Exotic Four-Manifolds, Corks, and Heegaard Floer Homology
A Dissertation Presented
by
Jonathan Hales
to
The Graduate School
in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
in
Mathematics
Stony Brook University
May 2013 Stony Brook University The Graduate School
Jonathan Hales
We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation.
Olga Plamenevskaya – Dissertation Advisor Associate Professor, Department of Mathematics
Dennis Sullivan – Chairperson of Defense Professor, Department of Mathematics
Oleg Viro Professor, Department of Mathematics
Robert Lipshitz Assistant Professor, University of North Carolina
This dissertation is accepted by the Graduate School.
Charles Taber Interim Dean of the Graduate School
ii Contents
Acknowledgements v
1 Introduction 1
2 Background on Heegaard Floer homology 11 2.0.1 Heegaard Floer homology ...... 11 2.0.2 Knot Floer homology ...... 15 2.0.3 Integer Surgery ...... 19 2.1 Lifted Heegaard Diagram ...... 22
3 Chain Complex 26 3.1 Reduction to Knot Surgery ...... 26 3.2 Heegaard Diagram ...... 29 3.3 Chain Complex ...... 34
4 Integer Surgery 50
5 Further Directions 69 5.1 Four-Manifold Invariant ...... 69 5.2 Cork Twist Cobordism ...... 70
iii 5.3 New Exotic Manifolds ...... 71 5.4 More corks ...... 73
Bibliography 79
iv Acknowledgements
I am extremely grateful to my advisors Olga Plamenevskaya and Robert Lip- shitz. I am deeply grateful for their guidance, support, and patience. I feel very lucky to have to have met and studied under each of them. The Mathematics community as a whole has been a joy to be a part of, I have befriended and learned from far too many to list here. I would like to thank Marco Golla, Jake Rasmussen, and Nathan Sanukjian in particular. Their insights and advice have had a large effect on this work. In addi- tion, there are many friends whose impact, while harder to classify, have been substantial to my development. From graduate school - Ali Aleyasin, Anant Atyam, Benjamin Balsam, Paul Cernea, Eitan Chatav, Ying Chi, Stephen Dal- ton, Shane D’Mello, Patricio Gallardo, Panagiotis Gianniotis,Jan Gutt, Mark Hughes, Claudio Meneses, Raquel Aguilar, Vamsi Pingali, Chaya Rosen, Lloyd Smith, Yury Sobolev, Dror Varolin, Evan Wright, Matthew Wroten - and from college - Dmytro Karabash, Andrew Lawrie, Michael Lim, and Anna Salamon. Finally, I would like to acknowledge the incredible support and encourage- ment I have received from my family. I cannot express how much they have done for me.
v Abstract of the Dissertation Exotic Four-Manifolds, Corks, and Heegaard Floer Homology
by
Jonathan Hales
Doctor of Philosophy
in
Mathematics
Stony Brook University
2013
Akbulut corks provide a topological avenue for understanding the smooth structures of a closed four-manifold. Computing the Hee- gaard Floer invariants is afirst step towards understanding them. To this end, the Heegaard Floer homology of the bounding three- manifolds of the Mazur corks is computed.
vi Chapter 1
Introduction
In this thesis, we calculate the Heegaard Floer homology of the three-manifolds bounding a family of Akbulut corks with the goal of shedding light on the smooth structures of four-manifolds. Four-manifolds exhibit a variety of unique exotic phenomena. For example, closed four-manifolds allow infinitely many diffeomorphism types for a single homeomorphism type.
One might think of dimension 4 as representing a phase transi- tion between low- and high-dimensional topology, where wefind uniquely complicated phenomena and diverse connections with other fields.[GS99, xii]
The low-dimensional equivalence between the smooth and topological cate- gories breaks down, while the powerful high-dimensional h-cobordism theorem has yet to take full effect. Akbulut corks come out of the failure to extend the smooth h-cobordism theorem to the fourth dimension.
1 In higher dimensions, the h-cobordism theorem states:
Theorem 1 [Sma61] IfW n+1 is an h-cobordism between the simply connected
n+1 n-dimensional manifoldsX andX +, andn 5, thenW is diffeomorphic − ≥ to the productI X . In particular,X is diffeomorphic toX +. × − −
Among other things, the h-cobordism theorem gives us the generalized Poincar´e conjecture for dimension greater than or equal tofive. The idea of the h-cobordism theorem is tofind a handle body decomposi- tion of the h-cobordism and then show the handles can all be cancelled. The difficulty with extending the h-cobordism theorem below thefifth dimension is the failure of the ‘Whitney trick.’ The Whitney trick is used to construct the handle body decomposition by resolving intersections between handle bodies by means of the embedded disks guaranteed by Whitney’s embedding theorem for dimensions greater than 5. Freedman was able to extend the h-cobordism theorem down to the fourth dimension for topological manifolds.
Theorem 2 [Fre82, Theorem 1.3] A compact, 1-connected, smooth, 5-dimensional h-cobordism(W;M,M �) (which is a product over the possibly empty boundary ∂M) is topologically a product, i.e.,W is homeomorphic toM [0, 1]. ×
Freedman did this by using Casson handles in place of the standard two- handles used in the original h-cobordism theorem. In the fourth dimension, the Whitney trick still allows us to remove an intersection, but one cannot be sure a new intersection isn’t created. The idea of a Casson handle is to iterate the process infinitely with the hope that the difficulties with creating
2 new intersections vanish in the limit. Indeed, Freedman showed this was true topologically:
Theorem 3 [Fre82, Theorem 1.1] Any Casson handle is homeomorphic as a
pair to the standard open 2-handle D2 (D 2)◦ ,∂D 2 (D 2)◦ . × × � � A smooth extension of the h-cobordism theorem to the fourth dimension was shown to be impossible by Donaldson. Thefirst step was his diagonal- izablity theorem, which put severe restrictions on the intersection form of four-manifolds supporting a smooth structure.
Theorem 4 [Don83, Theorem 1] IfX is a smooth, compact, simply-connected oriented 4-manifold with the property that the associated formQ is positive definite. ThenQ is equivalent, over the integers, to the standard diagonal form.
Then, in [Don85] and [Don87], he used a new invariant of smooth struc- tures, Donaldson polynomials, to show that the h-cobordism theorem could not extend to the fourth dimension for smooth manifolds. Namely, he showed two manifolds, which Freedman’s theorem said were homeomorphic, had dif- ferent Donaldson polynomials and were thus not diffeomorphic:
Theorem 5 [Don87, Theorem 3.24] The Dolgachev surface is not diffeomor- 2 phic to the rational surfaceCP 2#9CP .
Interestingly, the difficulties responsible for the failure of the smooth four dimensional h-cobordism theorem are, in some sense, contained in a sub-h- cobordism. In the context of four dimensional manifolds, this implies the
3 smooth structures on a given four-manifold can all be realized by means of a contractible co-dimension zero submanifold and an involution on it’s boundary.
Theorem 6 [Kir97] LetM 5 be a smoothfive-dimensional h-cobordism be- tween two simply connected, closed four-manifolds,M 0 andM 1. Then there exists a sub- h-cobordismW 5 M betweenW M andW M with the ⊂ 5 0 ⊂ 0 1 ⊂ 1 properties:
5 1.W 0 and henceW andW 1 are compact contractible manifolds, and
2.M intW 5 is a product h-cobordism, i.e.(M intW ) [0, 1]. − 0 − 0 ×
3.W 0 is diffeomorphic toW 1 by a diffeomorphism which, restricted to
∂W0 =∂W 1 , is an involution.
Definition 1.0.1 The contractible co-dimension zero submanifold with an in- volution of it’s boundary is a cork (W,τ) of a differentiable simply connected smooth four-manifoldM ifM=M W W is homeomorphic, but not − ∪ Id diffeomorphic toM � =M W W. − ∪ f The act of cutting and regluingW with the involutionτ is called a cork twist.
Corks have been found in many pairs of homeomorphic, but non-diffeomorphic, four-manifolds. Thefirst example of a cork was found by Akbulut in [Akb91]. 2 The Mazur manifold (W1,τ 1) was shown to be a cork inE (2) # CP . Sub- sequently, many more examples have been found. See for example [AY08] in which they show:
4 Theorem 7 Forn 1, ≥
(W2n 1,τ 2n 1) and(W 2n,τ 2n) are corks inW 2n. − −
The ‘Mazur’ corks, (Wn,τ n) are the focus of this thesis and are the compact four-manifolds given by the Kirby diagram infigure 1.1.τ n is given by the homeomorphism induced by swapping 0 on each component. Since the ↔· link is symmetric,τ n is an involution. 0
n n+1
Figure 1.1: Akbulut corkW n
The presence of a cork in a manifold does not, by itself guarantee the cork twist will produce an exotic copy. Smooth invariants are needed to distinguish smooth structures on four-manifolds with the same topological type. The Don- aldson polynomials, mentioned above in theorem 5, were thefirst such invari- ant. The Donaldson polynomial’s gauge theoretic approach to smooth four- manifolds was continued with the Seiberg-Witten invariant. Seiberg-Witten invariants were much easier to compute and largely superseded the use of Don- aldson polynomials. In many cases, it is possible to detect different smooth structures by using properties of the invariant with respect to some extra structure. For example, a symplectic structure on a four-manifold implies the Seiberg-Witten invariant is non-vanishing.
5 The Heegaard Floer invariant was inspired by the power of the Seiberg- Witten invariant (in fact they are conjectured to be equivalent cf. [KLT10]) and the hope to create an invariant whose computation did not rely so heavily on the fortuitous presence of some extra structure. The Heegaard Floer invariant is a diffeomorphism invariant for closed dif-
+ ferentiable four-manifolds withb 2 > 1. It is built off of the Heegaard Floer homology of three-manifolds. To a closed, oriented three-manifold, we can associate a tuple of groups :
+ HF −,HF ∞ andHF (we give a more detailed review in chapter 2). These groups are associated with a long exact sequence:
+ ... HF − HF ∞ HF ... −→ −→ −→ −→
which is abbreviated with the notationHF ◦. There is a fourth group,HF red which is defined to be either:
HF + k >>0 U k HF + ·
or
k ker U :HF − HF − k >>0. −→ � � HFred is well defined as the two groups are isomorphic. The isomorphism is given by the boundary map,∂ , of the above long exact sequence [OS04d, ∗ Lemma 4.6].
+ HFred thus gives us a way to compose maps onHF − andHF , which is needed in the definition of the four-manifold invariant.
6 LetW be cobordism between the closed oriented three-manifoldsM 0 and
c M1, andt a spin structure onW that restricts tos 0,s 1 onM 0 andM 1. In [OS06], they construct a map,
F ◦ :HF ◦ (M ,s ) HF ◦ (M ,s ) W,t 0 0 −→ 1 1
and show it is natural with respect toHF ◦ and a smooth invariant ofW and t. (A gap in the original proof of naturality wasfixed in [JT12].)
+ The Heegaard Floer invariant of a closed four-manifoldX withb 2 > 1 is defined as follows: Remove two four-balls fromX to get a cobordism X from S3 S 3. Cut X along an embedded three-manifoldN, −→ � � X=W W , 1 ∪N 2 � + + + such that bothb 2 (W1) andb 2 (W2)> 0 (recallb s (X)> 1). This condition on the Betti numbers are needed to keep the invariant independent of the cut N. The cobordism map:
3 F − :HF − S HF − (N) W1 −→ � �
factors through the inclusionHF − (N)� HF − (N) and red →
F + :HF + S3 HF + (N) W2 −→ � � factors through the projectionHF + (N) HF + (N). The isomorphism −→ red induced by∂ lets us compose these two maps. ∗
7 The closed four-manifold invariant is the coefficient of the image of the composition of these maps evaluated on the highest graded elementx ∈ 3 HF − (S ) = (−2): T −
1 + − Φ(X,t) =F W ∂ HF + F W− (x). 2 ◦ ∗| red ◦ 1 � �
(Here∂ is the isomorphism of theHF red coming from the long exact sequence). ∗ While the Seiberg-Witten invariants have been used with some success in studying corks, the hope is the Heegaard Floer invariants will facilitate subtler computations which do not rely on, for example, an associated symplectic structure. One hope would be to understand corks well enough to construct exotic pairs with their help. Toward this goal we compute the Heegaard Floer homology of the boundaries of the Mazur corks (figure 1.1). There has already been work done towards this goal. In [AD05] it was shown that the map,f onHF + induced by the boundary involution acted ∗ non-trivially on the Heegaard Floer homology offirst example of the Mazur
+ corks,HF ( ∂W1). More recently, in [AK11],f was shown to act non- − ∗ trivially on the contact invariant lying within the Floer homology. Last year an algorithm to compute the Heegaard Floer homologies of the Mazur corks were given in [AK12]. They realized that, if gradings are disre- garded, the homologies were found to be isomorphic to a class of plumbed three-manifolds which could be computed algorithmically using Nemethi’s method. This thesis gives a closed form solution for this class using a dif- ferent approach. This approach allows the calculation of absolute gradings, is applicable to all corks presented as two bridge links, and seems to have a
8 natural path to the computation of the four-manifold invariant. Also of related interest, the Instanton Floer homology of this family of corks was recently computed as well in [Har13].
+ Remark 1.0.2 For technical reasons (the preference ofHF overHF −) neg- atively oriented corks, W , are studied. − n
The Kirby diagram of the three-manifold bounding W is given by re- − n placing the 1-handle by a zero framed two handle: 0
n (n + 1) 0 − −
Figure 1.2: Oppositely oriented Akbulut cork boundary ∂W − n
In section 3.1 the calculation is reduced from, ∂W (a three-manifold − n n 3 n given by surgery on a link) toY = (S(+1)(K ))1 (a three-manifold given by surgery on a knot). A Heegaard diagram for this knot is found in chapter 3, section 3.2 and the chain complex in section 3.3. Finally, in chapter 4 we use the integer surgery formula from [OS08] to compute the Floer homology of their +1 surgeries,HF +(Y n) and thusHF + ( ∂W ). − n In chapter 5 we discuss further directions for this approach and sketch the first calculation for another class popular class of corks. The main result is:
9 Theorem 1.0.3
2 + + F[U] HF ( ∂Wn) = 0 m(s,0) − T ⊕ U [U] 2 2 n
Where sum is taken overU-modules distinguished by the exponent,m(s, i), of the quotient:
n i (s i)+1 min (0,(i s)) −| |− − ifn i (s i) is odd m(s, i) = − − 2 −| |− − n i (s i) min (0,(i s)) −| |− − ifn i (s i) is even − − 2 −| |− − (1.1) where, i , s 10 Chapter 2 Background on Heegaard Floer homology 2.0.1 Heegaard Floer homology Remark 2.0.4 LetF denoteZ/2Z. We will use the Floer invariant with coefficients inF. Notation 2.0.5 Recall the following commonly usedU-modules: 1 ∞ will denote theF[U] moduleF[U, U − ] T − will denote theF[U] moduleF[U] with maximum gradingd. T(d) 1 + will denote the [U] module F[U,U − ] with minimum gradingd. (d) F U F[U] T · Heegaard Floer homology is an invariant of a closed oriented three-manifold. A closed orientable three-manifold can be split into two handle bodies and a gluing mapφ: U U . α ∪φ β 11 This splitting can be encoded by a genusg surfaceΣ and twog-tuples of circles -α,β. The circles inα (andβ) are linearly independent in homology. Attaching two-handles to theα,β circles givesU α,U β respectively. If we distinguish a pointz on a Heegaard diagram we call it a pointed Heegaard diagram: (Σ,α,β,z). For technical reasons related to the Floer construction, we work with pointed Heegaard diagrams. From the pointed Heegaard diagram we form theg-fold symmetric product Symg (Σ). There are some technical conditions having to do with the almost complex structure we put on it, but it is shown that for a generic almost complex structure the invariant is independent of the choice of almost complex structure. Within Symg (Σ) we have two totally real tori: g Tα =α 1 ... α g Sym (Σ) × × ⊂ g Tβ =β 1 ... β g Sym (Σ). × × ⊂ The chain complexes are generated overF by the intersection points of Tα T β. ∩ Forx,y T α T β, a Whitney disk,φ π 2 (x,y), is an map of the unit ∈ ∩ ∈ diskD 2 C 2 into Symg (Σ) such that: ⊂ 1. i x − �→ 2.i y �→ 12 2 3.∂D x >0 T α ∩{ } �→ 2 4.∂D x <0 T β. ∩{ } �→ We study the moduli space, (φ), of psuedo-holomorphic embeddingsφ M with Maslov index equal to one,µ(φ) = 1. The Maslov index is the expected dimension of the moduli space. In general the Maslov index is hard to compute, but with the help of [Lip06] it can be computed from the diagram combinatorially. By restricting to the moduli space with Maslov index one and the help (φ) of Gromov’s compactness theorem, the quotient space M is shown to be R � a compact zero dimensional space. Thus the count of the holomorphic disks (φ) # M isfinite. R � Definition 2.0.6CF ∞(Σ,α,β,z) is the module freely generated overF by generators[x, i] (T α T β) F, endowed with a differential ∈ ∩ × ∂[x, i] = # (φ)[y,i n (φ)]. M − z y Tα Tβ ∈�∩ φ π2(x�,y) µ(φ)=1 { ∈ } � � � g 1 g Wheren (φ) is the algebraic intersection ofz Sym − withφ in Sym (Σ). z × In general, the differential is hard to compute asfinding the holomorphic disks require the solution of partial differential equations. In our case, we will be able to restrict to a special diagram, namely a surface with genus one, in which these disks can be computed combinatorially. Definition 2.0.7CF −(Σ,α,β,z) is the subcomplex withi<0: CF − =CF ∞ i <0 . { } 13 CF + is the quotient: CF CF + = ∞ . CF − CF(Σ,α,β,z) is the subcomplex ofCF +: � CF i 0 CF= { ≤ } . CF i <0 { } � These complexes are modules over the polynomial algebraF[U], where U [x, i] = [x, i 1]. · − We have inducedF[U]-actions on their homology groupsHF ∞(Y),HF −(Y), andHF +(Y). The complexes have a relativeZ-grading: gr(x,y)=µ(φ) 2n (φ). − z whereφ π (x,y). ∈ 2 In our case ofb 1(Y ) = 0 this can be lifted to an absoluteQ-grading. [OS03] + Theorem 8 [OS04d, Theorem 1.1] HF,HF −,HF ,HF ∞ thought of as modules overF[U] are topological invariants� ofY. c Whenb 1(Y)> 0 there is a mappings:T α T β spin (Y ). It can be ∩ −→ shown that the differential,∂, respects the partition, defined bys, ofCF ◦ by c spin structures. Thus the chain complexesCF ◦ and their homologies split into subcomplexes indexed by spinc structure. CF ◦ (Y)= CF ◦ (Y;s) s � 14 Note: the boundaries of corks are integer homology spheres, sob 1(Y)=0 and we can ignore issues with spinc structures. See [OS04d] and [OS04c] for details. 2.0.2 Knot Floer homology An invariant of a knotK inS 3 is constructed by defining afiltration on the Heegaard Floer chain complexes. In general, the invariant is defined for a null-homologous knot in a closed, oriented three-manifold, but only the case ofS 3 is used here. LetK be an oriented knot in an integer homology sphereY and let (Σ,α,β ), whereα= α , ...,α ,β = β , ...,β , be a Heegaard diagram 0 { 1 g} 0 { 2 g} for the knot complement. Then we will say (Σ,α,β µ), whereµ is the meridian ofK, is a Heegaard 0 ∪ diagram for (Y,K). Marking a pointp onµ, lets us definez andw. For a small enough neighborhoodN ofp, there are two conected components inN µ. Marka − z andw in each of the connected components. This gives us a doubly-pointed Heegaard diagram for a knot (Y,K), (Σ,α,β,z,w). The choice of which H component to placez andw in corresponds to the two possible orientations of the knot (so switchingz anw corresponds to switching the orientation of the knot). Seefigure 2.1 for an example whenY=S 3 andK is the right handed trefoil. Definition 2.0.8 Because the coefficients are inF=Z/2Z we can abuse notation and usex T α T β to also refer to the algebraic elements. ∈ ∩ 15 z w Figure 2.1: Doubly pointed Heegaard diagram for a right handed trefoil. Let be theZ Z bigraded chain complex generated overF byT α T β R ⊕ ∩ with differential, ∂x= # (φ)y. (2.1) M y Tα Tβ ∈�∩ φ π2(x�,y) µ(φ)=1 { ∈ } � � 3 � CFK∞ (S ,K) is then theF[U] module 1 F[U, U − ] . ⊗R We define afiltration function :CFK ∞ Z Z by demanding F −→ ⊕ 1. Forx,y , (x,y)=(n (φ), n (φ)), whereφ π (x,y). ∈R F z w ∈ 2 2. If we placex in the plane at coordinates (x), is symmetric ∈R F R across the liney=x. 3. (x,U x) = (1, 1). F ⊗ induces aZ Zfiltration onCFK ∞ by: (x) (y) if and only if F × F ≤F both the coordinates of (x) are less than are equal to the coordinates of (y). F F 16 Projection of thefiltration to the x-coordinate will be denoted by: (x,y). F F 1 Projection of thefiltration to the y-coordinate will be denoted by: (x,y). F F 2 This is equivalent to thefiltration from [OS04a, pg. 13], thefiltration will be encoded by placing the elements of the complex on aZ Z lattice at coordinates × given by their evaluation by . The differentials will represented by arrows. F Seefigure 2.2 for an example whenK is the right handed trefoil. Remark 2.0.9 If the Heegaard diagram for a knot has genus one, even the differentials can be computed combinatorially. This will be elaborated on in subsection 3.2. Remark 2.0.10 The initial definition ofCFK ∞ was given in the following equivalent form: 3 LetCFK ∞ (S ,K) be the chain complex of theF[U] module on the free abelian group generated by the triples[x, i, j],x T α T β and i, j Z with ∈ ∩ ∈ U-action: U [x, i, j]=[x, i 1, j 1], · − − with differential, ∂[x, i, j]= # (φ)[y,i n (φ), i n (φ)]. (2.2) M − z − w y Tα Tβ φ π2(x,y) µ(φ)=1 ∈�∩ { ∈ �| } � s(x) + (i j)PD[µ] =t . − Heres corresponds to the spinc structures of the knot complement. Theorem 2.0.11 [OS04a, Theorem 3.1] Let(S 3,K) be an oriented knot. 17 i=0 i=0 j=1 j=0 j=0 j= 1 − 3 3 Figure 2.2: The chain complexCFK ∞ (S ,K) (left) and j CFK�(S , K, j) (right) for right handed trefoil. � 18 3 Then thefiltered chain homotopy type of the chain complexCFK ∞ (S ,K) is a topological invariant of the oriented knotK. Just as with the three-manifold invariant, the homology of the infinity variant is not particularly interesting. Knot Floer homology is generally used to refer to the hat variant, HFK�. Although the homology ofCFK ∞(Y,K) is generally not interesting, its chain homotopy type carries a lot of information. In particular it can be used to compute the Floer homology of knot surgeries alongK (section 4). See [OS04a] for more details on the construction ofCFK ∞ and the com- + putation ofHF (Yn) for sufficiently large n . For the case of alln Z see | | ∈ [OS08]. 2.0.3 Integer Surgery 3 Given the chain homotopy type ofCFK ∞ (S ,K), [OS08] describes how to calculate the Floer homology of an integer surgery along (S3,K). We review the details for the case of +1 surgery. Theorem 2.0.12 [OS08, Theorem 4.1] LetY be an integral homology three- sphere. Then the homology of the mapping coneX +(1) of + + + D1 :A B −→ + is isomorphic toHF (Y1(K)). (The definition of the mapping cone follows below 2.0.14). 19 Remark 2.0.13 The above theorem is abridged. The details about the grading shift and spinc structures are skipped as the surgery coefficient of +1 makes the grading shift zero and the spinc structures are trivial on an integer homology sphere. Definition 2.0.14 Let: m m m = U x CFK ∞ µ(U x)<0 and (U x) m m C i <0 = U x CFK ∞ µ(U x)<0 . { } { ⊗ ∈ | ⊗ } Define the following quotients: + CFK∞ As = Cs CFK B+ = ∞ s C i <0 { } and maps between them: v+ :A + B + s s −→ s and h+ :A + B + . s s −→ s+1 The mapv + is projection ontoB +.h + is projection ontoC j s , which is s s s { ≥ } + s identified withB s+1 by multiplying byU , and then using the chain homotopy equivalence fromC j 0 toC i 0 . [OS08, pg. 2] { ≥ } { ≥ } + Note thatB s is not dependent on the indexs. The index is relevant to the 20 absolute grading defined below. X+(1) is then the mapping cone of + + + D1 :A B −→ where + + A = As s � + + B = Bs s � + andD 1 is the direct sum of the chain maps: D+ = v+ h +. 1 s ⊕ s s � + + The quotients ofCFK ∞,A s andB s inherit its absolute grading. We shift this old grading by: d d+s(s 1) + 1. (2.3) �→ − The integer surgeries formula says the resulting grading on the mapping cone X+(1) gives us the absolute grading of the surgered manifold. + + + + Remark 2.0.15 The1 inD 1 ,X (1),A s , andB s denotes we are doing +1 surgery. It is included to keep notation consistent with [OS08]. The computation of the mapping cone is simplified greatly by truncating the infinite sumsA +,B + to thefinite ones: + + A (b) = As b s b −�≤ ≤ 21 + + B (b) = Bs . 1 b s b −�≤ ≤ The truncated mapping cone is denoted: X+(1;b). The following lemma tells us we can always use the truncated mapping cone: Lemma 2.0.16 [OS08, Lemma 4.3] Forb sufficiently large,X +(1;b) is quasi- + + + isomorphic toX (1). In particular, they are quasi-isomorphic ifv s andh s are isomorphisms for s >b. | | 2.1 Lifted Heegaard Diagram When the genus of the Heegaard diagram is one, we can computeCFK ∞ combinatorially. This is done by looking at the lift of the Heegaard diagram, . H This is very straight forward. The one subtlety is determining the combi- natorial conditions for a Whitney disk to have a holomorphic representative. The condition for counting the holomorphic disks is given in the definition for the differential below. For the proof that this givesCFK ∞, see [OS04a]. Definition 2.1.1 The cover (K)of a genus one Heegaard Diagram (K): H H 1. LetD be the unit square [0, 1] [0, 1] marked at two points: ( 1 , 1 ),( 2 , 1 ) × { 3 2 3 2 } 2. Let R2 be a tiling ofR 2 byD. α, β be embeddings ofR. � � 22 3. 1 1 z= (Z+ ,Z+ { 3 2} � 4. 2 1 w= (Z+ ,Z+ { 3 2} � 5. Then denote = R2, α,β, w, z . H � � 6. If the quotient by integer� translation� � � is the Heegaard Diagram for a knot, H = (K), Z Z ∼ H ⊕ we write (K)= . H H Definition 2.1.2 The knot Floer complexCFK ∞ (K) of the cover (K): H H � � 1. The generators: Let s : α β spin c (S3 K) be a map to the spin c structures on • ∩ → − the knot complement defined by x s(x) =s (π(x)). We use this � � �→ to define the generators of the complex: � � � CFK ∞( (K)) is generated by [ x, i, j] x α β, i, j Z, s (x)+ • H { | ∈ ∩ ∈ (i j)=0 − } � � � � � 2. Thefiltration: Define aZ Zfiltration by • ⊕ [x, i, j]=(i, j) F � 23 and the ordering (i, j)<(i �, j�) i 3. The differential: 2 For x, y α β letπ 2(x, y)= φ:D R φ is an embedding, φ ∂ • ∈ ∩ { −→ | | D ⊂ α β, φ( i) = x, φ(i) = y ∪� � −� � � � } � � � For �x �α β, de�fine the local multiplicity of a disk φ at x as • � ∈ ∩ � � follows. Given φ π 2(x, y), there are four regions inC α β � � � ∈ −� −� which contain x as a corner point. Choosing interior pointsz , ..., z � � � �1 �4 in these four regions, define � n (φ) +n (φ) +n (φ) +n (φ) n (φ) = z1 z2 z3 z4 . (2.4) x 4 � � � � � � We define ny(φ) similarly, only now choosing the points in the four regions which� have� y as a corner point. let the index of a disk φ π (x, y) be given by: • � ∈ 2 � � � µ(φ) = 2(nx(φ) + ny(φ)). � � � � � The differential is then given by summing over disks with index one: • ∂x= [y,i n , j n ]. − w(φ) − z(φ) y φ π2(x,y) µ(φ)=1 � { ∈ �| } � � � � � � � � � � � In [OS08] it is proven that this this is indeed a chain complex and that it is 24 Figure 2.3: of the unknot. H chain homotopic toCFK ∞ ( (K)). H Proposition 2.1.3CFK ∞ (K) is a chain complex and we have afiltered H chain homotopy equivalence:� � CFK∞ (K) =CFK ∞ ( (K)). H H � � Proof. This is shown in [OS04a, Lemma 6.5, Proposition 6.4] Example 2.1.4 Let α R 2 bey+x=0 and Im( β) bey=0. Then (K) is ⊂ H a lifted diagram for the unknot (Figure 2.3). � � 25 Chapter 3 Chain Complex 3.1 Reduction to Knot Surgery ∂W is a surgery on a two bridge link. Instead of calculating the Floer − n homology of ∂W , we focus on an associated three-manifoldY n, which is − n given by surgery on a knot,K n. The topology ofY n is similar enough to ∂W that they have isomorphic Floer homology (proposition 3.1.2). Since − n Y n is given by a knot surgery, we can computeHF +(Y n) with the formula for integer surgery discovered in [OS08]. Definition 3.1.1 Let(S 3,Ln) be the underlying link in the Kirby diagram for ∂W . Since each of the components ofL n is individually unknotted, 1 − n ± surgery gives backS 3. We define a knot(S 3,Kn) by blowing down one of the components by 1 surgery (Figure 3.1). (We don’t need to specify the − component we are blowing down asL n is symmetric). LetY n be the three-manifold given by +1 surgery onK n. NoteY n is also given by the kirby diagramL n with framings (0, 1). − 26 1 − n n − − −→ (n + 1) (n + 1) − − Figure 3.1: Changing a link into a knot inS 3. Proposition 3.1.2 There is a gradedU-isomorphism between the Floer Ho- mologies of ∂W and a homology sphere given by +1 surgery on the knot in − 1 S3. HF +( ∂W ) = HF +(Y n) − n ∼ Proof.Y n fits into +1 surgery exact sequence with ∂W (Figure 3.2). − n Since ∂W anY n are integral homology three-spheres, they have a well − n definedd-invariant [OS03] as well as the splittingHF + = + HF . ∼ T(d) ⊕ red Thed-invariant is zero since they bound contractible four-manifolds (for Y n this can also be seen directly from the computations below) so they can + only differ onHF red. The claim will follow if we show∂ :HF red( ∂Wn) ∗ − → + n HFred(Y ) is aU-module isomorphism. The surgery exact sequence for +1 surgery on a homology sphere gives us: ∂ f g ∂ ∗ + ∗ + + ∗ + n ∗ ... HF ( ∂Wn) 1 1 HF (Y ) ... (3.1) 1 2 −2 1 −→ − −→−2 T ⊕T −→−2 −→ 27 0 0 0 n (n + 1) n (n + 1) n (n + 1) − − 0−→ − − −→0 − − 0 0 Handle Cancellation 0 0 0 n (n + 1) 1 − − − Handle Cancellation 0 1 Figure 3.2: From left to right we have ∂W ,S 1 S 2, and thenY n. − n × Since the Floer homology ofS 1 S 2 is supported entirely in its torsion × c 1 spin structure, the cobordism mapsf andg are homogeneous of degree − . ∗ ∗ 2 + + Since there is only a single inHF ( ∂Wn),f cannot be surjective T − ∗ and thus the cokernel off will contain a subcomplex isomorphic to a +. ∗ T Thus the image ofg must contain at least one subcomplex isomorphic to +. ∗ T This implies that the coker(g ) and thus the image of∂ isfinitely generated. ∗ ∗ + + Going back tof , this showsf is non-zero on 0 HF ( ∂Wn). ∗ ∗ T ⊂ − + + Letx 0 HF ( ∂Wn) such thatf (x)= 0. Then letk be such that ∈T ⊂ − ∗ � U k f (x)= 0 butU k+1f (x) = 0. Sincef isU-equivariant,U k x= 0. Since · ∗ � ∗ ∗ · � + + k 1 1 f maps into 1 1 , the degree ofU f (x) is either 2 or −2 . ∗ T 2 ⊕T −2 · ∗ 1 k + f drops degree by , so deg U x = 1 or 0. Since 0 takes only even val- ∗ 2 · T k + + ues, deg U x = 0. This shows� that�f is injective on 0 HF ( ∂Wn). · ∗ T ⊂ − � � 28 + + Sincef is injective on 0 HF ( ∂Wn), we know that the ker(g ) 1 ∗ T ⊂ − ∗ ⊂T 2 + + and thus Im(g ) 0 HF ( ∂Wn). Letx be the element with minimum ∗ ⊂T ⊂ − degree in coker(f ). If deg(x)> 1 , then deg(g (x))> 0. Since thed-invariant ∗ 2 ∗ ofHF + (Y n) = 0,U g (x)= 0. This gives the contradiction: · ∗ � 0 =g (0) =g (U x) =U g (x)= 0. ∗ ∗ · · ∗ � 1 + + 1 2 Thus deg(x)= 2 andg is injective on 1 HF (S S ). So coker(g ) = ∗ T 2 ⊂ × ∗ n n HFred (Y ) and ker(f ) =HF red (Y ). i.e.∂ is an isomorphism between the ∗ ∗ HFred groups of the two manifolds. Remark 3.1.3 In [AK12, Proposition 1.2] a more general but weaker, in the context of gradings, statement is proven. They show that there is an isomor- phismHF red(Y) andHF red(Yp) for any p. However, the isomorphism only preserves gradings for the casep=1 so the Floer groups of the plumbed mani- folds in their paper while isomorphic to ∂W for alln, only give information − n about the absolute gradings forn=1. 3.2 Heegaard Diagram [OS04a] shows how to construct (K) whenK is the result of blowing down H one of the components of a two bridge link. The following is a formalization of their algorithm: +1 i>0 Notation 3.2.1 Let sgn(i) = 1 i<0 − 29 γ2 γ2 Longitude z w A A z w Longitude Longitude A A z w z w γ γ3 3 Figure 3.3: Comparison of the action on the torus: (ψ, (K)) (left side) To H the action on the fundamental domain of the cover, ψ, (K) (right side). H 2 The rotatedA’s denote one handles. The top row corresponds� � toτ 2 . The bottom row toτ 3. Definition 3.2.2 Define two actions on these lifted Heegaard Diagrams (see Figure 3.3): Rotation byπ: On each image ofD (K), perform a half Dehn twist • ⊂ H along a circle containing both marked points. Finger move of lengthk: (K) along the vertical lines x=n+ 2 n • H { 3 | ∈ Z . Start at an intersection point of β and one of the vertical lines } γ x=n+ 2 n Z , isotope β upward (or downward, if sgn(n) = 1) ∈{ 3 | ∈ } � − alongγ such that β crosses exactly thefirst n marked points ( z or w) � | | onγ. � � � (K) can be obtained by repeated applications of the above two moves: H 30 Lemma 3.2.3 LetK be obtained by blowing down a component of a two bridge linkL. Letσ be an element of the braid group on four strands whose plat closure is the two bridge link, σ=L. Factorσ into the generating set τ , whereτ gives a positive crossing { i}i=1,2,3 i to the strandsi andi+1. If we blowdown with a 1 surgery, start with the lifted Heegaard Diagram − for an unknot, (K), given in example 2.1.4. For a +1 framed blowdown, let H Im(β)= y x=0 instead. { − } Proceeding� from left to right along the braid word obtained by factoring σ, we apply vertical shifts or rotations according to the following identifications: 2n 2n τ − orτ finger move of length n or+ n. • 2 2 ↔ − 1 τ − orτ rotation byπ or π. • 3 3 ↔ − Proof. [OS04a, Proposition 6.3] proves this for the (K). In particular, they H show, forγ 1,2 in Figure 3.3, that: 2n 2n τ − =τ Full Dehn twist along the circleγ • 1 2 ↔ 1 τ Half Dehn twist along the circleγ • 3 ↔ 3 To see it for , we check that it commutes with the covering mapπ: H ψ (K) (K) H −→ H ↓�↓ ψ (K) (K) H −→H 31 Since the unit squareD is a fundamental domain for R2, ψ is determined by its restriction toD. The support of the half Dehn twist,ψ 3 induced byτ 3 doesn’t intersect the boundary of the fundamental domainD so the lift ψ is still a half 3|D Dehn twist. (Bottom right of Figure 3.3). The full Dehn twist,ψ 1 induced by 2n 2n τ − τ unwinds to become a linear push on the cover asψ ’s support is a 1 ↔ 2 1 2 neighborhood of the circleγ 3, which lifts toR R (Top right of Figure 3.3). ⊂ 3 n Corollary 3.2.4CFK ∞ (S ,K ) is chain homotopic to the combinatorial chain complex associated to the Heegaard Diagram (Kn) (Figure 3.4). H n 2n 2(n+1) Proof.L factors into the braid wordτ − τ τ − . Following lemma 1 ◦ 2 ◦ 1 3.2.3, we begin with Figure 2.3 and apply the sequence of moves: 2n τ − corresponds to making afinger of length n • 1 − Theτ is a rotation byπ which transforms the single longfinger, created • 2 by the previous move, inton pairs offingers. 2(n+1) τ − , afinger move of length n 1, stretches the 2nfingers to a • 1 − − length ofn + 1. (see Figure 3.5). 32 R C B Figure 3.4: (Kn). See Figure 3.14 for an example whenn = 2. The labelB is used for pointsH that will function as boundaries in the homology computation. Likewise,R for relations andC for generators 33 2n 2(n+1) Figure 3.5:τ − τ τ − . 1 → 2 → 1 3.3 Chain Complex Recall that 1 CFK∞ = F[U, U − ] . ∼ ⊗R We prove several propositions about and combine them at the end of the R section to getCFK ∞ in Theorem 3.3.14. Abusing notation, the elements of α β as well as the elements they corre- ∩ spond to in chain complex will be denoted by the same symbol. This does R � � 34 not cause any difficulties as the coefficients are inF=Z/2Z. Notation 3.3.1 Let A denote the free module overA with coefficients inF � � Infigure 3.4, α β was split into three sets:R,C andB foreshadowing ∩ their roles as the Relations, Cycles, and Boundaries of . A precise definition � � R ofR,C andB: Definition 3.3.2 Partition α β into three classesR,C, andB: ∩ � 1. An element x α β �has four regions, in (K n) α β , that have ∈ ∩ H − ∪ x as a corner. If two or more of these regions are not� compact,� � � � � � � x R ∈ � EnumerateR: R= r (n 1), ..., r0, ..., r(n 1) { − − − } 2. Put an orientation on α andβ that makes at least one intersection point r R α β have a positive sign. (In fact allr R have the same ∈ ⊂ ∩ � � ∈ sign). � � x B(respectivelyC) if x has positive (negative) sign with the orienta- ∈ tion. � � 3. LetS i be the subset of Tα Tβ which lie on α betweenri andr i+sgn(i), ∩ � � i= 0. Fori=0, LetS 0 be the subset betweenr 1 andr 0 andS 0+ be � �− � � − the subset betweenr 0 andr 1. 35 1 The awkward notation ofS 0± is used to highlight the identification of ther i withS i. DenoteC =S C andB =S B i i ∩ i i ∩ R C B r 1 r0 r1 − S 1 S0 S0+ S1 − − Figure 3.6: A classification of the intersection points on α β ∩ � � Lemma 3.3.3 µ(b) + 1 =µ(g)=µ(r) 1 − b B,g C,r R ∀ ∈ ∈ ∈ Proof. Within a particularS i the alternating upward and downwardfingers show the alternatingg andb points to be split into two Maslov degrees. The two compact domains (the other two are always non compact) with a corner at a givenr i can be completed by addingfingers to form a Maslov index one bigon to the two adjacent (along α) points inR (Figure 3.12). This shows theg, and thus also theb of differentS andr , to have (respectively) � i i the same relative Maslov degree. The difficult part in computations ofCFK ∞ is generally the differential. For ‘lifted’ diagrams , [OS04a, section 6.2]. shows the differentials can be H computed combinatorially. It particular, if φ π (x, y) withµ( φ) = 1: ∈ 2 � � � � 36 φ is unique and a bigon. (Follows from genus being one.) • #� (φ) = 1 ( φ) 0 [OS04a, proposition 6.4] • M ⇔D ≥ n �= 1� = n [OS04a,� proposition 6.5] • x 4 y � � Proposition 3.3.4 The differential respects theS i: 1. Forr R,∂r S S i ∈ i ∈ i+sgn(i) ∪ i � � 2. If x S ,∂ x S ∈� i� ∈� i� � � Proof. This follows from the inability of positive bigons to cross the points r (n 1), ..., r0, ..., r(n 1) . Suppose a bigon has anr i on its boundary,r i ∂ φ. { − − − } ∈ Thenr must be a corner of φ. i.e.r = x or y where φ π (x, y) i i ∈ 2 � Inspecting the diagram, we see that for everyr R there are four domains � � � i ∈ � � � which haver as a corner point (Figure 3.7). Sincer ∂ φ at least one of these i i ∈ must have a non-zero positive coefficient in φ . Only the two of these are D � compact and can have a non-zero coefficient in� � φ . Since the two compact D� � � domains are oppositely oriented with respect to the� α and β curves, their coefficients in φ must have opposite signs. As we are only interested in D � � φ 0 exactly� one� of the domains has a non-zero positive coefficient. D ≥ � � � � Definition 3.3.5 The two compact domains of Figure 3.7 with corners at ri, will be called gloves. The name was chosen tofit in with the following definitions offingers domains and palm domains. Lemma 3.3.6 For a givenS i, there are only two domains out of which all of the other φ withµ φ = 1 are composed: D � � � � � � 37 ri Figure 3.7: Four domains with a corner atr i. The two compact domains are darkly shaded. The two non-compact domains are lightly shaded. R C B Figure 3.8: Upward pointingfingers on the left and downward pointingfingers on the right 38 1. Fingers with a corners at adjacentc C andb B (Figure 3.8). ∈ ∈ 2. Palms Bigons with one corner inR and the other corner at thec C ∈ which precedes the next element ofR (Figure 3.9), R C B Figure 3.9: Upward pointing Palms on the left and downward pointing Palms on the right Definition 3.3.7 The union of a all of the upwardfingers and the downward palm will be called, of course, an upward hand. Likewise for a downward hand. Proof. (lemma 3.3.6) It suffices to prove this forfingers and gloves instead - By adding and removingfingers, we can construct palms from gloves and vice versa. Thus together with thefinger domains, they generate the same set of domains φ : D � � � palms, fingers gloves, fingers � � ⇔ � � Clearly if φ gloves, fingers then φ S . (i.e. x, y S ). On D ∈ � � D ∈ i ∈ i the other hand,� � if φ is compact but/ gloves,� � fingers then it must pass � D ∈� � � � � contain one of the gloves� � of proposition 3.3.6. Otherwise φ would have � D � � � 39 more than one connected component and have a Maslov index greater than one. However proposition 3.3.6 concluded that this isn’t possible. We introduce some simple chain complexes commonly referred to as rails. [OS04b]. In [OS04b, Theorem 12.1], they showed CFL� could be given a sim- ple description as a direct sum of rails and some other simple complexes. Unfortunately,CFK ∞ does not have as simple of a description, but the rails nonetheless allow a construction of a tractable complex. Definition 3.3.8 A rail is an indecomposable bigraded chain complex with elements x 2i, y2i 1 a 2i,2i 1 b, a, b Z, with differentials: { ± } ≤ ± ≤ ∈ ∂y2i 1 = 0 ± and ∂x2i =y 2i 1 +y 2i+1 − (If eithery 2i 1 ory 2i+1 don’t exist replace them with0 in the above differen- − tial). The relativefiltration levels betweenx 2i andy 2i 1,y 2i+1 are given by: − (x 2i, y2i 1) = (1, 0) (x 2i, y2i+1) = (0, 1). F − F The length of the rail is max(# x 2i ,# y 2i 1 ). { } { ± } Remark 3.3.9 A rail has homologyF if# x 2i, y2i 1 a 2i,2i 1 b is odd and is { ± } ≤ ± ≤ acyclic otherwise. See the right hand side offigure 3.10 for an example. As afirst application of Lemma 3.3.6, we show there is a rail induced by eachS i. 40 Proposition 3.3.10 c, b S generates a subcomplex isomorophic to a rail � ∈ i� (seefigure 3.10). Proof. (proposition 3.3.10) Lemma 3.3.6, immediately gives us: R C B Figure 3.10: A rail of length two. The darkly shaded regions of the Heegaard Diagram are thefingers that induce the differentials in the chain complex on the right. The arrows inside thefingers are a visual aid denoting the direction of the differential. ∂(S ) S . i ⊂ i ThusS i is a subcomplex. Once again, by lemma 3.3.6, we can get away with only looking at linear combinations offingers and palms. Since the palms have a corner inR, andS i was explicitly defined not to contain them there are no differentials in the subcomplexS induced by a φ containing a palm. This i D � � leaves only thefingers, which all contribute differentials.� (Figure 3.10) Taken by itself, the homologyH (Si,∂) of a single rail isF and is generated ∗ by the cycle c C c. The missing elements and differentials, (R,∂), have the ∈ i effect of equating� these cycles. 41 Proposition 3.3.11 ∂ri = c c C C ∈ i∪�i+sgn(i) If eitherS i orS i+sgn(i) does not exist, take it to be∅. Proof. The argument is very similar to the previous proof. Corollary 3.3 together with the Maslov index, lemma 3.3.3 show: ∂r C C . i ⊂ i ∪ i+sgn(i) � � We can make this an equality by explicitly constructing bigons. Start with a palm and successively add pairs of upward and downwardfingers (Figure 3.11). The pairs must be adjacent for the Maslov count to be unchanged. Lemma 3.3.6 tells us that all of the domains contributing differentials to the subcomplex S are within the vertical strip bounded by the lines perpen- � i� dicular tor i andr i+1 (Figure 3.14). Lemma 3.3.12 There are two gloves having a givenS i as a border. The upward pointing hand φ has: � nz(φ) = 0 n � (φ) = max(0, i) w � − � � If φ is pointing downward: � nz(φ) = max(0, i) nw� (�φ) = 0 � � 42 R C B Figure 3.11: Compare with Figure 3.10 Proof. By inspection offigure 3.4. Counting the upward pointing hands: 1. The upward hands lying above S haven (φ) = 0 andn (φ) = 0. { i}i>0 z w � � 2. The upward hands lying above S haven (φ�) = 0 butn (φ�) = i { i}i<0 z w | | � � Analogously for downward pointing hands: � � 1. The downward hands lying below S haven (φ) = 0 andn (φ) = 0. { i}i<0 z w � � 2. The downward hands lying below S haven �(φ) = 0 butn (�φ) =i { i}i>0 w z � � � � 43 Figure 3.12: The shaded region on the left is an upward hand, the shaded region on the right is a downward hand. Proposition 3.3.13 (r , r ) = (i, i + 1) F i+1 i Proof. Since the diagram is invariant when rotated byπ and switching z w, ↔ it suffices to prove the result for r i 0 . { i| ≥ } � � - 3 Figure 3.13: A vertical strip in the diagram forY betweenr 1 andr 2. The left domain is φ. The middle, ψi The right is their difference φi ψi. The lighter shaded region is negative. − � � � � The relativefiltration levels betweenr i andr i+1 is calculated with the domain given by (Figure 3.13): φ ψ i − i � � 44 By inspection, all buti of thefingers cancel each other out. Giving us: n =i. Lemma 3.3.12 tells us the upper glove contributes nothing and the |{ z}| lower� has n =i + 1. |{ w}| Thus we have:� (r , r ) = i F1 i i+1 − (r , r ) = (i + 1) F2 i i+1 − 1(ri, ri 1) = (i + 1) F − − 2(ri, ri 1) = i F − − R C B Figure 3.14: The alternating stripes distinguish four hands for the diagram 2 2 (K ). Seefigure 3.14 for its associated chain complexCFK ∞ (K ) . H H � � We summarize this sections calculations and giveCFK ∞ an absolute grad- 45 ing: n Theorem 3.3.14 Thefiltered chain complexCFK ∞ (K ) is given by H 1 � � F[U, U − ] where is a chain group generated overF by the setR C B. ⊗R R ∪ ∪ SinceF=Z/2Z we can and will abuse notation by identifying the intersec- tion pointsR C B with the algebraic elements they induce in . is then ∪ ∪ R R just the chain complex generated from the intersection points in the diagram with differentials corresponding to the Whitney disks. Seefigure 3.15 for an example. 1. The differentials are ∂r i = c C C c • ∈ i∪ i+sgn(i) Fori=0�,∂r = c 0 c C C + ∈ 0− ∪ 0 � If eitherS i orS i+sgn(i) does not exist, take it to be∅. The differentials of thec andb are best understood graphically. They • form a rail 3.3.8 withc denoting the higher grading generators. 2. Thefiltration levels are: forr R: • i ∈ i2 i i2 +i (r ) = − , F i 2 2 � � The c, b S , form a rail of lengthm=n i withfiltration level • ∈ i − i2 i i2+i i2 i i2+i stretching from( − m, ) to( − , m) 2 − 2 2 2 − 3. The homology is generated by any of the homologous cycles:[ c C c] ∈ i � 46 for any (n 1) i n 1. − − ≤ ≤ − F=H (Si) = c ∗ �c C � �∈ i R C B 2 Figure 3.15: for the complexCFK ∞ (K ) . The differentials fromr 1 R H ± to the rails centered onr 0 are drawn curved� and� with a light hue. This is only to add clarity to a clutteredfigure and doesn’t have any mathematical significance. Seefigure 3.14 for the lifted Heegaard Diagram this complex is derived from. Proof. (Theorem 3.3.14) The claim about the differentials is just a restate- ment of propositions 3.3.11 and 3.3.10. The absolute grading onCFK ∞ is determined by looking at CFK� ([OS04a]). SinceH CF(S 3) =F, there is a unique class of cycles generating the ho- ∗ � � mology. Assigning� Maslov grading 0 to the image of the inclusion of these 47 cycles intoCFK ∞, and then extending linearly gives the absolute Maslov grading. Recall CF=CFK ∞ i=0 and so (x,y)=0, x,y CF . Thus we { } F 1 ∀ ∈ ignore all but� the completely vertical differentials in analyzing CF� . We show that CF is a direct sum of two simple subcomplexes (figure 3.16).� � Figure 3.16: A cyclic and acyclic complex For afixedm,U m r CF,∂(U m r ) now consists of only thec ⊗ i ∈ ⊗ i ∈ U m S with (U m r ,U m c) = 0. ⊗ i F 1 ⊗ i �⊗ 1. Ifi= 0,∂U m r consists of exactly one element, theU m c at the � ⊗ i ⊗ bottom of a rail. Since the only differential from thisU m c shifts ⊗ thefiltration level horizontally, theU m r andU m c=∂(U m r ) ⊗ i ⊗ ⊗ i generate an acyclic subcomplex (leftfigure offigure 3.16). m 2. Ifi = 0, There are two railsS ,S + and so∂U r consists two different 0− 0 ⊗ i U m c and together they form a cyclic subcomplex (rightfigure offigure ⊗ 3.16). The only other differentials are those inside a railU m S . Since these ⊗ i alternate from shifting thefiltration level horizonatally and vertically, their image in CF is composed of acyclic subcomplexes (rightfigure of Figure 3.17). � 48 Figure 3.17: The remnants of a rail in HFK�. The homology is then generated by the subcomplex of CF generated by U m r ,U m c . Thus (U m c) = (U m r ) = 0. { ⊗ 0 ⊗ } F 1 ⊗ F 1 ⊗ 0 � The symmetry ofCFK ∞ then forces (r ) = 0. F 2 0 Thefiltration levels of the remainingr i are given by the calculation in proposition 3.3.13 of the relativefiltration levels of ther i and the identity n n(n+1) k=1 k= 2 . � The claim about the homology amounts to computingH (U m ). ∗ ⊗R Consider the subcomplex generated by the railU m S . ⊗ i m m H (U S i) = U c =F ∗ ⊗ ⊗ �c C � �∈ i m m proposition 3.3.11 tells us that∂U r i equates the cycles: c C U c ⊗ ∈ i ⊗ ∼ m c C U c. � ∈ i+1 ⊗ m m � Thus the induced cycles are homologous: c C U c = c C U c ∈ i ⊗ ∈ j ⊗ for all i, j. �� � �� � 49 Chapter 4 Integer Surgery Definition 4.0.15 Recall from 2.0.14: m m m = U x CFK ∞ (U x)<0 and (U x) m m C i <0 = U x CFK ∞ (U x)<0 . • { } { ⊗ ∈ |F1 ⊗ } A + = CFK∞ s s • C + CFK∞ B s = C i<0 • { } X +(1) is then the mapping cone of • + + + D1 :A B −→ where + + A = As s � + + B = Bs s � 50 + andD 1 is the direct sum of the chain maps: D+ = v+ h +. 1 s ⊕ s s � Also, denote, byp :CFK ∞ , the projection map. s −→C s Using the integer surgery formula [OS08],HF + ( ∂W ) will be computed − n from the mapping coneX +(1). Definition 4.0.16 For convenience, we define two small generalizations of HFRed± for k >>0: + Red + H (As ) ∗ H As = k + ∗ U H (As ) ⊗ ∗ � � Red k H ( s) = Ker U :H ( s) H ( s) ∗ C ∗ C −→ ∗ C � � Red Note, theseHF are defined identically toHF red± . The only difference ∗ being they are defined on different subcomplexes ( vs.CF −) and quotients Cs + + (As vs.CF ) ofCFK ∞. In particular, 1. There is a long exact sequence induced by the short exact sequence: + 0 CFK ∞ A 0. −→C s −→ −→ s −→ 2. It is also clear that they arefinitely generated as they differ fromCF −, CF + in onlyfinitely many elements. 51 Since this is all that is used to prove the isomorphism betweenHF red− and + HFred, [OS04d, Proposition 4.8], we can also conlude: Proposition 4.0.17 There is a homogeneous graded isomorphism which drops Red Red + degree by 1 betweenH ( s) withH (As ) ∗ C ∗ Red + The computation ofH (X(n)) will be reduced to computing theH (As ) ∗ ∗ Red + Red in proposition 4.0.21 below.H (As ) can be calculated fromH ( s) by ∗ ∗ C Red Red + means of an isomorphism ofH ( s) withH (As ), proposition 4.0.17. ∗ C ∗ Understanding is thus the key part of the calculation. C s We investigate the structure of the chain complex with lemma 4.0.18, C s show the homology of ∂W can be derived from it, andfinish off with the − n calculation of the homology ofHF + ( ∂W ). − n Lemma 4.0.18 Letp :CFK ∞ be the projection. s −→C s Then is a direct sum of theU-modules (1) and (2) below. C s 1. A direct sum of rails (see Figure 4.1): p (U m S ) s ⊗ i m ∈�I(n,s) 2. and a subcomplex with homology −: T p (U m S ),p (U m r ) . � s ⊗ i s ⊗ i � m ∈�J(n,s) Where m (n,s) = m Z 0> max( (U r i) (0, s)) (4.1) I K∩{ ∈ | F ⊗ − } 52 and m (n,s) = m Z 0 max( (U r i) (0, s)) (4.2) J K∩{ ∈ | ≤ F ⊗ − } for i2 i n i (s i)+1 −| |− − m > 2− 2 ifn i (s i) is odd = m Z − −| |− − K ∈ i2 i n i (s i) m > − −| |− − ifn i (s i) is even � 2 − 2 −| |− − � (4.3) The max is between the two coordinates of thefiltration level for afixedi and m s. See Figure 4.1 for an example of m ps (U S i). ∈I(n,s) ⊗ � i=0 p (U m r ) s ⊗ i p U m+1 r s ⊗ i p �U m+1 r � s ⊗ i � j=s � UU p (U m S ) s ⊗ i p U m+1 S s ⊗ i p �U m+1 S � s ⊗ i � � Cs m Figure 4.1: An example of m ps (U S i) with (n,s) = 3. The un- ∈I(n,s) ⊗ |I | shaded area is . C s � Proof. Recall is defined as the subcomplex ofCFK ∞ generated by ele- C s 53 ments withfiltration level less than (0, s) and that theU-action lowers the filtration level by (1, 1), (U m a) = (a) (m, m) F ⊗ F − Proposition 3.3.13 gives thefiltration level ofp (U m r )= 0 explicitly: s ⊗ i � i2 i i2 +i (U m r ) = ( − , ) (m, m). F ⊗ i 2 2 − This gives us a simple way to predict the vanishing ofp (U m r ): s ⊗ i p (U m r )= 0 s ⊗ i � (4.4) ⇔ 0> max ( (U m r ) (0, s)) F ