Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group

Transcription

1 Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group Anar Akhmedov University of Minnesota, Twin Cities February 19, 2015 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

2 Outline 1 Lefschetz fibrations Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas 2 Surgery on symplectic 4-manifolds Luttinger Surgery Symplectic Connected Sum 3 Construction of Exotic Lefschetz fibrations and Stein fillings Luttinger surgeries on product 4-manifolds Σ n Σ 2 and Σ n T 2 Construction of Lefschetz fibration via Luttinger Surgery Exotic Stein fillings Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

3 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, where Σ 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 z 2 2. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

4 Lefschetz fibrations Vanishing cycle α X f S 2 Figure : Lefschetz fibration on X over S 2 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

5 Lefschetz fibrations Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

6 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

7 Lefschetz fibrations The genus of the regular fiber of f is defined to be the genus of the Lefschetz fibration. Let p 1,, p s denote the critical points of Lefschetz fibration f : X Σ h. e(x) = e(σ h )e(σ g )+s The signature σ(x) well understood for fibrations over S 2. There are formulas by Y.Matsumoto, H.Endo, B.Ozbagci, I.Smith,...) σ(x) = g+1 2g+1 n 0 + ( ) ( g 2 ) 4h(g h) h=1 2g+1 1 n h Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

8 Lefschetz fibrations Monodromy of Lefschetz fibration Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

9 Lefschetz fibrations 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. This element is a right-handed (or a positive) Dehn twist along a simple closed curve on Σ g, called the vanishing cycle. Vanishing cycle α X f S 2 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

10 Lefschetz fibrations Global monodromy of Lefschetz fibration Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

11 Lefschetz fibrations Global monodromy of Lefschetz fibration For a genus g Lefschetz fibration over S 2, the product of right handed Dehn twists t αi along the vanishing cycles α i, for i = 1,, s, determines the global monodromy of the Lefschetz fibration, the relation t α1 t α2 t αs = 1 in M 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 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

12 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

13 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 ) 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

14 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 ) 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 = 1 in M 1. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

15 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 ) 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 = 1 in M 1. The total space of this fibration is the elliptic surface E(n). E(1) = CP 2 #9CP 2, 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 over S 2. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

16 Lefschetz fibrations Example (Families of 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental GroupFebruary 19, / 39

17 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, the complex projective plane blown up at 4g + 5 points. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

18 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, the complex projective 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). Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

19 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, the complex projective 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 ge 1 e 2 e 4g+5, where e 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. The exceptional 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

20 Lefschetz fibrations Important Theorems on Lefschetz fibrations Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

21 Lefschetz fibrations Important Theorems on 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. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

22 Lefschetz fibrations Important Theorems on 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, and let [F] denote the homology class of the fiber. Then X admits a symplectic structure with symplectic fibers iff [F] 0 in H 2 (X;R). If e 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. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

23 b + 2 (X(G)) is very large. These examples don t contain essential tori. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39 Lefschetz fibrations Important Theorems on 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, and let [F] denote the homology class of the fiber. Then X admits a symplectic structure with symplectic fibers iff [F] 0 in H 2 (X;R). If e 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, M. Korkmaz) For any finitely presented group G, there exist a Lefschetz fibration X(G) over S 2 with π 1 (X(G)) = G.

24 Lefschetz fibrations Main Theorems Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

25 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. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

26 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 )) = G that can be obtained from X n (G) via knot surgery along K i, where K i are inf. family of genus g 2 fibered knots with distinct Alexander polynomials. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

27 Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas Lefschetz fibrations by Y. Matsumoto and M. Korkmaz Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

28 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 + 4 singular fibers. This was shown by Yukio Matsumoto for k = 1, and in the case k 2 by Mustafa Korkmaz, 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

29 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, the follwing relations between right handed 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

30 Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas Lefschetz fibrations by Y. Gurtas Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

31 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. The total space of these genus g = 2k + n 1 Lefschetz fibration over S 2 is Y(n, k) = Σ k S 2 #4nCP 2. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

32 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 involutionθ of the surface Σ 2k+n 1 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

33 Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas The branched-cover description for Σ k S 2 #4nCP 2 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

34 Lefschetz fibrations by Y. Matsumoto, M. Korkmaz and Y. Gurtas The branched-cover description for Σ k S 2 #4nCP 2 A generic horizontal fiber is the double cover of S 2, branched over two points. Thus, we have a sphere fibration on Y(n, k) = Σ k S 2 #4nCP 2. A generic fiber of the vertical fibration is the double cover of Σ k, branched over 2n 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

35 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

36 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 involutionθ of the surface Σ 2k+n 1 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

37 Construction Tools Surgery on symplectic 4-manifolds Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

38 Construction Tools Surgery on symplectic 4-manifolds Symplectic Connected Sum (1995) (M. Gromov, R. Gompf, J. McCarthy- J. Wolfson) Luttinger Surgery (1995) (K. Luttinger) Knot Surgery (1998) (R. Fintushel- R. Stern) Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

39 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 λ on 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 X will be X Λ,λ (1/m) = (X ν(λ)) φ (S 1 S 1 D 2 ), the 1/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)), where µ 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

