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DATE DUE DATE DUE DATE DUE 5108 K‘lProyAccaPres/CIRC/DateDue.indd HEEGAARD FLOER HOMOLOGY OF CERTAIN 3—MANIFOLDS AND COBORDISM INVARIANTS By Daniel Selahi Durusoy A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Mathematics 2008 ABSTRACT HEEGAARD FLOER HOMOLOGY OF CERTAIN 3—MANIFOLDS AND COBORDISM INVARIANTS By Daniel Selahi Durusoy In this thesis, we study the Heegaard F loer homology of 3—manifolds which is in- troduced by Peter Ozsvath and Zoltan Szabé. We first look at certain Brieskorn homology spheres. We then consider the question of which 3—manifolds bound ra- tional homology 4-balls, and give partial answer in the case of Brieskorn homology spheres 2(1), q, r) with small values for p, q, 7“ using the obstruction d defined through cobordism maps between Heegaard Floer homologies. To my parents iii Acknowledgments I would like to thank my advisor, Dr. Selman Akbulut for his excellent guidance, help and encouragement throughout all of my study at l\«'Iichigan State University. Many thanks to the members of my advisory committee Dr. Ronald F intushel, Dr. Nikolai Ivanov, Dr. Lawrence Roberts and Dr. Michael Shapiro. I would also like to thank my friends and fellow graduate students, in particular, Svetlana Roudenko, Onur Agirseven, Jens von Bergmann, Firat Arikan, Jim Carter, Steve Miller, Susanne Kleff, Don and Becky McMahon for useful discussions and keeping me motivated. Finally, I would like to thank Dr. Sergey Finashin, Dr. Mustafa Korkmaz, Dr. Ylldlray Ozan and Dr. Turgut Onder from Middle East Technical University who have introduced me to the field of geometric topology. iv Table of Contents List of Tables .................................. vi List of Figures .................................. vii 1 Introduction ................................ 1 2 Background on Heegaard Floer homology .............. 3 2.1 Heegaard diagrams ............................ 3 2.2 Symmetric product and the chain complex ............... 9 2.3 L-spaces .................................. 15 2.4 Absolute gradings ............................. 15 3 Floer homology of certain 3-manifolds ................ 17 3.1 Plumbing diagrams ............................ 17 3.2 Floer homology of certain homology spheres .............. 19 3.3 A combinatorial model .......................... 19 3.4 Floer homology of 2(2, 5, 7) ....................... 22 3.5 The family 2(2, 3, 6n + 1) ........................ 24 4 Invariants of cobordisms ........................ 28 4.1 d-invariant ................................. 28 5 Action of diffeomorphisms ....................... 33 5.1 An involution ............................... 33 A Appendix: Plumbing graphs ...................... 35 Bibliography ................................. 40 List of Tables 4.1 Values of d and [L for r g 11 ....................... 31 4.2 List of Brieskorn spheres with d = 0 with r g 13 ............ 32 A.1 Plumbing graphs for 2(1), q, 7‘) with d = 0 and 7‘ S 13 ......... 35 vi List of Figures 2.1 T3 = nd(I‘) o nd(I"). ........................... 4 2.2 Standard system of curves on 8H9 ................... 6 2.3 A Heegaard diagram for S3 ....................... 7 2.4 Sliding 132 over 31 ............................. 8 2.5 Destabilizing a Heegaard diagram .................... 8 2.6 Pointed Heegaard diagrams for S3 and S1 x S2 ............ 9 2.7 Whitney disk representing an element of n2(a:, y) ............ 10 2.8 Whitney triangle representing an element of 7rg(:1:, y, z) ........ 12 2.9 Heegaard triples .............................. 13 2.10 Three views for the standard genus 1 Heegaard diagram for S3 . . . . 14 2.11 Three views for the standard genus 1 Heegaard diagram for S1 x 5'2 . 14 3.1 Rational surgery description of 2(a1, a2, a3) .............. 17 3.2 Plumbing graph for 2(2, 5, 7) ...................... 22 3.3 Equivalences in K+(G) .......................... 23 5.1 Zid=XU—X and Zf=XUf—X ................... 33 vii Chapter 1 Introduction Floer homology was introduced by Andreas Floer [F11, F12] which was built on the ideas of Morse theory of critical points. Since then several homology theories have been defined using the same basic principles, the so called Floer-type homologies. Starting in 2001, Peter Ozsvath and Zoltan Szabo [081] have defined and extended Heegaard Floer homology for 3 and 4 manifolds, and eventually for knots in 3- manifolds, whose setup is similar to the Lagrangian intersection Floer homology. The generators of the chain group are intersections of two Lagrangians which are the products of Heegaard curves, inside the symmetric product of the Heegaard surface [081, Pe] and the boundary map counts holomorphic disks connecting the intersection points. Heegaard Floer homology is well suited for calculatons and has provided new proofs of famous results such as the following: symplectic 4-manifolds have non-trivial in- variant, Milnor conjecture for torus knots, and recently it has been used in finishing the proof of the Property P conjecture. Also, the version for knots can detect the genus of a knot, and also fiberedness of a knot. This new setup is also based on gauge theory, but recently it has been shown to admit some purely combinatorial desctiptions, too. One of them [SW] is based on