Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via Luttinger Surgery

Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via Luttinger Surgery Anar Akhmedov University of Minnesota, Twin Cities June 20, 2013, ESI, Vienna Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 1 / 42

Outline 1 Symplectic 4-manifolds Lefschetz fibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas 2 Surgery on symplectic 4-manifolds Luttinger Surgery Symplectic Connected Sum 3 Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σ n Σ 2 and Σ n T 2 Construction of Lefschetz fibration via Luttinger Surgery Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 2 / 42

Symplectic 4-manifolds Symplectic manifolds Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 3 / 42

Symplectic 4-manifolds Symplectic manifolds Definition A(compact)symplectic2n-manifold (X, ω) is a smooth 2n-manifold with a symplectic form ω Ω 2 (X) (i.e., ω is closed (dω = 0) andnon-degenerate (ω n = ω ω > 0everywhere)2-form. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 3 / 42

Symplectic 4-manifolds Symplectic manifolds Definition A(compact)symplectic2n-manifold (X, ω) is a smooth 2n-manifold with a symplectic form ω Ω 2 (X) (i.e., ω is closed (dω = 0) andnon-degenerate (ω n = ω ω > 0everywhere)2-form.Adiffeomorphism f :(X 1, ω 1 ) (X 2, ω 2 ) is a symplectomorphism if ω 1 = f (ω 2 ). Examples X = R 2n with linear coordinates x 1,, x n, y 1,, y n and with the 2-form ω 0 = n i=1 dx i dy i. If (X 1, ω 1 ) and (X 2, ω 2 ) are symplectic manifolds, then π 1 ω 1 + π 2 ω 2 gives asymplecticstructureonx 1 X 2. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 3 / 42

Symplectic 4-manifolds Symplectic manifolds Definition A(compact)symplectic2n-manifold (X, ω) is a smooth 2n-manifold with a symplectic form ω Ω 2 (X) (i.e., ω is closed (dω = 0) andnon-degenerate (ω n = ω ω > 0everywhere)2-form.Adiffeomorphism f :(X 1, ω 1 ) (X 2, ω 2 ) is a symplectomorphism if ω 1 = f (ω 2 ). Examples X = R 2n with linear coordinates x 1,, x n, y 1,, y n and with the 2-form ω 0 = n i=1 dx i dy i. If (X 1, ω 1 ) and (X 2, ω 2 ) are symplectic manifolds, then π 1 ω 1 + π 2 ω 2 gives asymplecticstructureonx 1 X 2. Σ n Σ m are symplectic 4-manifolds Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 3 / 42

Symplectic 4-manifolds Symplectic manifolds Definition A(compact)symplectic2n-manifold (X, ω) is a smooth 2n-manifold with a symplectic form ω Ω 2 (X) (i.e., ω is closed (dω = 0) andnon-degenerate (ω n = ω ω > 0everywhere)2-form.Adiffeomorphism f :(X 1, ω 1 ) (X 2, ω 2 ) is a symplectomorphism if ω 1 = f (ω 2 ). Examples X = R 2n with linear coordinates x 1,, x n, y 1,, y n and with the 2-form ω 0 = n i=1 dx i dy i. If (X 1, ω 1 ) and (X 2, ω 2 ) are symplectic manifolds, then π 1 ω 1 + π 2 ω 2 gives asymplecticstructureonx 1 X 2. Σ n Σ m are symplectic 4-manifolds Every Kähler manifold is also a symplectic manifold. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 3 / 42

Symplectic 4-manifolds Symplectic manifolds Definition A(compact)symplectic2n-manifold (X, ω) is a smooth 2n-manifold with a symplectic form ω Ω 2 (X) (i.e., ω is closed (dω = 0) andnon-degenerate (ω n = ω ω > 0everywhere)2-form.Adiffeomorphism f :(X 1, ω 1 ) (X 2, ω 2 ) is a symplectomorphism if ω 1 = f (ω 2 ). Examples X = R 2n with linear coordinates x 1,, x n, y 1,, y n and with the 2-form ω 0 = n i=1 dx i dy i. If (X 1, ω 1 ) and (X 2, ω 2 ) are symplectic manifolds, then π 1 ω 1 + π 2 ω 2 gives asymplecticstructureonx 1 X 2. Σ n Σ m are symplectic 4-manifolds Every Kähler manifold is also a symplectic manifold. AclosedcomplexsurfaceS is Kähler iff the first Betti number b 1 (S) is even. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 3 / 42

