Monopole Floer Homology, Link Surgery, and Odd Khovanov Homology Jonathan Michael Bloom Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2011 c 2011 Jonathan Michael Bloom All Rights Reserved ABSTRACT Monopole Floer Homology, Link Surgery, and Odd Khovanov Homology Jonathan Michael Bloom We construct a link surgery spectral sequence for all versions of monopole Floer homology with mod 2 coefficients, generalizing the exact triangle. The spectral sequence begins with the monopole Floer homology of a hypercube of surgeries on a 3-manifold Y , and converges to the monopole Floer homology of Y itself. This allows one to realize the latter group as the homology of a complex over a combinatorial set of generators. Our construction relates the topology of link surgeries to the combinatorics of graph associahedra, leading to new inductive realizations of the latter. As an application, given a link L in the 3-sphere, we prove that the monopole Floer homology of the branched double-cover arises via a filtered perturbation of the differential on the reduced Khovanov complex of a diagram of L. The associated spectral sequence carries a filtration grading, as well as a mod 2 grading which interpolates between the delta grading on Khovanov homology and the mod 2 grading on Floer homology. Furthermore, the bigraded isomorphism class of the higher pages depends only on the Conway-mutation equivalence class of L. We constrain the existence of an integer bigrading by considering versions of the spectral sequence with non-trivial Uy action, and determine all monopole Floer groups of branched double-covers of links with thin Khovanov homology. Motivated by this perspective, we show that odd Khovanov homology with integer coef- ficients is mutation invariant. The proof uses only elementary algebraic topology and leads to a new formula for link signature that is well-adapted to Khovanov homology. Table of Contents 1 Introduction 1 1.1 Background on monopole Floer homology . 4 2 The link surgery spectral sequence 13 2.1 Hypercubes and permutohedra . 15 2.2 The link surgery spectral sequence: construction . 22 2.3 Product lattices and graph associahedra . 32 2.4 The surgery exact triangle . 42 2.5 The link surgery spectral sequence: convergence . 47 2.5.1 Grading . 48 2.5.2 Invariance . 49 2.6 The Uy map and HMg •(Y )............................ 56 3 Realizations of graph associahedra 69 4 Odd Khovanov homology and Conway mutation 76 4.1 A thriftier construction of reduced odd Khovanov homology . 78 4.2 The original construction of reduced odd Khovanov homology . 82 4.3 Branched double-covers, mutation, and link surgery . 86 5 Link signature and homological width 89 6 From Khovanov homology to monopole Floer homology 96 6.1 Khovanov homology and branched double-covers . 98 i 6.1.1 Grading . 102 6.1.2 Invariance . 107 6.2 The spectral sequence for a family of torus knots? . 108 7 Donaldson's TQFT 114 7.1 The algebraic perspective . 114 7.2 The geometric perspective . 117 7.3 Monopole Floer homology and positive scalar curvature . 120 8 Khovanov homology and Uy 127 8.1 An integer bigrading? . 131 8.2 The Brieskorn sphere −Σ(2; 3; 7) . 134 8.3 Beyond branched double-covers . 141 9 Appendix: Morse homology with boundary via path algebras 144 Bibliography 149 ii List of Figures 2.1 The cobordism for the hypercube f0; 1g3. ................... 17 2.2 The hexagon of metrics for f0; 1g3........................ 21 2.3 The permutohedron of metrics for f0; 1g4.................... 21 2.4 The associahedron of metrics for f0; 1; 1g2................... 35 2.5 The polyhedron of metrics for f0; 1; 1g × f0; 1g2. 36 2.6 The graph associahedron of the 3-clique with one leaf. 38 2.7 From link surgeries to lattices, graphs, and polytopes. 40 2.8 The cobordism for f0; 1; 1; 00g and the pentagon of metrics. 44 2.9 The cobordism used to the construct the homotopy equivalence for f0; 1g2. 50 2.10 The hexagon of metrics used in the homotopy equivalence for f0; 1g2. 52 2.11 The cobordism used to construct the Uy map andm ~ (W ). 58 2 2.12 The cobordism and hexagon of metrics for the HMg • version of f0; 1g . 60 2.13 The associahedron of metrics for f0; 1; 1; 00g × f0; 1g. 64 3.1 Sliding the sphere to construct realizations of graph associahedra. 70 3.2 Realizations of the permutohedron and a graph associahedron. 70 3.3 The graphs G and G0 for Λ and Λ × f0; 1g. .................. 71 3.4 Realizations of the associahedra. 72 4.1 Oriented resolution conventions. 79 4.2 Linking number conventions. 80 4.3 The Kinoshita-Terasaka and Conway knots. 83 4.4 Constructing a surgery diagram for the branched double-cover. 88 iii 5.1 A dealternator connected, 2-almost alternating diagram of T (3; 7). 