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Symplectomorphism Group of Rational 4-Manifolds Symplectomorphism Group of Rational 4-Manifolds A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Jun Li IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Tian-Jun Li June, 2017 c Jun Li 2017 ALL RIGHTS RESERVED Acknowledgements First I would like to thank my thesis advisor Tian-Jun Li for his generosity in sharing his time and knowledge over the years. It is his constant support, help and patience for various aspects of my life made this work possible. Thanks also goes to my academia brother Weiwei Wu, who gave me many aca- demic advices and shared with me many ideas throughout the years. I also appreciate the opportunity to learn from many other mathematicians during my graduate stud- ies. Particularly, I am very grateful to Professor Anar Akhmedov, Yu-jong Tzeng and Alexander Voronov for their great patience explaining details on various different top- ics. Conversations with Professor Hao Fang, Richard Hind, Keiko Kawamuro, Mark Mclean, Yongbin Ruan, Michael Usher, and Weiyi Zhang are greatly helpful for my understandings of mathematics. I would also like to thank all the friends throughout the years, in particular, Denis Bashkirov, Yang Cao, Pak Yeung Chan, Guosheng Fu, Fei He, Chung-I Ho, Qifeng Li, Cheukyu Mak, Nur Saglam, Sumeyra Sakalli, and Zonglin Jiang for all the sharing and discussions we have. Finally, thank Fan for being part of my life, and thank my beloved parents and their never ending encouragement and support. i Abstract We develop techniques for studying the symplectomorphism group of rational 4- manifolds. We study the space of tamed almost complex structures J! using a fine decom- position via smooth rational curves and a relative version of the infinite dimensional Alexander duality. This decomposition provides new understandings of both the varia- tion and stability of the symplectomorphism group Symp(X; !) when deforming !. In particular, we compute the rank of π0(Symp(X; !)), with Euler number less than 8 in terms of the number N of -2 symplectic sphere classes. In addition, using the above decomposition and coarse moduli of rational surfaces with a given symplectic form, we are able to determine π0(Symp(X; !)), the symplectic mapping class group (SMC). Our results can be uniformly presented regarding Dynkin diagrams of type A and type D Lie algebras. Applications of π0(Symp(X; !)) and π0(Symp(X; !)) includes the classification of symplectic spheres and Lagrangian spheres up to Hamiltionian isotopy and a possible approach to determine the full rational homotopy type Symp(X; !). ii Contents Acknowledgements i Abstract ii List of Tables vi List of Figures vii 1 Introduction 1 1.1 A fine decomposition of the space of almost complex structures . .2 1.2 Application to symplectomorphism group . .3 1.2.1 Symplectic Mapping Class Group(SMC) . .6 1.2.2 π1(Symph(X; !)) and Topological Persistence . .9 2 The normalized reduced symplectic cone 12 2.1 Normalized reduced symplectic cone: . 12 2.1.1 Reduced symplectic forms . 12 2.1.2 Combinatorics: Normalized Reduced symplectic cone as polyhedron 15 2.1.3 Lie theory: Wall and chambers labeled by Dynkin diagram . 17 2.1.4 Identifying edges with roots . 18 2.1.5 A uniform description for reduced cone of M when χ(M) < 12 . 19 2 2.2 Examples: CP #kCP 2; k = 1; 2; 3 and remarks for k > 9 . 21 2 2.2.1 Examples: CP #kCP 2; k = 1; 2; 3 ................. 21 2.2.2 Symplectic cone and Normalized reduced cone, and discussion for general cases . 24 iii 3 The space of almost complex structures 27 3.1 Decomposition of J! via smooth rational curves . 27 3.1.1 General facts of J-holomorphic curves and symplectic spheres . 27 3.1.2 Prime submanifolds . 29 3.2 Constraints on simple J−holomorphic curves for a reduced form . 34 3.2.1 The key lemma . 34 3.3 Negative square classes and their decompositions . 38 3.4 Codimension 2 prime submanifolds . 41 3.4.1 Level 2 stratification . 41 3.4.2 Enumerating the components by −2 symplectic sphere classes . 43 4 Symplectic rational 4-manifold with Euler number less than 8 49 4.1 Strategy . 49 4.1.1 Groups associated to a configuration . 50 4.1.2 Choice of the configuration in each case . 51 4.2 Connectedness of the Torelli symplectic mapping class group . 52 4.2.1 Reduction to the connectedness of Stab(C)............ 53 0 4.2.2 Reduction to the surjectivity of : π1(Symp(C)) ! π0(Stab (C)) 54 4.2.3 Three types of configurations . 56 4.2.4 Criterion . 60 2 2 4.2.5 Contractibility of Sympc(U) and the proof in the case of CP #4CP 61 4.3 The fundamental group of Symp(X; !) when χ(X) ≤ 7 . 65 4.3.1 Proof of Theorem 1.2.6 . 65 4.3.2 Discussion in each case . 71 5 Rational surfaces with Euler number grater or equal to 8 77 2 5.1 Symplectic -2 spheres and Symp(CP #5CP 2;!)............. 77 5.1.1 Basic set-up and pure braid groups on a sphere . 78 5.2 A semi-toric Ball-swapping model and the connecting homomorphism . 89 5.3 Forget one strand map when ΓL = D4 ................... 95 5.4 Torelli Symplectic mapping class group for a general form . 108 5.5 Fundamental group and topological persistence of Symp(X; !)..... 113 iv References 122 v List of Tables 2 2 2.1 Reduced cone of X3 = CP #3CP ..................... 22 4.1 The quantity Q on the persistence of Symp(X; !)............. 71 2 2 4.2 ΓL and π1(Symph(CP #2CP )) ...................... 71 2 2 4.3 ΓL and π1(Symph(X; !) for CP #3CP .................. 72 2 2 4.4 ΓL and π1(Symph(X; !) for CP #4CP .................. 73 2 5.1 Reduced symplectic form on CP #5CP 2 .................. 120 5.2 number of geometric intersection points (g.i.p.) for J 2 JC........ 121 vi List of Figures 2 2.1 Normalized Reduced cone of CP #3CP 2 ................. 21 4.1 3 types of configruations . 58 4.2 Configuration of 4-point blow up . 61 5.1 The Artin generator σi and the standard generator Ai;j ......... 85 5.2 Standard toric packing and ball swapping in O(4) . 90 5.3 Configuration of two minimal area exceptional classes for ! 2 MA: ... 96 5.4 Configuration of exceptional classes for ! 2 MA: ............. 99 vii Chapter 1 Introduction A symplectic manifold (X; !) is an even dimensional manifold X with a closed, nonde- generate two form !. A symplectic submanifold S 2 (X; !) is a submanifold such that !jS is a symplectic form. A Lagrangian submanifold L 2 (X; !) is a submanifold such that !jL = 0: Let (X; !) be a closed simply connected symplectic manifold. the symplectomor- phism group with the standard C1-topology, denoted as Symp(X; !), is an infinite dimensional Fr´echet Lie group. Understanding the homotopy type of Symp(X; !) is a classical problem in symplectic topology initiated by [20]. Let J! be the space of !−tame almost complex structures. It is known that the stratification structure of J! is closely related to the topology of Symp(X) when dim(X) = 4 [20, 2, 4, 26] and [7], etc. However, the study of the whole stratification of J! is usually formidable even when X is relatively simple [5][7]. Among all homotopy groups of Symp(X), π0(Symp(X)) and π1(Symp(X)) have more direct geometric meaning. π0(Symp(X)), which we also call the symplectic mapping class group (SMCG), is closely related to isotopy problems of symplec- tic/Lagrangian submanifolds in X. π1(Symp(X)) is tied to Hofer geometry of Symp(X) (cf. [51]) and quantum cohomology (cf.[53]). Also, generator of π1(Symp(X; !)) is also the generator of the rational homotopy groups of Symp(X; !); for some rational surfaces with small Euler number, as shown in [7, 6] Hence it is essential in determining the full homotopy type of Symp(X; !): This dissertation is a summary of a series [31], [32] and [29], which studies the relation 1 2 2 2 between πi(Symp(Xk)) for Xk = CP #kCP and i = 0; 1, the lower stratification of J! and negative symplectic curves in symplectic rational manifolds. 1.1 A fine decomposition of the space of almost complex structures When (X; !) is a rational symplectic 4-manifold, we consider the following natural decomposition of J! via smooth !−symplectic spheres of negative self-intersection. <0 ≤−2 Let S! (S! respectively) denote the set of homology classes of embedded !- symplectic sphere with negative self-intersection (square less than -1 respectively). ≤−2 Definition 1.1.1. A subset C ⊂ S! is called admissible if C = fA1; ··· ;Ai; ··· ;AnjAi · Aj ≥ 0; 8i 6= jg; Given an admissible subset C, we define the real codimension of the label set C as cod = 2 P (−A · A − 1). And we define the prime subset R Ai2C i i JC = fJ 2 J!jA 2 S has an embedded J−hol representative if and only if A 2 Cg: In particular, for C = ;, it has codimension zero, and the corresponding J; is often called Jopen. In section 2 (see Proposition 3.1.13 and Remark 3.1.15 for details), we will show that a prime subset is either empty or a submanifold with real codimension of its labeling set under a reasonable Condition 3.1.9. Notice that these prime subsets are disjoint. Clearly, we have the decomposition: J! = qCJC: Hence we can define a filtration according to the codimension of these prime subsets: ; = X2n+1 ⊂ X2n(= X2n−1) ⊂ X2n−2 ::: ⊂ X2(= X1) ⊂ X0 = J!; where X := q J is the union of all prime subsets having codimension no less 2i CodR≥2i C than 2i.
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