1), (1, 1, . . . , 1)) is a trivial. Thegluing diffeomorphism: φ = ta1

tb1· · · tag

tbg.

If we set pi = 1 and qi = 0, for all 1 ≤ i ≤ k and pi = 1 and qi = 1, for all

k + 1 ≤ i ≤ g, the fundamental group of Xg,n((1, 1, . . . , 1), (1, 1, . . . , 0)) isa a free group of rank k . The gluing diffeomorphism:

φ = ta1· · · tak

tak+1tbk+1

· · · tagtbg

.


Construction of Symplectic 4-Manifolds Exotic Stein fillings

Stein fillings from Lefschetz fibrations

Definition

A complex surface V is Stein if it admits a proper holomorphic embedding

f : V → Cn for some n. For a generic point p ∈ C

n, consider the map

φ : V → R defined by φ(z) = ||z − p||2. For a regular value a ∈ R, the level set

M = φ−1(a) is a smooth 3-manifold with a distinguished 2-plane field

ξ = TM ∩ iTM ⊂ TV . ξ defines a contact structure on M, and S = φ−1([0, a])is called a Stein filling of (M, ξ).

Theorem (S. Akbulut - B. Ozbagci)

Let f : X → S2 be a Lefschetz fibration with a section σ and let Σ denote a

regular fiber of this fibration. Then S = X \ int(ν(σ ∪ Σ)) is a Stein filling of its

boundary equipped with the induced (tight) contact structure, where ν(σ ∪ Σ)denotes a regular neighborhood of σ ∪ Σ in X.



Finiteness Results on Stein Fillings

The tight contact structure on S3 has a unique Stein filling (Y. Eliashberg,1989).

All tight contact structures on lens spaces L(p, q) have a finite number of

Stein fillings (D. McDuff, P. Lisca, 1992).

Finiteness results also have been verified for simple elliptic singularities

(H. Ohta and Y. Ono, 2002).

Finiteness results on symplectic fillings of Seifert fibered spaces over S2

(L. Starkston, 2013).



Infiniteness Results on Stein Fillings

B. Ozbagci and A. Stipsicz, and independently I. Smith showed that

certain contact structures have an infinite number of Stein fillings (2003).Their examples have non-trivial fundamental groups.

Infinitely many simply-connected exotic Stein fillings (Akhmedov - Etnyre- Mark - Smith, 2007).

Small exotic Stein fillings (S. Akbulut - K. Yasui, 2008).

Exotic Stein fillings with π1 = Z⊕ Zn (Akhmedov - Ozbagci, 2012).




For any finitely presented group G, there exist an infinite family of exotic Stein

4-manifolds Sn(G,Ki) with π1(Sn(G,Ki)) = G, where Ki are inf. family ofgenus g ≥ 2 fibered knots with distinct Alexander polynomials.



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Construction of Lefschetz fibrations via Luttinger surgery

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