Symplectic 4-manifolds Kodaira-Thurston manifold Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 4 / 42

Symplectic 4-manifolds Kodaira-Thurston manifold Example Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 4 / 42

Symplectic 4-manifolds Kodaira-Thurston manifold Example Consider R 4 with the 2-form ω 0 = dx 1 dy 1 + dx 2 dy 2. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 4 / 42

Symplectic 4-manifolds Kodaira-Thurston manifold Example Consider R 4 with the 2-form ω 0 = dx 1 dy 1 + dx 2 dy 2. Let Γ be the discrete group generated by the following symplectomorphisms: Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 4 / 42

Symplectic 4-manifolds Kodaira-Thurston manifold Example Consider R 4 with the 2-form ω 0 = dx 1 dy 1 + dx 2 dy 2. Let Γ be the discrete group generated by the following symplectomorphisms: γ 1 :(x 1, x 2, y 1, y 2 ) (x 1, x 2 + 1, y 1, y 2 ) γ 2 :(x 1, x 2, y 1, y 2 ) (x 1, x 2, y 1, y 2 + 1) γ 3 :(x 1, x 2, y 1, y 2 ) (x 1 + 1, x 2, y 1, y 2 ) γ 4 :(x 1, x 2, y 1, y 2 ) (x 1, x 2 + y 2, y 1 + 1, y 2 ) Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 4 / 42

Symplectic 4-manifolds Kodaira-Thurston manifold Example Consider R 4 with the 2-form ω 0 = dx 1 dy 1 + dx 2 dy 2. Let Γ be the discrete group generated by the following symplectomorphisms: γ 1 :(x 1, x 2, y 1, y 2 ) (x 1, x 2 + 1, y 1, y 2 ) γ 2 :(x 1, x 2, y 1, y 2 ) (x 1, x 2, y 1, y 2 + 1) γ 3 :(x 1, x 2, y 1, y 2 ) (x 1 + 1, x 2, y 1, y 2 ) γ 4 :(x 1, x 2, y 1, y 2 ) (x 1, x 2 + y 2, y 1 + 1, y 2 ) M = R 4 /Γ admits both symplectic structure and complex structures, but non-kähler. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 4 / 42

Symplectic 4-manifolds Symplectic 4-manifolds via surface bundles over surfaces Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 5 / 42

Symplectic 4-manifolds Symplectic 4-manifolds via surface bundles over surfaces Theorem (W. Thurston (1976)) Assume that Σ g, Σ h are closed, oriented, 2-dimensional surfaces. If f : X Σ h is a bundle with fiber Σ g and the homology class of the fiber is nonzero in H 2 (X; R), thenxadmitsasymplecticstructure. Theorem (J. Bryan - R. Donagi) For any integers n 2, thereexisitsmoothalgebraicsurfacex n that have signature σ(x n )=8/3n(n 1)(n + 1) and admit two smooth fibrations X n BandX n B such that the base and fiber genus are (3, 3n 3 n 2 + 1) and (2n 2 + 1, 3n) respectively. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 5 / 42

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 Σ h,whereσ h 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 : C 2 C is given by f (z 1, z 2 )=z 1 2 + z 2 2. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 6 / 42

Lefschetz fibrations Vanishing cycle α X f S 2 Figure: Lefschetz fibration on X over S 2 Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 7 / 42

Lefschetz fibrations Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 8 / 42

Lefschetz fibrations The genus of the regular fiber of f is defined to be the genus of the Lefschetz fibration. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 8 / 42