92 5.2 A local move to ensure dealternator connectedness. 93 6.1 The short arc between the strands of a resolved crossing. 100 6.2 A surgery diagram for the branched double cover of the trefoil. 101 6.3 The branched double-cover of the cube of resolutions of the trefoil. 103 6.4 Four types of crossings in an oriented diagram with checkerboard coloring. 106 6.5 Conjectural HMg • spectral sequences for T (3; 6n ± 1). 110 7.1 Commutative diagram relating the functors HMg • and CKh. 114 7.2 Commutative diagram for the functor Λ∗H1. 116 8.1 Conjectural HMd • and other spectral sequences for T (3; 7). 137 8.2 Conjectural HMd • spectral sequences for T (3; 6n ± 1) . 139 8.3 Conjectural HMd • spectral sequences for T (3; 19) . 140 8.4 Comparison of Khf (T (3; 7)) and Khf (P (−2; 3; 7)). 141 9.1 Path algebras and Morse homology for manifolds with boundary. 145 9.2 The differential and identities as elements of a path algebra. 147 9.3 The path algebra of a cobordism with fixed metric. 148 9.4 Chain map for a cobordism with three ends. 150 iv Acknowledgments I am deeply grateful to my advisor, Peter Ozsv´ath,for his unwavering and enthusiastic support over the last five years, and for providing wonderful problems to explore. Even before I arrived at Columbia, he arranged for me to attend the 2006 Summer School at the Park City Mathematical Institute, a formative experience that kindled my interest in low-dimensional topology and geometry. I started graduate school with a focus on Heegaard Floer homology and Khovanov homology. At Professor Ozsv´ath'ssuggestion, and with his invaluable guidance, I begin to pursue connections with monopole Floer theory. This has taken me down an exciting and rewarding path. I am indebted to Tim Perutz and Tom Mrowka for generous mentoring in monopole Floer theory. I have also benefitted from the expertise of Adam Knapp, Peter Kronheimer, and Max Lipyanskiy. I want to thank Peter Kronheimer and Tom Mrowka for investing the time and energy to write their foundational book Monopoles and Three-Manifolds. My copy has surely seen better days. It has been a privilege to be a member of the mathematical community at Columbia. I am grateful for the friendship and support of John Baldwin, Ben Elias, Allison Gilmore, Josh Greene, Eli Grigsby, Adam Levine, Tom Peters, Ina Petkova, Thibaut Pugin, Vera Vertesi, and Rumen Zarev. My research has benefitted from discussions with Professors Dave Bayer, Mikhail Khovanov, Robert Lipshitz, and Dylan Thurston at Columbia, and Professors Satyan Devadoss, Danny Ruberman, James Stasheff, and Zolt´anSz´abo abroad, among many others. I had the good fortune of spending the spring of 2010 at the Mathematical Sciences Research Institute for the program Homology Theories of Knots and Links. I want to thank my uncle Steven Greenfield for providing a warm home away from home during my stay in Berkeley. I have spent my final year of graduate school as an exchange scholar at MIT, and v would like to thank the department for welcoming me. I am also grateful for support from the National Science Foundation (grant DMS-0739392). I want to express my appreciation to several people who nurtured my interest in math- ematics in my youth: my mentor Greg Bachelis, my calculus teacher Joe Brandell, and my cousin David Goss, who inspired me to attend the Ross Program. Through it all, my parents, sister, and grandparents have been a constant source of love and support. Finally, I want to thank Masha for her endless encouragement and so much more. vi CHAPTER 1. INTRODUCTION 1 Chapter 1 Introduction Monopole Floer homology is a gauge-theoretic invariant defined via Morse theory on the Chern-Simons-Dirac functional. As such, the underlying chain complex is generated by Seiberg-Witten monopoles over a 3-manifold, and the differential counts monopoles over the product of the 3-manifold with R. We review this construction in Section 1.1. In [27], a surgery exact triangle is associated to a triple of surgeries on a knot in a 3-manifold (for a precursor in instanton Floer homology, see [11], [18]). In Chapter 2, we construct a link surgery spectral sequence in monopole Floer homology, generalizing the exact triangle. This is a spectral sequence which starts at the monopole Floer homology of a hypercube of surgeries on Y along L, and converges to the monopole Floer homology of Y itself. The differentials count monopoles on 2-handle cobordisms equipped with families of metrics parameterized by polytopes called permutohedra. Those metrics parameterized by the boundary of the permutohedra are stretched to infinity along collections of hypersur- faces representing surgered 3-manifolds.