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Applications of 3-Manifold Floer Homology Tth\ Applications of 3-Manifold Floer Homology by Ilya Elson Bachelor of Arts University of Pennsylvania, May 2001 Master of Arts University of Pennsylvania, May 2001 Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Master of Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2008 @ Ilya Elson, MMVIII. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author ..............'''''''•'' ' . i ''''':~,_ "1- .. J .... £ Ar_..L.I~~ .... .L:_"_ Department of Mathematics February 8, 2008 Certified by........................................ .............. Tomasz S. Mrowka Professor of Mathematics tTh\ Thesis Supervisor Accepted by................ .V . David S. Jerison MASSACHUSTTS INSTITUTE OF TEOHNOLOGY Chairman, Departmental Committee on Graduate Students MAR 1,0 2008 ARCHfVES LIBRARIES Applications of 3-Manifold Floer Homology by Ilya Elson Submitted to the Department of Mathematics on February 8, 2008, in partial fulfillment of the requirements for the degree of Master of Science Abstract In this thesis we give an exposition of some of the topological preliminaries neces- sary to understand 3-manifold Floer Homology constructed by Peter Kronheimer and Tomasz Mrowka in [16], along with some properties of this theory, calculations for specific manifolds, and applications to 3-manifold topology. Thesis Supervisor: Tomasz S. Mrowka Title: Professor of Mathematics Contents 1 Topological Preliminaries 1 1 Surgery 1.1.1 Handle Decompositions . 1.1.2 Surgery in any Dimension 1.1.3 Cobordism ...... .... 1.1.4 Rational Surgery on a Knot 3-manifold 1.2 Taut Foliations .... ....... 1.3 Spin Structures ........... 1.4 Plane fields . ............ 1.5 Special surgeries ........... 2 Floer Homology 24 2.1 Morse Homology ..... ... 24 2.2 Monopole Floer Homology ........ 26 2.2.1 General Definitions and Theorems 26 2.2.2 The Surgery Exact Triangle . .. 27 2.3 L-spaces . ................. 28 2.4 Some Calculations . ..... ...... 31 References . ................... 34 Chapter 1 Topological Preliminaries 1.1 Surgery 1.1.1 Handle Decompositions A detailed exposition of handle decompositions can be found in [7, Chapter 4]. Handle decompositions are highly analogous to CW-decompositions. The differ- ence is that the cells are "thickened" to have the same dimension as the manifold we are building. An n-dimensional k-handle h is simple an n-ball thought of as Dk x Dn- k . Dk x {0} is called the core, and {0} x Dn- k is called the co-core. The boundary of the core, aDk x {0} • S k - 1 x {0} is called the attaching sphere and k - {0} x aD - k = {0} x Sn- i is called the belt sphere. Suppose we have an n-manifold X with non-empty boundary, and an S k - 1 em- bedded in aX with trivial normal bundle. A choice of trivialization 7 of the normal k - bundle gives a diffeomorphism of a tubular neighborhood N of the S 1, called the attaching region, : N -- S k - 1 x Dn- k. We can then form a new manifold with boundary, X' = X ud h. There is a technicality, which is that X' is well defined as a topological space, but not as a smooth manifold with boundary because there is a "corner," in other words, X' has the structure of a manifold with boundary every- where except along the boundary of the attaching region in aX. However, there is a canonical way to smooth corners, see [24, section 7.51. Figure 1-1: Attaching a 3-dimensional 1-handle to a 0-handle The pair (S k - l, T) specified up to isotopy determines X' up to diffeomorphism. Handle decompositions arise from Morse functions, so the existence of Morse func- tions implies that every smooth, compact manifold admits a handle decomposition. See [7, Chapter 4] for a very terse overview. See [19, Part I, §1-3] for more details, though, in this reference everything is formulated in terms of cells rather than handles, the modifications are obvious. More precisely, the following are elementary facts, which altogether can be called the fundamental preliminary lemma of Morse theory. The proof of these occupies a number of pages in [19], but the proof is nothing more than some vector calculus. Every closed, smooth n-dimensional manifold admits a smooth function f : M -- [-1, n] such that the critical points, i.e. points where df = 0, are isolated and non- degenerate, i.e. 0 is not an eigenvalue of Hess f, where Hess f is the Hessian of f, which is well-defined at a critical point. The number of negative eigenvalues of Hess f is called the index of a critical point. Such a function is called a Morse function. Further, there exists a Morse function, such that the critical values are 0, 1,..., n and f- 1({k}) are the critical points of index k. Such a Morse function is called self- indexing. Finally, around a critical point Pk e M of index k,there exist coordinates (Xl,...,Xn), p = (0, ... , 0), such that f(xI, ... ,x) = f(p) - EI X2 + E k+l: This innocuous statement, which is absolutely elementary to prove, allows one to conclude that f -1(-1, k + 1/2) is just f-l(-1l, k - 1/2) u k-handles. 1.1.2 Surgery in any Dimension See [7, Sections 5.2] and [22, Theorem 3.12] for details. Suppose we have Sk embed- ded in some n-dimensional manifold M with trivial normal bundle v. A choice of trivialization 7 of v is called a framing, and gives a diffeomorphism q of a tubular neighborhood of the embedded Sk in M, which we call N, with S k x Dn- k . We can then form a new manifold M' by excising N and gluing in D k+1 x Sn- k- 1, since the two have diffeomorphic boundaries, 9!a) : N - a (sk x Dn-k) a (Dk+1 x Sn-k-1) M' = M\N IJ0N Dk+1 x Sn-k-1 . The manifold M' is said to be obtained from M by surgery on Sk with framing T. Note that the framing plays a crucial role; a different choice of framing will generally yield a different manifold. Note also that the class in 7Fk(M) represented by the embedded Sk is killed in M'. Thus, surgery can be used to successively simplify a manifold. The normal bundle v is canonically trivial over the two disks Dk and Dk which are glued along S k- 1 to form the Sk along which the surgery is performed. Hence, the difference between any two trivializations of v gives rise to a map S k - 1 -- GL(n - k), and so to an element of 7 k-1 (SO (n - k)). The notion of surgery is one of the most powerful in differential topology, and allows a sort of classification of manifolds of dimension greater than 4. The classic references for this topic, [1, 29], are very difficult to read, but the first triumphs of surgery theory, the Smale's h-cobordism theorem as explained in [22], and the Kervaire-Milnor classification of homotopy spheres in high dimension, [12] are very readable. Also, [27] is an accessible introduction to the subject; in particular, in that Chapter 13 of [27] one can find the classical surgery exact sequence, which is reminiscent of the surgery exact sequence for Floer Homology (2.6). Surgery was introduced by Milnor in [21] and independently by Wallace, who called it "spherical modification" in [30]. 1.1.3 Cobordism Two n-dimensional manifolds M1 and M2 are said to be cobordant if there exists an n + 1-dimensional manifold W such that aW = Mr1 _ M 2. Cobordism is clearly an equivalence relation, since W = M x [0, 1] is a cobordism of M with itself, and if W1 is a cobordism between M 1 and M 2 and W2 is a cobordism from M 2 to M 3,then W = W1 IHM2 2 is a cobordism from M 1 to M 3. Further, with the empty manifold as the neutral element and disjoint union as the addition the set of equivalence classes of manifolds upto cobordism becomes an abelian group Qn. We can actually further equip the union Q = Un Qn with the structure of a ring by using ordinary cartesian product as the multiplication. Indeed, if W1 is a cobordism from M 1 to M3 and W2 is a cobordism from M2 to M 4,then W1 x M 2 IHMxM3 2 M 3 x W2 is a cobordism from M 1 x M 2 to M 3 x M 4.The cobordism ring Q is known to be generated by CP', n > 2. A refinement of the notion of cobordism is the notion of oriented cobordism. Namely, an oriented cobordism between oriented manifolds M 1 and M2 , is an ori- ented manifold W, such that aW = M1 H[ M 2 and the orientation induced from W on M 1 agrees with the orientation of M1, while the orientation induced on M2 is the opposite orientation. A crucial, though trivial, point is that two manifolds related by surgery are cobor- dant. Indeed, if M' is obtained from M by performing surgery on some embedded Sk with some framing T, then we can form W = M x [0, 1] and attach an n + 1- dimensional k + 1-handle h along this Sk embedded in M x {1} with the specified framing. Then W usk h is a cobordism from M to M'. Conversely, two manifolds are cobordant if and only if they are related by a finite number of surgeries.
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