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On the Twisted Floer Homology of Mapping Tori of Periodic Diffeomorphisms

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On the Twisted Floer Homology of Mapping Tori of Periodic Diffeomorphisms On the twisted Floer homology of mapping tori of periodic diffeomorphisms Evan Fink Submitted in partial fullfillment of the Requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2010 ABSTRACT On the twisted Floer homology of mapping tori of periodic diffeomorphisms Evan Fink Let K ⊂ Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y . We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the twisted Heegaard Floer homology HF + of the mapping torus of a diffeomorphism of a closed Riemann surface whose mapping class is periodic, giving an almost complete description of the structure of these groups. When the surface is of genus at least three and the mapping class is nontrivial, we find in particular that in the \second-to-highest level" of Spinc structures, this is isomorphic to a free module (over a certain ring) whose rank is equal to the Lefschetz number of the diffeomorphism. After taking a tensor product with Z=2Z, this agrees precisely with the symplectic Floer homology of the diffeomorphism, as calculated by Gautschi. Contents Acknowledgements iv 1 Introduction 1 1.1 Results . 3 2 Standard Heegaard Diagrams 14 2.1 Standard Heegaard diagrams and cobordisms . 15 2.2 Relative Spinc structures . 18 2.3 Intersection points and relative Spinc structures . 19 2.4 Spinc structures on cobordisms . 23 2.5 Squares of Chern classes . 27 3 Families of Standard Heegaard Diagrams 31 4 A Twisted Long Exact Sequence 37 4.1 Additive functions on polygons . 37 4.2 Standard diagram coefficients . 40 4.3 The long exact sequence . 46 5 Twisted Knot Floer Homology 53 5.1 The knot filtration . 53 5.2 Relationship with large N surgeries . 55 i 5.3 The K¨unneth Formula . 58 6 Twisted Surgery Formula 60 7 Computations 67 7.1 The Borromean knots Bg ........................... 67 7.2 O-knots . 72 7.3 The full filtered complex for connected sums . 75 8 Proof of Main Theorem 77 c 8.1 The relative Spin structures ξi ........................ 77 8.2 The homology of C+(Z ;B ; ξk )....................... 79 g g Bg 8.3 The homology of C+(Y; K; ξ) ......................... 86 8.4 An absolute Z-grading for the mapping cone . 86 8.5 The homology of the mapping cone . 88 8.6 The modules Ωg(k)............................... 94 9 Proofs of Theorem 1.1 and Corollary 1.2 97 10 Sample Calculations 104 10.1 0-surgery on the (2,7) torus knot . 104 1 10.2 S × Σg .....................................106 Bibliography 110 Appendix 113 ii List of Figures 2.1 Local picture of a standard diagram . 16 2.2 Vector fields in the neighborhood of a flowline . 21 2.3 Arrangement of spiders . 25 7.1 A doubly-pointed Heegaard diagram for (Z1;B1) . 68 0 7.2 A standard Heegaard diagram for WK , K = O5;3 . 73 10.1 O2;1#O7;3#O14;1 ................................105 10.2 Graphs of some well functions . 107 10.3 Graph of another well function . 108 A.1 Kirby calculus for XN .............................118 iii Acknowledgements First and foremost, I would like to thank my advisor, Peter Ozsv´ath.I will never cease to be amazed how he has time to manage all the things he has on his plate, while still being an exceptionally attentive advisor. This thesis would clearly not have been possible without the wealth of ideas he patiently explained to me; but even more importantly, his encouragement and support was vital to my success during the frustrating periods. My gratitude to him is immeasurable. Were it not for the Informal (Informal) Floer Homology seminar at Columbia, I probably would have learned during my graduate studies but a fraction of what I know now. I owe a great deal of thanks to all its participants, for making the seminar a lively and friendly place to exchange ideas. I hope that I was able to give back a small piece of the immense amount I gained by attending. The staff of the Columbia Mathematics department were extremely helpful in making sure things got done, even when I was not (which was most of the time). I am especially grateful to Mary Young and Terrence Cope for watching over me. To my friends, the whole lot of you, I am pleased with who I am today, and this is entirely due to your influence. To my aunts and uncles, it is wonderful to have a family who will help you in any way possible, and give you anything that you need, from career advice to cassette tapes. You've helped me grow more than you could know. Mama and Papa, it would be absurd to try and enumerate what you've done for me over the years, including all that you continue to do. So let me just thank you for