40 Surgery on symplectic 4-manifolds Luttinger Surgery Example Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

41 Surgery on symplectic 4-manifolds Luttinger Surgery Example Let T 4 = a b c d = (c d) (a b). Let K n be an n-twist knot. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

42 Surgery on symplectic 4-manifolds Luttinger Surgery Example Let T 4 = a b c d = (c d) (a b). Let K 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). The tori c 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 = 1 the second surgery is also a Luttinger surgery. a a ~ d ~ b b Figure : The 3-torus d a b Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

43 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. Suppose that [F 1 ] 2 +[F 2 ] 2 = 0 and the genera of F 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. Using Ψ, we glue X 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, determined by Ψ. 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

44 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. Let X# ψ X denote the fiber sum of X with itself by a self-diffemorphism ψ of the generic fiber Σ. Then X# ψ X has the vanishing cycles α 1,α 2,,α s,ψ(α 1 ),ψ(α 2 ),,ψ(α s ). Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

45 Construction of Exotic Lefschetz fibrations and Stein fillings 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 0 and q i 0, where 1 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+4 Luttinger surgeries on Σ n Σ 2. 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/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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

46 Construction of Exotic Lefschetz fibrations and Stein fillings 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) are the standard loops that 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 b i x y c j 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

47 Construction of Exotic Lefschetz fibrations and Stein fillings 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 4 and its 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, 2 and j = 1,...,n) and the following relations 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 1, b 2] = d q 1 1, [d 1 2, b 1 1 ] = cp 2 2, [c 1 2, b 1] = d q 2 2, [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 ] = dq 3 3,..., [a 1 1, d 1 n ] = cpn n, [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 ] = dqn n, Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

48 Construction of Exotic Lefschetz fibrations and Stein fillings 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 ). They are symplectic submanifolds in Y n (1/p 1, 1/q 1,, 1/p n, 1/q n ). Denote their images by Σ 2 and Σ n. Note that [Σ 2 ] 2 = [Σ n ] 2 = 0 and [Σ 2 ] [Σ n ] = 1. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

49 Construction of Exotic Lefschetz fibrations and Stein fillings 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

50 Construction of Exotic Lefschetz fibrations and Stein fillings 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 ). Denote by M 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) and c, d denote the standard generators of π 1 (Σ g ) and π 1 (T 2 ), respectively. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

51 Construction of Exotic Lefschetz fibrations and Stein fillings 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, and the following relations hold in M 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 g, d] = b g qg, [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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

52 Construction of Exotic Lefschetz fibrations and Stein fillings 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, where t 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

53 Construction of Exotic Lefschetz fibrations and Stein fillings 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, where t 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 1 fibration over S 2 and an elliptic fibration over S 2. 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

54 Construction of Exotic Lefschetz fibrations and Stein fillings Construction of Lefschetz fibration via Luttinger Surgery Exotic Non-holomorphic Lefschetz fibrations over S 2 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+4 pairwise disjoint sphere 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 1 Lefschetz fibration on Σ g S 2 #4nCP 2. This follows from the fact that the symplectic sum of 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 1 fibration on Σ g S 2 #4nCP 2. The gluing φ diffeomorphism can be described explicitly using the curves along which we perform our Luttinger surgeries. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

55 Construction of Exotic Lefschetz fibrations and Stein fillings 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 = 1 and q i = 0, for all 1 i g, we see that the fundamental group of X g,n ((1, 1,...,1),(0, 0,...,0)) is a free group of rank g. The gluing diffeomorphism: φ = t a1 t ag. By setting p i = 1 and q i = 1, for all 1 i g, we see that the 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 = 1 and q i = 0, for all 1 i k and p i = 1 and q i = 1, for all k + 1 i g, the fundamental group of X g,n ((1, 1,...,1),(1, 1,...,0)) is a a free group of rank k. The gluing diffeomorphism: φ = t a1 t ak t ak+1 t bk+1 t ag t bg. Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

56 Construction of Exotic Lefschetz fibrations and Stein fillings Exotic Stein fillings Stein fillings from Lefschetz fibrations Definition A complex surface V is Stein if it admits a proper holomorphic embedding f : V C n 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). Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

57 Construction of Exotic Lefschetz fibrations and Stein fillings Exotic Stein fillings Stein fillings from Lefschetz fibrations Definition A complex surface V is Stein if it admits a proper holomorphic embedding f : V C n 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 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

58 Construction of Exotic Lefschetz fibrations and Stein fillings Exotic Stein fillings 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). Finiteness results on Stein fillings of Seifert fibered spaces over S 2 (L. Starkston, 2013). Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

59 Construction of Exotic Lefschetz fibrations and Stein fillings 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 and the fillings are pairwise non homotopy equivalent. Infinitely many simply-connected exotic Stein fillings (Akhmedov - Etnyre - Mark - Smith, 2007). Small exotic Stein fillings (S. Akbulut - K. Yasui, 2008). Infinitely many exotic Stein fillings with π 1 = Z Z n (Akhmedov - Ozbagci, 2012). Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

60 Construction of Exotic Lefschetz fibrations and Stein fillings 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 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)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39

61 Construction of Exotic Lefschetz fibrations and Stein fillings Exotic Stein fillings THANK YOU! Anar Akhmedov (University of Minnesota, Minneapolis)Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group February 19, / 39