modifying a given Heegaard diagram so that it will have only simple regions except one, which is enough to recover hat theory combinatorially, and the other one is based on starting with a grid diagram for a link [MOL MO2] which gives a simple Heegaard diagram to start with, but probably of very high genus. In both versions number of generators tend to be very large, but. in many cases the homology groups can be computed using a computer. In this thesis we use the combinatorial approach given by Ozsvath and Szabo [034] to compute H F + for negative definite plumbings with at. most one bad vertex. In particular, we use these techniques to compute Theorem 1.1. HF+(—Z3(2, 5, 7)) = T; 69 2%0). Theorem 1.2. HF+(—::(2, 3,672. + 1)) = T0+ e z?” The Brieskorn spheres in Theorem 1.2 are obtained as —1/n surgery on the right handed trefoil knot also as +1 surgery on the twist knot with 2n + 2 crossings. In [033] Ozsvath and Szabo have computed H F +(2(2, 3, 6n. + 1)) (denoted by Z_1/n in that paper) using surgery exact sequence and that +5 surgery on the trefoil is a lens space. By reversing orientation we show that we get the same result. We also investigate values of the d invariant, which is an invariant of spinc ratio— nal homology cobordisrn. In particular, for integral homology spheres it gives an obstruction to bounding rational homology 4-balls. Recently Lisca [Li] has applied the celebrated theorem of Donaldson to list all lens spaces that smoothly bound rational homology 4-balls. We investigate the intersection forms for the canonical negative definite 4-manifolds bounding Brieskorn spheres and provide a list of results. Chapter 2 Background on Heegaard Floer homology 2. 1 Heegaard diagrams We will be working with smooth closed oriented 3-manifolds, unless stated otherwise. A solid handlebody of genus g is the compact 3—manifold homeomorphic to a tubu- lar neighborhood of a wedge of g circles in R3. The boundary of a solid handlebody H g is the closed surface of genus g, which we denote by 29. Definition 2.1. A Heegaard splitting of Y3 is a decomposition of Y into a union of two solid handlebodies along their common boundary: for some 9 2 0. The embedding of HO and H [3 into Y induces a gluing map (25 : (91-10 —> (9H5 on their common bounday 29 up to conjugation by an element of the mapping class group. Proposition 2.2. Each closed connected 3-manifold Y admits a Heegaard splitting. Proof. We recall the sketch of proof from [R0], page 240. The main ingredient is the fact that each 3-manifold is triangulable. Let H be the tubular neighbourhood of the 1-skeleton of a triangulation. Then its complement is also the tubular neighbourhood of a 1-complex (the dual 1-complex), which is also a solid handlebody. Cl Example 2.3. We describe a Heegaard splitting of the 3—torus T3. Consider the graph F in T3 where we visualize T3 as the quotient of the cube obtained by glu- ing the opposite faces together via translation. Since F is a connected graph, its tubular neighborhood is a solid handlebody. Note that its complement is the tubular neighborhood of the graph I" obtained by the image of the edges of the cube in the quotient, hence we get a genus 3 splitting of T3. \ / Figure 2.1: T3 = nd(I‘) U nd(F'). A Heegaard splitting can be encoded by a pair of sets of curves on the Heegaard surface 29. These curves will be the boundaries of a set of compression disks for the solid handlebodies Ha, H 5, which cut the corresponding solid handlebody into B3. Definition 2.4. A Heegaard diagram of a 3—manifold Y is a. triple (Eg, a, ,3) where 29 is a Heegaard surface and a and ,3 are sets of pairwise disjoint 9 simple closed curves on 239 that cut 29 into punctured 52. Remark 2.5. The curves 01, . . . ,ag are essential curves and their homology classes are linearly independent in H1(Eg). We will call such set of curves a system of curves. They are maximal in the sense that any other curve disjoint from given ones has to be linearly dependent. Same statement is true for B curves. In fact, switching the a and fl curves corresponds to changing the orientation of the 3-1nanifold. To recover a 3-manifold Y3 from a given Heegaard diagram, we look at 29 x I and add 2-handles along ((1,0) and also 2-handles along (fl, 1). Since a and ,6 are systems of curves, this 3-manifold has boundary 5'2 II 5'2, and we cap these off by gluing copies of B3. It is instructional to look at the same concept from the Morse theoretic point of view. Given a self indexing Morse function f : Y3 ———> R, with only one index 0 and one index 3 critical point, we obtain a. Heegaard splitting by taking Ha = f "1 ([0, 23]) and H 3 = f ’1 ([%,3[). Since the Morse function gives a handle decomposition for Y3, Ha- will be precisely the union of the unique 0-handle and the g 1-handles, which is a solid handlebody. We see that H 5, the complement of Ha, is also a solid handlebody by turning Y3 upside down, i.e. f r—> —- f. The compression disks will correspond to the ascending and descending disks at. the index 1 and 2 critical points of f, respectively. Proposition 2.6. A fter a homeormorphism of the surface, we can always take the (1 curves to be the standard system of curves for the model solid handlebody Hg as in figure 2. 2. Proof. Given a Heegaard diagram (29, a, H), cutting 29 along or gives a sphere with 29 holes. The same is true with the (1 curves given in Figure 2.2. This gives us a homeomorphism from Eg\a to (8H9)\a. Now observe that this homeomorphism (I (I 1 02 9 Figure 2.2: Standard system of curves on 8H9 extends to the closed surface. [3 Due to this proposition, any Heegaard diagram can be drawn with only 5 curves on the model solid handlebody. Even though not drawing 0 curves might seem useful at first, it doesn’t fit out purposes since we will need to see where the a and ,6 curves intersect. Remark 2.7. The information carried in a Heegaard diagram is how Ha. is glued to H 5, which can be considered as an element (/5 of the mapping class group of the Heegaard surface after identifying 8H0 with a standard genus g handlebody. For a different identification, we get conjugates of Q5. Composing d) with an element of the Torelli group gives the same 3-manifold. Example 2.8 (A Heegaard diagram for S3). In Figure 2.3, we see E32 and two pairs of curves on it (adapted from [Ro], page 243). Since (11 and (12 are linearly independent, and similarly 51 and fig, this gives a Heegaard diagram. We can cut along a1 and 02 to get 4 times punctured 32, hence we can represent the Heegaard diagram as a planar diagram as shown in the middle of Figure 2.3. Now it becomes obvious that we can slide the feet. of the 1—handles to obtain the obvious Heegaard diagram for S3#S3 where we see two cancelling pairs. Example 2.9. The only 3-manifolds which admit genus 1 Heegaard splittings are a 0‘1 0’2 1 Figure 2.3: A Heegaard diagram for S3 the Lens spaces L(p, q). In particular L(p,0) = S3 and L(0,q) = S1 x S2. Their Heegaard diagrams have (11 representing (0,1) and fil representing (—19, q) on the torus. The Reidemester-Singer theorem tells that any two Heegaard diagrams of Y3 are stably isotopic. This can be formulated as follows: Theorem 2.10. Any two Heegaard diagrams for Y can be connected by a sequence of the following moves: 1. isotopies of a and ,3, through one parameter family of systems of curves, 2. handle slides among a and handle slides among 3, 3. stabilization and destabilization. For the proof, we refer the reader to Proposition 2.2 in [081]. A handle slide among (1 curves means isotoping one of the attaching circles oz,- over the core disk in Ha of another o'j. This is illustrated in Figure 2.4. As a final step we can isotope fig and get a completely symmetric diagram, which corresponds to the 3-manifold (51 x s2)#(sl x 52). —-—> 01 (1'2 (11 ()2 Figure 2.4: Sliding [32 over {31 Stabilization is the process of taking the connected sum of the Heegaard surface with a torus containing curves ag+1,l3g+1 lying as the standard longitude-meridian pair. In terms of the 3-manifold this corresponds to a connected sum with S3. Destabilization is the inverse operation, namely if only fig intersects cry, and geo- metrically once, then the diagram can be simplified by compressing 29 along the disk that 09 bounds in HO and erasing fly from the diagram, see Figure 2.5. Figure 2.5: Destabilizing a Heegaard diagram For the purposes of Heegaard Floer homology, we will need marked Heegaard di- agrams. Marking a point on 29 avoiding a and fl gives rise to a choice of spinC structure on Y3 for each generator. Furthermore two distinct. points on a Heegaard diagram encode a knot in Y3 as follows: given a Heegaard diagram compatible with a Morse function f with a unique maximum and a unique minimum and two points z, w E 29 — (a U ,8), consider the flow-lines connecting index 0 and index 3 critical points to z and w. These four arcs make up an oriented closed embedded curve in Y3 (start traveling at the maximum, visit 2 first). 1, “X “’n? [31 51 Figure 2.6: Pointed Heegaard diagrams for S3 and S1 x 82 2.2 Symmetric product and the chain complex Heegaard F loer homology is inspired by Lagrangian intersection Floer homology in- troduced by Andreas Floer in [F11]. The symplectic manifold that is used in Heegaard Floer homology is the symmetric product of the Heegaard surface, and the generators for the chain complex will be the intersections of two totally real tori in the symmetric product. Definition 2.11. The kth symmetric product of a surface 29 is the quotient of the product 2g x x 239 (k factors) under the action of the symmetric group on k letters. It is denoted by Symeg. A point x E S ymk 29 means unordered k points {171, . . . ,ark} (not necessarily distinct.) on 29. Unless mentioned otherwise, we will use I: = g. In particular we will denote Symgilg by 2(9). Note that the symmetric product of a surface is a 2k-dimensional manifold, whereas in general symmetric products of manifolds of other dimensions have orbifold singularities. The curves 0: and ,8 define g-dimensional tori in 2(9) as follows: let 'll‘a be the set of points x with at, E a, for 1 S i S g. This is indeed an imbedded g-dimensional torus since the curves oi,- are disjoint. Similarly we have the torus T3. 2.2.1 Holomorphic disks The chain groups CF (Y) are freely generated by intersection points of Ta and T3. Similarly The chain groups CF 00(Y) are generated by [agi] for a: 6 Ta 0 T3 and i E Z. The boundary operator is defined via counting holomorphic Whitney disks: Definition 2.12. A Whitney disk it between :13, y E To 0 T3 is a continuous map u : D ———> 2(9) with u({0} X IR) C T23, u({1} x R) C Ta, tlim u(s,t) = a: and —-+—oo tlim u(s, t) = y. The set of homotopy classes of Whitney disks from .r to y is denoted —)00 by 7r2017,31)- Figure 2.7: Whitney disk representing an element of 7T2(.’13, y) Let. p E 7rg(a:,y). The local multiplicity nw(¢) of ob is defined to be the algebraic intersection number of 9501)) with {u} x Symg’lZ. The set of holomorphic disks which represent (25 is the moduli space M(qb), and its dimension is the Maslov index, denoted by mo). There is an action of R on M (o) given by precomposing u with a vertical translation in ID). We will denote the quotient Mos/1R by We. Proposition 2.13. Ifqb has a homolorphic representative u, then nu,(gb) _>_ 0. 10 Proof. Because of the almost complex structures chosen, {w} x 2(9_1) is a holomor- phic variety and the intersection of two holomorphic varieties is non-negative. El Remark 2.14. If nw(¢) < 0 then M(q‘)) is empty. Proposition 2.15 ([OSI], Prop 2.15). When 9 > 1, if 7rg(:1:,y) 75 (0 then Ma, y) 2 Z 69 H1(Y; Z). The boundary operator is defined by (9hr, i] = Z #A/T (<15) - [y, i — nw(¢)]. ¢E7r2($ay)a #(¢):1 Recall figure 2.6 given in page 9. From the Heegaard diagram we see that 577(83) is generated by a single element. For S1 x 82 there are two generators yl and y2. In general when b1 (Y) > 0 a special class of Heegaard diagrams are used so that the resulting homology is an invariant. Note that in that figure 51 could be isotoped to get rid of the intersection points, and that would mean there would be no generators for the F loer homology. But that is not an admissible Heegaard diagram in the sense of [081]. We will revisit these examples in section 2.2.4. 2.2.2 Heegaard Floer chain complex There is a Z action on CF 00(Y) given by U [.1: i] = [:r, i— 1]. By definition it decreases the homological grading by 2. The subcomplex generated by [:r, i] for i < 0 is called CF - (Y). The inclusion CF _(Y) C CF 00(Y) fits into the following short exact. sequence of chain complexes, where CF +(Y) is the quotient complex: 0 ——+ CF’(Y) —> CF00(Y) —-—> CF+(Y) ———> 0 On the other hand the U map has a kernel which we call CF (Y): 0 —> Kim ——» CF+(Y) 11—. CF+(Y) ——> 0 11 These short exact sequences induce the following long exact sequences in homology: ._. HF—(Y) —-(-—> HF°C(Y) i. HF+(Y) i. (2.1) . —+ RFC”) —l——> HF+(Y) _U_. HF+(Y) i. (2.2) All these chain groups can be obtained from CF _(Y) as follows: 0 CF00 is the localization CF_(Y) ’8: Z[U,U-1], 0 CF +(Y) is the cokernel of the localization map, 0 CF(Y) is the quotient CF‘(Y)/U - CF—(Y). 2.2.3 Holomorphic triangles Figure 2.8: W'hitney triangle representing an element. of 7rQ(:c, y, 2) Counting holomorphic triangles gives rise to a. chain map Fa,f3,7 3 CF(Yafletafil 8" CFiyfle/a tfiy) __’ CFO/07: to?) $®y+—> Z #M(¢l‘3- eternal/e). #(O)=0~ 321?(¢)=3~71vu‘(0)=0 12 Figure 2.9: Heegaard triples In figure 2.9 there is a small triangle #1 E 7rg(:1:,6’,y') but. it is not orientation pre- serving, hence can’t be holomorphic. Hence we have FW(:1:) = y, and FW increases absolute grading by %, hence gily) = —%, and there is a holomorphic disk from y’ to e, heeee era) = i. 2.2.4 Two examples: S3 and S1 x 52 We quote the Heegaard F loer homology of two basic 3-manifolds that we will need in later sections. Since S3 has a unique spine structure, we will ommit it from the notation. Lemma 2.16. The Heegaard F loer homology of S3 is given by 173.93) = Z Z, deuen HFWSB) = [0, dodd 0 Z, d 2 0 and even HF;(S3) = 0 , otherwise HFeee(S3> = The chain groups 577(53) are generated by the single intersection point :c in fig— ure 2.10, hence boundary map is trivial. The chain groups CF 00(S3) are generated by [:r,i] for i E Z, and U is an isomorphism decreasing grading by 2, hence we get 13 the mentioned homology groups. U : H F +(S3) ——) H F +(S3) is surjective, hence HFre(1(53) = H F +(S3) / I m(U k ) = 0 where k is any large positive integer. 01 131 Figure 2.10: Three views for the standard genus 1 Heegaard diagram for S3 Lemma 2.17. The Heegaard Floer homology of S1 x S2 is given by fi(81 X52,t0) = Z 1€EZ1 —2 2 Z,d€%+Z 0, otherwise HFf(Sle2,t0) = [ 0w . ‘- a y .- 3 1 Oti . X31 fil J, 01 [/31 Figure 2.11: Three views for the standard genus 1 Heegaard diagram for S1 x 32 There are two holomorphic disks connecting :c to y with opposite signs, hence we get PIP (S1 X 52) g Z +29 Z. The homological grading is initially defined as a relative grading, but in section 2.4 we tell how to obtain absolute gradings, and the rational gradings come from the degree shift formula. 14 2.3 L-spaces Definition 2.18. A rational homology sphere Y is called an L-space if rank of UP (Y) is same as the cardinality of H1(Y; Z). Example 2.19. Lens spaces are L-spaces. If 53(18’) is an L-space with n > 0, then Sg+1(K) is also an L—space. The Poincare homology sphere 2(2, 3, 5) is an L-space. Plumbed 3-manifolds with no bad vertices are L-spaces. Also branched double covers of 5'3 along non-split alternating links are L-spaces. 2.4 Absolute gradings Ozsvath and Szabo ([OS2], theorem 7.1) have shown that there is a. consistent way to extend the relative gradings on rational homology 3-spheres to an absolute Q-grading. Definition 2.20. Heegaard F loer homology groups of rational homology spheres carry an absolute grading g7“ : H F°(Y, t) —+ Q characterized as follows: 1. (normalization) PIP(S3) is supported in grading 0. 2. (lifting) g~r lifts relative grading: gr(£, 7)) = g~7(€) — 97(7)) 3. (cobordisms) gv-(FW-,,(§)) — 97(5) = 211— (01(s)2 — 2x(W) — 30(W)) for a cobor- dism (IV, .9) between (Y1, t1) and (Y2, t2) with torsion spine structures 4. the maps i, 1. and 7r in equations 2.1 and 2.2 (see page 12) preserve g~r and 6 decreases g? by 1. More concretely, if L is a framed link so that Y = S3(L), and W = I'V(L) is the 15 corresponding cobordism, then for y 6 CF (Y), ~ arty) = #16") + 2nuv('l£~’) + (01(8)2 - 2><(W) - 300W) - 1 4 There is a duality pairing between on Heegaard F loer chain groups given as follows [083]: .I._ 1r=y,j=—i~—1 ([232], [y’JD _ { 0 otherwise When c1(t) is torsion, (-,o) induces an isomorphism CF +(Y,t) ——> C F_(—Y, 8), hence an isomorphism HFi+(Y, t) ——+ HF:i_2(—Y, t) given by [1“, i] r—+ —[:r, —i— 1]*. 16 Chapter 3 Floer homology of certain 3-manifolds 3. 1 Plumbing diagrams A weighted graph is a graph with integer weights associated to its vertices. Given a weighted graph G', there is a corresponding smooth 4—manifold X (G') obtained as follows: for each vertex form the D2 bundles over 5'2 with Euler number given by the weight, and corresponding to each edge, plumb the corresponding D2 bundles (and corners are smoothed). The boundary of X (G) is a 3—manifold which we will denote by Y(G). Consider the Seifert fibered homology spheres with 3 singular fibers: they are the Brieskorn homology spheres 2(a1, a2, a3) for relatively prime positive integers ai: 53 b2 Figure 3.1: Rational surgery description of 2(a1, a2, a3) 17 Here, the unknowns are determined from the following equations: b l) b- al-ag-a3-(—b+511+;:-+;:-)=l where b S —-1 and b,- S —2. These conditions determine a negative definite minimal plumbed 4-manifold with its boundary 2(a1, (12, a3). For example the following weighted graph G describes Y(G) = 2(2, 3, 6n + 1), where the total number of vertices is n + 3 and vertices where the weight is not written have weight ——2. 2(2, 3, 6n + 1) The intersection form on H2(X(G)) is given by the adjacency matrix of the graph with the weights on the diagonal entries. For example, using the above graph with n = 3 we get the following intersection matrix: [-11110 01 1-2 0 0 0 0 10—3 0 0 0 10 0—7 1 0 0 0 01—2 1 _0 0 0 01—2] If this matrix is negative definite, the weighted graph is called negative definite. Remark: The negative definite minimal plumbing diagram for 2(a1, a2, a3) is char- acterized by —a,' :(2,3,6n+ 1); Theorem 3.2. HF+(—Z(2, 3, 6n +1)) = T0+ EB Z8». This family of homology spheres can be obtained by doing —1/n surgery on the right handed trefoil knot. This family and some more were considered in [F S3] where their instanton Floer homology is calculated. For the Heegaard-Floer homology computa- tions for the family 2(2, 2n + 1, 4n + 3) see [Rul]. Another combinatorial description for calculating H F + has been given by Némethi in [Ne]. 3.3 A combinatorial model In this section G will denote a negative definite weighted graph with at. most one bad vertex. X (G) is the plumbing 4-manifold and Y(G) = (9X (G), W'(G) = X (G) — B4 cobordism from Y(G) to S 3. Let. Char(G) denote the set of characteristic vectors in H2(X(G)) with respect to the intersection form. 19 Given an element. 5 E HF+(—Y(G)), we can define a map 6.55 E lilap(Char(G), 76+) by declaring ¢€(K) = Fli"(G) (.5) where c1(sK) = K and FW is the map on Floer .8 K homology induced from the Spine cobordism (W(G), 85'). Definition 3.3 ([OS4]). ltll+(G) C AIap(Char(G), 76+) is the subset of finitely sup- ported maps satisfying Um+n - f(K + 2PD[v]) = Um - f(K) if min{m,m + n} 2 0 and K - v + v - v = 2n In [OS4] the following is shown: Theorem 3.4 ([OS4]. Thrn 1.2). For such a graph G, for each Spinc structure t over —Y(G)2 HF+(—Y(G),t) a: IHI+(G,t). We will be working with homology spheres, so we suppress the S pin" structure from the notation. In the computations instead of working with elements of lHl+(G), [OS4] suggests working with elements of K +(G), which is the set of equivalence classes of elements in Z20 X Char(G), with the equivalence relation defined by (m, K) ~ (m + n, K + 2PD[v]) where v is a vertex in G with K - v + v - v 2 2n and min(m, m + n) 2 0. Equivalence class of (m, K) will be denoted by U 7" 8 K. For an equivalence class U 7’ l 8 K, define its U—depth as the largest number l so that. (l, K ' ) is a representative of U m 8 K for some vector K ' . K +(G) is determined by elements of U -depth 0 and the U action on K+(G), which follows from 20 Proposition 3.5 ([OS4], Prop 3.2). For an equivalence class Um (8 K of U-depth 0, there is a unique representative (0, K) satisfying 1),: -v,' + 2 S K - v,- 3 —v,' - v,- for each vertex 17,-. (3.1) Conversely if a vector K satisfies (3.1), then K has U-depth 0 if and only if K supports a good full path ( in this case K will be called a basic vector). In the above, full path stands for a. path of vectors K1, K2, ..., Kn in Char(G) with K1 satisfying (3.1) obtained by adding 2PD[v,-] if Ki - vj + vj - vj = 0 for some j, until —Kn satisfies (3.1) or Kn -vj + vj - vj > 0 for some j. It is called good if —Kn satisfies (3.1). In the proof of [OS4], Proposition 3.2, it is shown that given a characteristic vector Al, the final vector of any full path is identical, hence we observe the following useful Remark 3.6. Observe that if a vector supports a good full path, then all full paths are good, hence finding one bad full path means the initial vector is not basic. Secondly observe that bad full paths are hereditary, hence if a vector for a subgraph has a bad full path, then so does the containing vector and graph. This will reduce the number of vectors that we need to check if they support a good full path or not. We will express the vectors K E Ghar(G) as sequences (K - 11,-). Lemma 3.7. For the linear graph A3 with 5 vertices and each weight —2, there are no good full paths starting at vectors K satisfying (3.1) with K - '12,: = 2 for more than one i. Proof. we will use induction on s, observing that we can use hereditary property of bad full paths. (3.1) implies K - v, 6 {0,2}. For 3 = 2, (2. 2) for A2 has a bad full 21 path obtained by adding 2PD(vl). For 8 > 2, observe that if K - v,- = K - vj = 2 for some i # j, then K is equivalent. to a vector containing (2, 2) as shown below, hence has a bad full path. (*,2,0.0, ...,0,0, 2, e’) ~ a”, —2, 2,0, ...,0, 0,2, e’) ~ (*”, —2,0, —2,2,2, ..J), 3.4 Floer homology of 2(2, 5, 7) From [AK] we know that 2(2, 5, 7) is the boundary of the negative definite plumbing of disk bundles over 2—spheres described by the following plumbing graph: Figure 3.2: Plumbing graph for 2(2, 5, 7) This graph is negative definite and has only one bad vertex vl in the sense of [OS4], i.e., only [vl] - [vl] > —d(vl) where d(v) counts edges containing v. In the notation of [OS4] we write 2(2, 5, 7) = (9X(G) = Y(G), and we compute HF+(-Y(G)) '5 H+(G) where H + (G) is a group which can be calculated by the algorithm of [OS4] for negative definite (possibly disconnected) trees G with at most one bad vertex. Here we follow this algorithm (for the terminology, we refer to [OS4]): 22 First we need to find characteristic vectors K satisfying inequality (3.1). There are only finitely many such vectors since G is a definite graph, in particular there are 80 characteristic vectors satisfying (3.1) for the given graph. Proposition 3.2 in [OS4] characterizes a spanning set for K +(G) which is the dual point of view for H +(G). Among the 80 characteristic vectors satisfying (3.1) we look for those vectors K which carry full paths ending at vectors L for which —L satisfies (3.1). There are precisely three such vectors K and we list them and also the full paths for convenience (paths are obtained by consecutively adding 2PD(v.z-) for the numbers i listed): K1 =2 12111 + 6122 + 3113 + 4114 + 2v5 path: 1,2,1 K2 = —8v1 — 41.22 — v3 —- 2v4 — 211,5 path: 1,2,1,5,4,1,2,1,3,1,2,1,4,1,2,1,5 K3 = —16vl — 8172 — 3113 — 4114 — 2v5 path: 1,2,1 All other equivalence classes in K +(G) can be obtained using the U -action from these vectors. \Ve must check which are distinct. From now on, we will write the vectors as 5-tuples with entries (K, [vi]), hence Ki will be denoted as (1,0,—3.—2,0), (1,0,—3,—2,2), and (1,0,—1,—2,0). The following diagram shows how U 8- K1, U 8 K2 and U 8 K3 are equivalent to (—3, 4, 1, 2,0), and hence to each other: —3,4,1,2,0 ( 1 . ) 1 1 2 4 l 3 1 2 1 1 2 1 2 3 1 4 1 1(_3,4’1,2,0) K1 o—o/ K2 1 1 2 5 4 1 5 1 12 3 1 l 1 2 4 1 3 1 2 1 1 2 12 3 l 4 1(_3 41 2 0) K3 1 2 4 1 2 5 4 1 1 5 Figure 3.3: Equivalences in K +(G) In the above figure, vertices correspond to characteristic vectors and an edge ending with a. number i means 2PD(v,-) was added to previous vector to reach the new one. 23 Edges with positive slope mean tensoring with U. Hence first diagram represents the vectors (1,0, —3, —2,0), (—1, 2. —1,0,0), (—3,4, 1,2,0) which tells us that. (1.0, —3, —2, 0) ~ (—1,2, —1.0,0) and U a (—1,2,—1,0.0) ~ (—3,4,1,2,0) Hence each K, lies on the same grading level and we can check this by calculating 1 E (K - K + |G|) which is zero for each Ki. Therefore we conclude: H F +(—Y(G)) g TO+ @Zm) @Zm). By Prop 7.11 of [082] we can convert this to the Heegard-Floer homology of Y(G): HF,+(Y(G)) = HF:*—2(—Y(G)) = T0+ e Z(_1)ea 2H) To do this, we first use the following long exact sequence to compute H F _(-Y(G )) .. _e arr-(4(0)) _. HF°°(—Y(G)) _. HF+(—Y(G)) —-—> then we adjust gradings by changing the signs and substracting —2. Note that here we adapted the convention of [OS4] by denoting Ts"L = @2020 Z(s+2k)' So we have: HF+(2(2, 5, 7)) = T3 are z(_1)e Z(_1) 3.5 The family E(2,3,6n + 1) Consider the family of Brieskorn spheres Y(n) = 2(2, 3, 6n+1). The negative definite plumbing graph defining Y(n) is a tree with weights —1 on central node, —2, —3, —7 on adjacent. nodes and a —2 chain of length n — 1 starting at —7 as follows: 2(2, 3, 6n + I) --.-—-O 24 The Heegaard-Floer homology of the first member of this family, H F +(—E(2, 3, 7)), has been studied in [OS4], [Rul]. Lemma 3.8. For arbitrary n, the basic vectors for Y(n) = 2(2, 3, 6n + 1) are K1 2 (1,0, -—1, —5,0,0,0, ...,0) K2 = (1,0, —1, —3, 0,0,0, ...,0) K3 2 (1,0,—1,—5,2,0,0,...,0) K4 = (1,0,—1,—5,0,2,0,...,0) K5 = (1,0,—1,—5,0,0,2,...,0) Kn+1 = (1,0,—1,—5,0,0,0,...,2) Proof. Clearly for each j, K j satisfies (3.1). Next we need to see that among all characteristic vectors satisfying (3.1) these are the only ones supporting good full paths. For n = 1 this is done in [OS4], and we verified it by computer. By remark 3.6, for n > 1 first 4 entries of a basic vector has to coincide with one of (1,0 — 1, —3), (1,0,—1,—5) which were computed in [OS4] for n = 1. Other entries are either 0 or 2. Moreover lemma 3.7 implies that for basic vectors K for Y(n) there can be at most one vertex with K - v,- = 2. Claim. (1, 0, —1, —3, *) has a bad path if * has a non—zero entry. Proof of claim. As in lemma 3.7, we can find a vector (1,0, —1, —3, 2, *') equivalent to K. But (1, 0, —1, —3, 2) has a bad path obtained by addind 2PD(v,-) in the order i = 1, 2, 1, 3, 1, 2, 1, 5, 4, 1, 2,1 and bad paths are hereditary. Cl Hence the only basic vector with initial segment (1, 0, —1, -—3) is (1, 0, —1, —3, 0, ..., 0). Next we need to show that K1, ..., K7111 support good paths. This we do by explicitly giving the paths. First, 1, 2, 1,3,1,2,1 is a good full path for both K1 and K2. For others, the path starts the same, but continues as: 5, 6, 7, ..., n+3 forK3 6,5, 7,6, 8,7, n+3,n+2 forK4 7,6,5, 8,7,6, 9,8,7, n+3,n+2,n+1 forK5 n+3,n+2,n+1,...,5 for Kn+1 This finishes the proof of the lemma. [:1 For each K 2'» when we compute the renormalized lengths KR HG , each time we get 0. Next we investigate relationships between U powers of Ki. Lemma 3.9. U®Ki ~ U®Ifj ~ L =(—3,2,5,I,0,0,...,0)f07‘1_<_i,j S n+1 Proof. For K1, the sequence 1,1,2,1 leads to L. For i > 1, the path from K ,- leading to L is of the following form: 17172717273317 A71,l3 B” where Ben is 1,2, 3, 1,4, 1,2,1 followed by 5,6, ...,n + 3 followed by 1 and Ana‘ is of the following form: Cn—i+1aCn—i+2e "'3 C0- In the above, Ck denotes the sequence 1,2,3,1,4,1,2,1,3,1,2,1. 5, 6, 4 + k if k > 0 and empty path if k = 0. As an example, for n = 4, the path from K2 to L is given 26 1,1,2,1,2,3,1.A4,2,B4 = 1,1,2,1,2,3,1,C3,CQ,C1,B4 = 1,1,2,1,2,3,1, 1,2,3,1,4,1,1,2,1,2,3,1, 1,2,3,1,4,1,2,1,3,1,2,1,5, 1,2,3,1,4,1,2,1,3,1,2,1,5,6, 1,2,3,1,4,1,2,1,5,6, 7,1 It is straightforward to check that these paths end at L. Next, one observes that. as in [AD], throughout these paths the U-depth stays between 0 and 1, hence we get U 8 K,- N L as announced. C] Proof of Theorem 3.2. Now we know that K+(G) consists of U0 8 K1, U0 8 K2 and Um 8 K1 for m > 0. For any f in lHl+(G), f(K1) E T; determines the images f(m, K1) of the induced map f : Z x Char(G) —> Tdf. The remaining values f(K,') for i > 1 are also determined up to addition of an element of Z(0). This finishes the proof of the theorem. 1:] Note that Ozsvath and Szabé have computed H F + of this family with opposite orientation using a different method. After orientation reversal we see that the two computations agree. 27 Chapter 4 Invariants of cobordisms 4. 1 d—invariant Using absolute gradings, for rational homology 3-spheres Ozsvath and Szabo have defined [083] an invariant of spinc rational homology cobordism as follows: Definition 4.1. Given a rational homology sphere (Y, t), d(Y, t) is the least. absolute grading of the elements in H F +(Y, t) that. lie in the image of H F QC(Y, t) under the map induced by the projection 71. Theorem 4.2 ([083], Theorem 1.2). d is an invariant of spine rational homology cobordism between rational homology 3-spheres endowed with spine structures. Since there is only one spine structure on an integral homology sphere, if it bounds a rational homology 4—ball, it is rational homology cobordant to S3, hence we have Corollary 4.3. If an integral homology 3-sphere E bounds a smooth rational homology 4-ball, then d(Z) = 0. In [AK], Akbulut and Kirby [AK] have shown Z(2,5,7),Z(3,4, 5) and Z(2,3,13) bound contractible 4-manifolds. Also Casson and Harer [CH] and Stern [St] have 28 shown families of Seifert fibered homology spheres that bound contractible 4-manifolds. Since then some more sporadic examples have been shown. More recently Lisca has identified connected sums of lens spaces that bound rational homology balls. The Neumann-Siebenmann invariant [1. gives another obstruction for Seifert fibered homology spheres to be homology cobordant to S3, which was proven by Saveliev [Sa3]. Also the invariant R((11, - ~ ,an) defined by Fintushel and Stern [FS2] tells the central node in a. minimal negative definite plumbing diagram has to be —1 if a Seifert fibered homology sphere bounds a Zg-acyclic 4-rnanifold. Remark 4.4. Fintushel and Stern [FS1] have shown that 2(2, 3, 7) bounds a rational homology 4—ball via constructing a rational homology cobordism using Kirby calculus. For this example [1 = 1 whereas d = 0. Since [1 7E 0, Z(2,3, 7) doesn’t bound an integral homology 4—ball. Given a negative definite plumbing diagram with at most one bad vertex, Ozsvath and Szabo [OS4] show that d(Y, t) can be computed as the minimum of —-]1(K- K+ |o|) over all characteristic vectors K corresponding to the spinc structure t, where |G| stands for the number of vertices in the plumbing graph G. This minimum is achieved among basic vectors, and using hereditary property we can reduce the number of characteristic vectors to check. This way we computed the values of d for the first few Brieskorn homology spheres. We present these data together with values of [i in Table 4.1. There are 79 Brieskorn homology spheres Z(p,q,r) with 0 < p < q < r S 13. In Table 4.2 in page 32, we provide a list. of those with d = 0. In Appendix pages 35—39 we provide the corresponding plumbing graphs. 29 Also we investigated the intersection forms of the canonical negative definite 4- manifolds X (p, q, r) bounding the Brieskorn spheres given as the plumbed D2 bundles over SQ. Donaldson’s theorem tells that the intersection form of a negative definite closed smooth 4-manifold X is diagonalizable over the integers, i.e., there is an inte— gral matrix B with B - Q X - BT = i] d. This gives an obstruction for a homology sphere to bound a rational homology ball, in terms of intersection forms of negative definite fillings. We applied this to our particular examples: Theorem 4.5. Among all Brieskorn homology spheres B(p,q,r) with d = 0 and r < 13, the intersection form of X(p, q, r) is diagonalizable over the integers. Proof. The proof is given by explicitly constructing the matrices B(p,q, r). At the time being we didn’t obtain matrices B for (7, 8, 9) and (9, 10, 11), but since they fall into the category CH1, the intersection forms are diagonalizable over the integers. We provide a couple of examples below to give a taste of the process to the reader. The following denote the row-column operations to diagonalize the intersection form. 30 E(3,5,8) Z(5,7, 9) 2 ——> 2+1 2 ——+ 2+1 4 —> 4+1 4 ———> 4+1 5 —+ 5+1 6 ——> 6+1 3 —> 3+2 3 —+ 3+2 4 ——+ 4+2 4 —2 4+2 5 ——+ 5+2 6 —-> 6+2 4 ——> 4+3 5 -—> 5+4 5 —> 5+3 6 ——> 6+4 5 —> 5+4 3 -——> 3+5 4 —> 4+5 6 —-> 6+3 3 ——+ 3+6 5 —-> 5-3 4 —> 4-5 -1 —1 -1 -1 —1 -2 0 1 -1 0 -1 0 0 0 0 1 -1 -1 —1 -1 2 0 5 2 3 7 2 4 8 2 7 5 9 2 9 2 5 9 4 7 7102 9100 3 7 7112 2 2 3 3 3 3 9110 4112 5 11 2 7 11 0 8 11 2 10 11 0 0 3 4 4 4 5 5 1100 5 7 11 2 9 11 0 6 11 2 7112 91140 5 11 10 11 7 10 11 9 Table 4.1: Values of d and [1 for r _<_ 11 31 comments CDQOOOOONKIKINIO‘IH AK, CH1 It = 1 AK, CH2 fl=1 CH1 CH1 H .._1 r—ar—t OOerOCAt-‘l—‘KlONOOQDO‘IOKILDOTCDOOOOKJKIWODU‘Cflwi-lfi-Q i--|i—-‘ GHQ 11 8 11 12 11 10 11 11 12 Hoooooqxlxicnmesoooooomwxrmcncampawwmwuqmmwwmamwww F—‘H Table 4.2: List of Brieskorn spheres with d = 0 with r g 13 32 Chapter 5 Action of diffeomorphisms 5. 1 An involution In this section we show how Saveliev’s theorem ([Sal], Theorem 15) translates into Heegaard Floer homology. Theorem 5.1. Let W 4 be a contractible manifold whose double is 84, and Y = 6W be a homology sphere with non-trivial HF. Then the involution f : Y#Y ——+ Y#Y exchanging the summands acts non-trivially on HF(Y#Y). Proof. Consider the cobordism X = (Y x I)hW', where n denotes boundary connected sum. This cobordism induces a homomorphism F X : BF (Y) ——> HF(Y#Y). Y Y#Y Y Y Figure 5.1: Zid = X U —X and Zf = XUf —X 33 Let Zid = X Uy#y (—X) and Zf = X Uf (—X) (also compare to Matveyev [Mat] figure 2, page 576). Since IV U (—W) = S 4, Zid is diffeomorphic to the product Y x I relative to the boundary. By the functoriality of the induced maps, F(_ X) o F X = id : FIFTY) ——> KITTY), hence FX has to be injective. Z f is diffeomorphic to X Uy#y Cf Uy#y (—X), where Cf is the mapping cylinder of the involution f. Again by functoriality, F Z f is the composition F(_ X) OFCf oF X and F Zr we get FC f 7é id, hence the action induced by switching the summands acts non- factors through BFTS3), as pictured in figure 5.1. Since we assume HF (Y) aé Z, trivially on ITF(Y#Y). [:1 Remark 5.2. The assumptions for the theorem are satisfied for several contractible 4- manifolds W. By results of Rustamov [R112], the only Brieskorn homology sphere with trivial BF is 2(2, 3, 5), which doesn’t bound any contractible 4-manifold. Therefore any Mazur manifold IV with Brieskorn homology sphere boundary gives an example that satisfies the assumptions. This theorem also applies in the H F + setting: Theorem 5.3. Let W4 be a contractible manifold whose double is S4, and Y = 8W be a homology sphere with non-trivial BF. Then the involution f : Y#Y —> Y#Y erchanging the summands acts non-trivially on H F +(Y#Y). 34 Appendix A Plumbing graphs Below we provide the plumbing graphs for the Brieskorn homology spheres B(p, q, r) listed in Table 4.2, namely, those with d = 0 and r g 13. Vertices with no labels correspond to spheres of square —2. -1 2(3, 4, 5) —3 ‘1 _7 $12.3. 7) _1 ”—5 —4 _1 —3 _7 2(4,5, 7) —3 —6 -1 _3 2(5,6, 7) Table A.1: Plumbing graphs for B(p, q, r) with d = 0 and r S 13 35 Table A.1 (cont’d) 2(3, 5, 8) 2(5, 7,8) 2(5, 7, 9) 2(5, 8, 9) 2(7, 8, 9) 2(7,9, 10) 2(2,5,11) 2(2,9,11) 2(3,7,11) 2(3, 10,11) I H II {I}... l on Table A.1 (cont’d) | c: ' l .—-| O CO | I i—* {]\e~ e» l 00 || 4:»«1 II uses: 37 2(4,5,11) 2(4,9,11) E(5,8,11) 2(6,7,11) 2(9, 10,11) 2(5,7,12) 2(5,11,12) 2(7, 11,12) 2(2,3, 13) 2(2,9,13) l y-A l | | y—A $00 00 it) I «I l y-a l p-i l | | | Table A.1 (cont‘d) U! 2(3,4,13) Z(3,8,13) 2(3, 10,13) 2(4,11,13) Z(5,6,13) 2(5,11,13) 2(7813) 2(7,11,13) 2(7,12,13) 2(8, 11,13) Table A.1 (cont’d) —5 —4 3 2(9,10,13) —9 —4 —3 _3 _3 2(9, 11,13) —4 -6 _7 2(10,11,13) —3 —12 _3 2(11,12,13) 39 Bibliography [AD] [AK] [CH] [F81] [F82] [FS3] [F11] [F12] [Li] [M at] S. Akbulut, S. 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Abstract 78TG75 41 | ||||| || ||||||| | | || |||||| ||| | 23 02956 , l',. .' yr! ~ ' “'4‘“; '1—a»~ -