the two things that are most directly relevant here: playing Buzz, and letting me ransack your basement for books. Mom, Dad, Erica, I've never had the misfortune of knowing what it's like to have a family that is less than completely focused on my success and happiness. (Nor have I iv had the misfortune of knowing what it's like to have an insane family, although it took me a while to realize this.) I have you to thank for everything. Finally, Christine, my dear, thank you for being a constant source of joy over the last three years. v Chapter 1 Introduction Floer homology is a generic name for a vast array of invariants associated to manifolds equipped with various types of structures. In its original incarnation, it was introduced in the 1980's by Andreas Floer [1, 2, 3, 4] as a means of attacking the Arnol'd conjecture. This conjecture proposed that the number of fixed points of a Hamiltonian symplecto- morphism of a symplectic manifold would satisfy an inequality analogous to the classical Morse inequalities for the manifold. Floer realized that what was needed was a type of Morse theory for certain infinite-dimensional loop spaces, and furthermore that this theory could be realized by replacing the idea of gradient flow lines in the loop space with pseudoholomorphic curves in finite-dimensional manifolds associated to the original symplectic manifold itself. Since then, this idea has grown tremendously, leading to applications useful to prob- lems all over geometry and topology. One of the most interesting offshoots of this idea is Heegaard Floer homology. This is itself a generic name for a suite of invariants for three-manifolds [19], four-manifolds [24], and knots [17]. We briefly describe the idea of the original Heegaard Floer homologies, of three-manifolds. Any closed oriented three- manifold admits a Heegaard decomposition, into two three-dimensional handlebodies glued along their common boundary. These handlebodies can be specified by their at- 1 Chapter 1. Introduction 2 taching circles on the Heegaard surface; if the surface is of genus g, then specifically each handlebody can be specified by g circles in the surface which are homologically linearly independent. If we take the g-fold symmetric product of the surface, then these two g-tuples of circles form two g-dimensional tori in the symmetric product. Equipping everything with suitable structures, it makes sense to talk about the Floer homology of these tori in the manifold. This homology turns out to yield invariants of the original three-manifold. Of course, if it could only be studied in the above terms, it would be very difficult to use. However, computations in the theory can be understood by examining the curves in the Heegaard surface itself. This leads to the theory having a strong combinatorial flavor, making it much simpler than many of the other Floer homologies, and also simpler than the related gauge-theoretic techniques of Donaldson theory and Seiberg-Witten theory that were previously used to study three- and four-manifolds. As such, progress in Heegaard Floer has been quite rapid. It has, for example, been applied to questions of unknotting numbers [22], lens space surgeries [21], and fiberedness of knots and manifolds [6, 12, 13]; and it has strong connections with Seiberg-Witten theory (which agrees with Heegaard Floer in all cases where both have been computed) and Khovanov homology (see for example [23]). There are also connections with the symplectic Floer homology that it initially grew out of, but these are perhaps slightly less developed. One of these connections is con- jectural. Symplectic Floer homology may be defined for a symplectomorphism of a symplectic surface. The generators of this homology are the fixed points of the symplec- tomorphism, and the differential counts certain pseudoholomorphic curves connecting the non-degenerate fixed points of the diffeomorphism. This homology is known to de- pend only on the mapping class of the diffeomorphism. The conjecture, then, is that the Heegaard Floer homology of the mapping torus of the diffeomorphism determines the symplectic Floer homology; more precisely, a certain \level" of the former is conjectured Chapter 1. Introduction 3 to agree with the latter. This has previously been shown to hold for a large number of mapping classes, and no counter examples have been found. The main motivating purpose of this thesis is to show a version of this conjecture for periodic diffeomorphisms. For such a diffeomorphism φ of a closed Riemann surface Σ of genus gΣ, we compute the twisted coefficient Heegaard Floer homology of its mapping torus, and compare this with the symplectic Floer homology of φ.
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