Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 2 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Symplectic manifolds
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
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. A diffeomorphism
f : (X1, ω1) → (X2, ω2) is a symplectomorphism if ω1 = f ∗(ω2).
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
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. A diffeomorphism
f : (X1, ω1) → (X2, ω2) is a symplectomorphism if ω1 = f ∗(ω2).
Examples
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
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. A diffeomorphism
f : (X1, ω1) → (X2, ω2) is a symplectomorphism if ω1 = f ∗(ω2).
Examples
X = R2n with linear coordinates x1, · · · , xn, y1, · · · , yn and with the 2-form
ω0 =∑n
i=1 dxi ∧ dyi .
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
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. A diffeomorphism
f : (X1, ω1) → (X2, ω2) is a symplectomorphism if ω1 = f ∗(ω2).
Examples
X = R2n with linear coordinates x1, · · · , xn, y1, · · · , yn and with the 2-form
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
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. A diffeomorphism
f : (X1, ω1) → (X2, ω2) is a symplectomorphism if ω1 = f ∗(ω2).
Examples
X = R2n with linear coordinates x1, · · · , xn, y1, · · · , yn and with the 2-form
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. Σn × Σm are symplectic 4-manifolds
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
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. A diffeomorphism
f : (X1, ω1) → (X2, ω2) is a symplectomorphism if ω1 = f ∗(ω2).
Examples
X = R2n with linear coordinates x1, · · · , xn, y1, · · · , yn and with the 2-form
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. Σn × Σm are symplectic 4-manifolds
Every Kähler manifold is also a symplectic manifold.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
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. A diffeomorphism
f : (X1, ω1) → (X2, ω2) is a symplectomorphism if ω1 = f ∗(ω2).
Examples
X = R2n with linear coordinates x1, · · · , xn, y1, · · · , yn and with the 2-form
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. Σn × Σm are symplectic 4-manifolds
Every Kähler manifold is also a symplectic manifold.
A closed complex surface S is Kähler iff the first Betti number b1(S) is
even.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 3 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 4 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold
Example
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 4 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold
Example
Consider R4 with the 2-form ω0 = dx1 ∧ dy1 + dx2 ∧ dy2.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 4 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold
Example
Consider R4 with the 2-form ω0 = dx1 ∧ dy1 + dx2 ∧ dy2.
Let Γ be the discrete group generated by the following symplectomorphisms:
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 4 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold
Example
Consider R4 with the 2-form ω0 = dx1 ∧ dy1 + dx2 ∧ dy2.
Let Γ be the discrete group generated by the following symplectomorphisms:
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)
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 4 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold
Example
Consider R4 with the 2-form ω0 = dx1 ∧ dy1 + dx2 ∧ dy2.
Let Γ be the discrete group generated by the following symplectomorphisms:
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 4 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold continued
Example
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 5 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold continued
Example
Let φ = Db denote the right-handed Dehn twist on T2 = a × b along the curve
b.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 5 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold continued
Example
Let φ = Db denote the right-handed Dehn twist on T2 = a × b along the curve
b.
φ∗(a) = a + b
φ∗(b) = b
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 5 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold continued
Example
Let φ = Db denote the right-handed Dehn twist on T2 = a × b along the curve
b.
φ∗(a) = a + b
φ∗(b) = b
Let Zφ denote the mapping torus of φ.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 5 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold continued
Example
Let φ = Db denote the right-handed Dehn twist on T2 = a × b along the curve
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 5 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Kodaira-Thurston manifold continued
Example
Let φ = Db denote the right-handed Dehn twist on T2 = a × b along the curve
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〉.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 5 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
Symplectic 4-manifolds via surface bundles over surfaces
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 6 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 6 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 6 / 43
Symplectic 4-manifolds and Lefschetz fibrations Symplectic 4-manifolds
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 (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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 6 / 43
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 7 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
Vanishing cycle α
S2
X
f
Figure: Lefschetz fibration on X over S2
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 8 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 9 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
The genus of the regular fiber of f is defined to be the genus of the Lefschetzfibration.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 9 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
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, ...)
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 9 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
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
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 11 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
Example (Genus one Lefschetz fibrations)
M1 be the mapping class group of the torus T2 = a × b.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 12 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
Example (Genus one Lefschetz fibrations)
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 12 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
Example (Genus one Lefschetz fibrations)
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
(tatb)6n = 1 in M1.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 12 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
Example (Genus one Lefschetz fibrations)
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
(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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 12 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
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)
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 13 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 14 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 15 / 43
Symplectic 4-manifolds and Lefschetz fibrations Lefschetz fibrations
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.
Theorem (A. Akhmedov - B. Ozbagci, 2012)
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 16 / 43
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 17 / 43
Symplectic 4-manifolds and Lefschetz fibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 18 / 43
Symplectic 4-manifolds and Lefschetz fibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 19 / 43
Symplectic 4-manifolds and Lefschetz fibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 20 / 43
Symplectic 4-manifolds and Lefschetz fibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 21 / 43
Symplectic 4-manifolds and Lefschetz fibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 22 / 43
Symplectic 4-manifolds and Lefschetz fibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 23 / 43
Surgery on symplectic 4-manifolds
Construction Tools
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 24 / 43
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)
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 25 / 43
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 26 / 43
Surgery on symplectic 4-manifolds Luttinger Surgery
Example
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 27 / 43
Surgery on symplectic 4-manifolds Luttinger Surgery
Example
Let T4 = a × b × c × d ∼= (c × d)× (a × b). Let Kn be an n-twist knot.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 27 / 43
Surgery on symplectic 4-manifolds Luttinger Surgery
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),
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 28 / 43
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).
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 29 / 43
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).
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 30 / 43
Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn × Σ2 and Σn × T2
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
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 31 / 43
Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn × Σ2 and Σn × T2
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 32 / 43
Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn × Σ2 and Σn × T2
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 33 / 43
Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn × Σ2 and Σn × T2
Let pi , qi ≥ 0 : 1 ≤ i ≤ g be a set of nonnegative integers and let
p = (p1, . . . , pg) and q = (q1, . . . , qg).
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 34 / 43
Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn × Σ2 and Σn × T2
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 34 / 43
Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn × Σ2 and Σn × T2
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 35 / 43
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 36 / 43
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. 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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 36 / 43
Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 37 / 43
Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery
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
.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 38 / 43
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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 39 / 43
Construction of Symplectic 4-Manifolds Exotic Stein fillings
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).
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 40 / 43
Construction of Symplectic 4-Manifolds Exotic Stein fillings
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).
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 41 / 43
Construction of Symplectic 4-Manifolds Exotic Stein fillings
Theorem (A. Akhmedov - B. 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.
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 42 / 43
Construction of Symplectic 4-Manifolds Exotic Stein fillings
THANK YOU!
Anar Akhmedov (University of Minnesota, Minneapolis)Construction of Lefschetz fibrations via Luttinger surgery August 2, 2013 43 / 43