Lefschetz fibrations The genus of the regular fiber of f is defined to be the genus of the Lefschetz fibration. Letp 1,, p s denote the critical points of Lefschetz fibration f : X Σ h. e(x) =e(σ h )e(σ g )+s σ(x) well understood for fibrations over S 2 (Y.Matsumoto, H.Endo, B.Ozbagci, I.Smith,...) Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group viajune Luttinger 20, 2013 Surgery 8 / 42

Lefschetz fibrations Anar Akhmedov (University of Minnesota, Figure: Minneapolis)Lefschetz Vanishing Fibrationscycle and Exoticof Stein Lefschetz Fillings with Arbitrary fibration Fundamental Group viajune Luttinger 20, 2013 Surgery 9 / 42 Monodromy of Lefschetz fibration A singular fiber of the genus g Lefschetz fibration can be described by its monodromy, i.e., an element of the mapping class group M g.thiselementis aright-handed(orapositive)dehntwistalongasimpleclosed curve on Σ g, called the vanishing cycle. Vanishing cycle α X f S 2

Lefschetz fibrations For a genus g Lefschetz fibration over S 2,theproductofrighthandedDehn twists t αi along the vanishing cycles α i,fori = 1,, s, determinestheglobal monodromy of the Lefschetz fibration, the relation t α1 t α2 t αs = 1inM g. Conversely, such a relation in M g determines a genus g Lefschetz fibration over S 2 with the vanishing cycles α 1,, α s. Vanishing cycle α X f S 2 Figure: Lefschetz fibration on X over S 2 Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery10 / 42

Lefschetz fibrations The fundamental group of Lefschetz fibration Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery11 / 42

Lefschetz fibrations The fundamental group of Lefschetz fibration Lemma Let f : X S 2 be a genus g Lefschetz fibration with global monodromy given by the relation t α1 t α2 t αs = 1. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery11 / 42

Lefschetz fibrations The fundamental group of Lefschetz fibration Lemma Let f : X S 2 be a genus g Lefschetz fibration with global monodromy given by the relation t α1 t α2 t αs = 1. Supposethatfhasasection. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery11 / 42

Lefschetz fibrations The fundamental group of Lefschetz fibration Lemma Let f : X S 2 be a genus g Lefschetz fibration with global monodromy given by the relation t α1 t α2 t αs = 1. Supposethatfhasasection.Thenthe fundamental group of X is isomorphic to the fundamental group ofσ g divided out by the normal closure of the simple closed curves α 1, α 2,...,α s, considered as elements in π 1 (Σ g ).Inparticular,thereisanepimorphism π 1 (Σ g ) π 1 (X) Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery11 / 42

Lefschetz fibrations Example (Genus one Lefschetz fibrations) M 1 be the mapping class group of the torus T 2 = a b. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery12 / 42

Lefschetz fibrations Example (Genus one Lefschetz fibrations) M 1 be the mapping class group of the torus T 2 = a b. M 1 = SL(2, Z) is generated by ( ) 1 1 t a = 0 1 t b = ( 1 ) 0 1 1 Subject to the relations t a t b t a = t b t a t b (t a t b ) 6 = 1 Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery12 / 42

Lefschetz fibrations Example (Genus one Lefschetz fibrations) M 1 be the mapping class group of the torus T 2 = a b. M 1 = SL(2, Z) is generated by ( ) 1 1 t a = 0 1 t b = ( 1 ) 0 1 1 Subject to the relations t a t b t a = t b t a t b (t a t b ) 6 = 1 (t a t b ) 6n = 1inM 1. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery12 / 42

Lefschetz fibrations Example (Genus one Lefschetz fibrations) M 1 be the mapping class group of the torus T 2 = a b. M 1 = SL(2, Z) is generated by ( ) 1 1 t a = 0 1 t b = ( 1 ) 0 1 1 Subject to the relations t a t b t a = t b t a t b (t a t b ) 6 = 1 (t a t b ) 6n = 1inM 1.ThetotalspaceofthisfibrationistheellipticsurfaceE(n). E(1) =CP 2 #9CP 2,thecomplexprojectiveplaneblownupat9points,and E(2) is K 3surface.E(n) also admits a genus n 1Lefschetzfibrationover S 2. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery12 / 42

Lefschetz fibrations Example (Hyperelliptic Lefschetz fibrations) Let α 1, α 2,...,α 2g, α 2g+1 denote the collection of simple closed curves given in Figure, and c i denote the right handed Dehn twists t αi along the curve α i. α2 α4 α2g 2 α2g α1 α3 α2g 1 α2g+1 Figure: Vanishing cycles of the genus g Lefschetz fibration given by hyperelliptic involution The following relations hold in the mapping class group M g : Γ 1 (g) =(c 1 c 2 c 2g 1 c 2g c 2g+1 2 c 2g c 2g 1 c 2 c 1 ) 2 = 1. Γ 2 (g) =(c 1 c 2 c 2g 1 c 2g c 2g+1 ) 2g+2 = 1. Γ 3 (g) =(c 1 c 2 c 2g 1 c 2g ) 2(2g+1) = 1. (1) Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery13 / 42

Lefschetz fibrations The monodromy relation Γ 1 (g) =1, the corresponding genus g Lefschetz fibrations over S 2 has total space X(g, 1) =CP 2 #(4g + 5)CP 2,thecomplex projective plane blown up at 4g + 5points. It is known that for g 2, the above fibration on X(g, 1) admits 4g + 4disjoint ( 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 ge 1 e 2 e 4g+5,wheree i denotes the homology class of the exceptional sphere of the i-th blow up and h denotes the pullback of the hyperplane class of CP 2.Theexceptional spheres represented by the homology classes e 2, e 3,...,e 4g+5 are sections of the Lefschetz fibration X(g, 1) S 2. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery14 / 42

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#nCP 2 of X admits a Lefschetz fibration f : X#nCP 2 S 2. Theorem (R. Gompf) Assume that the closed 4-manifold X admits a genus g Lefschetz fibration f : X Σ h,andlet[f] denote the homology class of the fiber. Then X admits asymplecticstructurewithsymplecticfibersiff[f] 0 in H 2 (X; R). Ife 1,, e n is a finite set of sections of the Lefschetz fibration, the symplectic form ω can 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 S 2 with π 1 (X(G)) = G. b 2 + (X(G)) is very large, anddependsfromthepresentationofg. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery15 / 42

Lefschetz fibrations Main Theorems Theorem (A. Akhmedov - B. Ozbagci, 2012) For any finitely presented group G, there exist a closed symplectic 4-manifold X n (G) with π 1 (X(G)) = G, which admits a genus 2g + n 1 Lefschetz fibration over S 2 that has al least 4n + 4 pairwise disjoint sphere sections of self intersection 2. Moreover,X n (G) contains a homologically essential embedded 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 X n (G, K i ) over S 2 with π 1 (X n (G, K i )) = GthatcanbeobtainedfromX n (G) via knot surgery along K i,wherek i are inf. family of genus g 2 fibered knots with distinct Alexander polynomials. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery16 / 42

Symplectic 4-manifoldsand Lefschetzfibrations 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) =Σ k S 2 #4CP 2 is the total space of the genus g Lefschetz fibration over S 2 with 2g + 4singularfibers.Thiswasshownby Yukio Matsumoto for k = 1, and in the case k 2byMustafaKorkmaz,by factorizing the vertical involution θ of the genus 2k surface. 1 2 k θ k +1 2k 1 2k Figure: The involution θ of the genus 2k surface Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery17 / 42

Symplectic 4-manifoldsand Lefschetzfibrations 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 M g,thefollwingrelationsbetweenrighthanded Dehn twists hold: a) (t B0 t B1 t B2 t Bg t c ) 2 = θ 2 = 1 if g is even, b) (t B0 t B1 t B2 t Bg (t a ) 2 (t b ) 2 ) 2 = θ 2 = 1 if g is odd. B k, a, b, c are the simple closed curves defined as in Figure. B0 c B1 B2 Bg a Bg B0 B1 B2 b Figure: The vanishing cycles Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery18 / 42

Symplectic 4-manifoldsand Lefschetzfibrations 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 of involutions in the mapping class group M 2k+n 1 of a compact, closed, oriented 2-dimensional surface Σ 2k+n 1.Thetotalspaceofthesegenus g = 2k + n 1LefschetzfibrationoverS 2 is Y (n, k) =Σ k S 2 #4nCP 2. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery19 / 42

Symplectic 4-manifoldsand Lefschetzfibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas 1 2 k e2 e4 e6 e2n 6 e2n 4 e2n 2 θ e1 e3 e5 e2n 1 e2n 5 e2n 3 k +1 k +2 2k Figure: The involution θ of the surface Σ 2k+n 1 Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery20 / 42

Symplectic 4-manifoldsand Lefschetzfibrations Family of Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas The branched-cover description for Σ k S 2 #4nCP 2 AgenerichorizontalfiberisthedoublecoverofS 2,branchedovertwopoints. Thus, we have a sphere fibration on Y (n, k) =Σ k S 2 #4nCP 2.Ageneric fiber of the vertical fibration is the double cover of Σ k,branchedover2n points. Thus, a generic fiber of the vertical fibration has genus n + 2k 1. Σ g Σ g S 2 S 2 S 2 S 2 2n copies S 2 S 2 S 2 S 2 Figure: The branch locus for Σ k S 2 #4nCP 2 Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery21 / 42

Symplectic 4-manifoldsand Lefschetzfibrations 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 = e 2i+2 e 2n 2 e 2n 1 e 2i e 1 B 0 e 2n 1 e 2i+2 e 1 e 2i B 1 B 2 B 4k 1 B 4k e 2i+1. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery22 / 42

Surgery on symplectic 4-manifolds Construction Tools Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery23 / 42

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)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery24 / 42

Surgery on symplectic 4-manifolds Luttinger Surgery Luttinger surgery Definition Let X be a symplectic 4-manifold with a symplectic form ω, andthetorusλ be alagrangiansubmanifoldofx with self-intersection 0. Given a simple loop λ on Λ, letλ 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 X will be X Λ,λ (1/m) =(X ν(λ)) φ (S 1 S 1 D 2 ),the1/m surgery on Λ with respect to λ under the Lagrangian framing. Here φ : S 1 S 1 D 2 (X ν(λ)) denotes a gluing map satisfying φ([ D 2 ]) = m[λ ]+[µ Λ ] in H 1 ( (X ν(λ)), whereµ Λ is a meridian of Λ. X Λ,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)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery25 / 42

Surgery on symplectic 4-manifolds Luttinger Surgery Example Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery26 / 42

Surgery on symplectic 4-manifolds Luttinger Surgery Example Let T 4 = a b c d = (c d) (a b). LetK n be an n-twist knot. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery26 / 42

Surgery on symplectic 4-manifolds Luttinger Surgery Example Let T 4 = a b c d = (c d) (a b). LetK n be an n-twist knot. Let M Kn denote the result of performing 0 Dehn surgery on S 3 along K n. S 1 M Kn is obtained from T 4 =(c d) (a b) =c (d a b) =S 1 T 3 by first performing a Luttinger surgery (c ã, ã, 1) followed by a surgery (c b, b, n). Thetoric ã and c b are Lagrangian and the second tilde circle factors in T 3 are as pictured. Use the Lagrangian framing to trivialize their tubular neighborhoods. When n = 1thesecondsurgeryisalsoa Luttinger surgery. a a ~ d ~ b b Figure: The 3-torus d a b Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery26 / 42

Surgery on symplectic 4-manifolds Symplectic Connected Sum Symplectic Connected Sum Definition Let X 1 and X 2 are symplectic 4-manifolds, and F i X i are 2-dimensional, smooth, closed, connected symplectic submanifolds in them. Supposethat [F 1 ] 2 +[F 2 ] 2 = 0andthegeneraofF 1 and F 2 are equal. Take an orientation-preserving diffemorphism ψ : F 1 F 2 and lift it to an orientation-reversing diffemorphism Ψ : νf 1 νf 2 between the boundaries of the tubular neighborhoods of νf i.usingψ, wegluex 1 \ νf 1 and X 2 \ νf 2 along the boundary. The 4-manifold X 1 # Ψ X 2 is called the (symplectic) connected sum of X 1 and X 2 along F 1 and F 2,determinedbyΨ. e(x 1 # Ψ X 2 ) = e(x 1 )+e(x 2 )+4(g 1), σ(x 1 # Ψ X 2 ) = σ(x 1 )+σ(x 2 ), Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery27 / 42

Surgery on symplectic 4-manifolds Symplectic Connected Sum The vanishing cycles of twisted fiber sum of Lefschetz fibration Lemma Let f : X S 2 be a genus g Lefschetz fibration with global monodromy given by the relation t α1 t α2 t αs = 1. LetX# ψ XdenotethefibersumofXwith itself by a self-diffemorphism ψ of the generic fiber Σ. ThenX# ψ Xhasthe vanishing cycles α 1, α 2,, α s, ψ(α 1 ), ψ(α 2 ),, ψ(α s ). Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery28 / 42

Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn Σ 2 and Σn T 2 Luttinger surgeries on product manifolds Σ n Σ 2 and Σ n T 2 Fix integers n 2, p i 0andq i 0,where1 i n. Let Y n (1/p 1, 1/q 1,, 1/p n, 1/q n ) denote symplectic 4-manifold obtained by performing the following 2n + 4LuttingersurgeriesonΣ n Σ 2. These 2n + 4surgeriescompriseofthefollowing8surgeries (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/p 1), (a 2 d 1, d 1, +1/q 1), (a 1 c 2, c 2, +1/p 2 ), (a 1 d 2, d 2, +1/q 2 ), together with the following 2(n 2) additional Luttinger surgeries (b 1 c 3, c 3, 1/p 3), (b 2 d 3, d 3, 1/q 3),,, (b 1 c n, c n, 1/p n), (b 2 d n, d n, 1/q n). Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery29 / 42

Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn Σ 2 and Σn T 2 Here, a i, b i (i = 1, 2) and c j, d j (j = 1,...,n) arethestandardloopsthat generate π 1 (Σ 2 ) and π 1 (Σ n ). x y b i c j x y a i a i a i d j d j x y c j b i c j x a i x y d j y Figure: Lagrangian tori a i c j and a i d j Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery30 / 42

Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn Σ 2 and Σn T 2 The Euler characteristic of Y n (1/p 1, 1/q 1,, 1/p n, 1/q n ) is 4n 4andits signature is 0. The fundamental group π 1 (Y n (1/p 1, 1/q 1,, 1/p n, 1/q n )) is generated by a i, b i, c j, d j (i = 1, 2andj = 1,...,n) andthefollowingrelations hold in π 1 (Y n (1/p 1, 1/q 1,, 1/p n, 1/q n )): [b 1 1, d 1 1 ]=a 1, [a 1 1, d 1]=b 1, [b 1 2, d 1 2 ]=a 2, [a 1 2, d 2]=b 2, (2) [d 1 1, b 1 2 ]=cp 1 1, [c 1, [d 1 2, b 1 1 ]=cp 2 2, [c 1 2, b 1]=d q 2 2, 1, b 2]=d q 1 1 [a 1, c 1 ]=1, [a 1, c 2 ]=1, [a 1, d 2 ]=1, [b 1, c 1 ]=1, [a 2, c 1 ]=1, [a 2, c 2 ]=1, [a 2, d 1 ]=1, [b 2, c 2 ]=1, n [a 1, b 1 ][a 2, b 2 ]=1, [c j, d j ]=1, [a 1 1, d 1 3 ]=c p 3 3, [a 1 2, c 1 3 ]=d q 3 3,...,[a 1 1, d n 1 ]=cn pn, [a 1 [b 1, c 3 ]=1, [b 2, d 3 ]=1,...,[b 1, c n ]=1, [b 2, d n ]=1. j=1 2, c 1 n ]=d qn n, Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery31 / 42

Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn Σ 2 and Σn T 2 The surfaces Σ 2 {pt} and {pt} Σ n in Σ 2 Σ n descend to surfaces in Y n (1/p 1, 1/q 1,, 1/p n, 1/q n ).Theyaresymplecticsubmanifoldsin Y n (1/p 1, 1/q 1,, 1/p n, 1/q n ).DenotetheirimagesbyΣ 2 and Σ n.notethat [Σ 2 ] 2 =[Σ n ] 2 = 0and[Σ 2 ] [Σ n ]=1. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery32 / 42

Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn Σ 2 and Σn T 2 Let {p i, q i 0 : 1 i g} be a set of nonnegative integers and let p =(p 1,...,p g ) and q =(q 1,...,q g ). Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery33 / 42

Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn Σ 2 and Σn T 2 Let {p i, q i 0 : 1 i g} be a set of nonnegative integers and let p =(p 1,...,p g ) and q =(q 1,...,q g ).DenotebyM g (p, q) the symplectic 4-manifold obtained by performing the following 2g Luttinger surgeries on the symplectic 4-manifold Σ g T 2 : (a 1 c, a 1, 1/p 1), (b 1 c, b 1, 1/q 1), (3) (a 2 c, a 2, 1/p 2 ), (b 2 c, b 2, 1/q 2 ), (a g 1 c, a g 1, 1/p g 1), (b g 1 c, b g 1, 1/q g 1), (a g c, a g, 1/p g), (b g c, b g, 1/q g ). Here, a i, b i (i = 1, 2,, g) andc, d denote the standard generators of π 1 (Σ g ) and π 1 (T 2 ),respectively. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery33 / 42

Construction of Symplectic 4-Manifolds Luttinger surgeries on product 4-manifolds Σn Σ 2 and Σn T 2 The fundamental group of M g (p, q) is generated by a i, b i (i = 1, 2, 3, g) and c, d, andthefollowingrelationsholdinm g (p, q): [b 1 1, d 1 ]=a 1 p 1, [a 1 1, d] =b 1 q 1, [b 1 2, d 1 ]=a 2 p 2, [a 1 2, d] =b 2 q 2, (4),,, [bg 1, d 1 p ]=a g g, [a 1 q g, d] =b g g, [a 1, c] =1, [b 1, c] =1, [a 2, c] =1, [b 2, c] = [a 3, c] =1, [b 3, c] =1, [a g, c] =1, [b g, c] =1, [a 1, b 1 ][a 2, b 2 ] [a g, b g ]=1, [c, d] =1. Let Σ g M g (p, q) and T be a genus g and genus 1 surfaces that desend from the surfaces Σ g {pt} and {pt} T 2 in Σ g T 2. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery34 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery M g (p, q) is a locally trivial genus g bundle over T 2 where T is a section. The (a i c, a i, p i) or (b i c, b i, q i) Luttinger surgery in the trivial bundle Σ g T 2 preserves the fibration structure over T 2 introducing a monodromy of the fiber Σ g along the curve c in the base. Depending on the type of the surgery the monodromy is either (t ai ) p i or (t bi ) q i,wheret denotes a Dehn twist. E(n) can be obtained as a desingularization of the branched double cover of S 2 S 2 with the branching set being 4 copies of {pt} S 2 and 2n copies of S 2 {pt}. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery35 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery M g (p, q) is a locally trivial genus g bundle over T 2 where T is a section. The (a i c, a i, p i) or (b i c, b i, q i) Luttinger surgery in the trivial bundle Σ g T 2 preserves the fibration structure over T 2 introducing a monodromy of the fiber Σ g along the curve c in the base. Depending on the type of the surgery the monodromy is either (t ai ) p i or (t bi ) q i,wheret denotes a Dehn twist. E(n) can be obtained as a desingularization of the branched double cover of S 2 S 2 with the branching set being 4 copies of {pt} S 2 and 2n copies of S 2 {pt}. E(n) admits a genus n 1fibrationoverS 2 and an elliptic fibration over S 2.AregularfiberoftheellipticfibrationonE(n) intersects every genus n 1 fiber of the other Lefschetz fibration twice. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery35 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery Construction of Lefschetz fibrations over S 2 and exotic Stein fillings Let X g,n (p, q) denote the symplectic sum of M g (p, q) along the torus T = c d with the elliptic surface E(n) along a regular elliptic fiber. The symplectic 4-manifold X g,n (p, q) admits a genus 2g + n 1 Lefschetz fibration over S 2 with at least 4n + 4pairwisedisjointsphere sections of self intersection 2. Moreover, X g,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 X g,n (p, q) can also be constructed as the twisted fiber sum of two copies of a genus 2g + n 1Lefschetzfibration on Σ g S 2 #4nCP 2.Thisfollowsfromthefactthatthesymplecticsumof E(n) along a regular elliptic fiber with Σ g T 2 along a natural square zero torus is diffeomorphic to the untwisted fiber sum of two copies of the genus 2g + n 1fibrationonΣ g S 2 #4nCP 2.Thegluingφ diffeomorphism can be described explicitly using the curves along which we perform our Luttinger surgeries. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery36 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery The fundamental group of the symplectic 4-manifold X g,n (p, q) is generated by the set {a i, b i : 1 i g} subject to the relations: a p i i = 1, b q i i = 1, for all 1 i g, and Π g j=1 [a j, b j ]=1. By setting p i = 1andq i = 0, for all 1 i g, weseethatthe fundamental group of X g,n ((1, 1,...,1), (0, 0,...,0)) is a free group of rank g. Thegluingdiffeomorphism:φ = t a1 t ag. By setting p i = 1andq i = 1, for all 1 i g, weseethatthe fundamental group of X g,n ((1, 1,...,1), (1, 1,...,1)) is a trivial. The gluing diffeomorphism: φ = t a1 t b1 t ag t bg. If we set p i = 1andq i = 0, for all 1 i k and p i = 1andq i = 1, for all k + 1 i g, thefundamentalgroupofx g,n ((1, 1,...,1), (1, 1,...,0)) is aafreegroupofrankk. Thegluingdiffeomorphism: φ = t a1 t ak t ak+1 t bk+1 t ag t bg. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery37 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery Stein fillings from Lefschetz fibrations Definition AcomplexsurfaceV is Stein if it admits a proper holomorphic embedding f : V C n for some n. Foragenericpointp C n,considerthemap φ : V R defined by φ(z) = z p 2.Foraregularvaluea R, thelevelset 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, ands = φ 1 ([0, a]) is called a Stein filling of (M, ξ). Theorem (S. Akbulut - B. Ozbagci) Let f : X S 2 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)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery38 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery Finiteness Results on Stein Fillings The tight contact structure on S 3 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). The homemorphism types of Stein fillings of T 3 (A. Stipsicz, 2002). Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery39 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery 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 π 1 = Z Z n (Akhmedov - Ozbagci, 2012). Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery40 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery Theorem (A. Akhmedov - B. Ozbagci, 2012) For any finitely presented group G, there exist an infinite family of exotic Stein 4-manifolds S n (G, K i ) with π 1 (S n (G, K i )) = G, where K i are inf. family of genus g 2 fibered knots with distinct Alexander polynomials. Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery41 / 42

Construction of Symplectic 4-Manifolds Construction of Lefschetz fibration via Luttinger Surgery THANK YOU! Anar Akhmedov (University of Minnesota, Minneapolis)Lefschetz Fibrations and Exotic Stein Fillings with Arbitrary Fundamental Group via June Luttinger 20, 2013 Surgery42 / 42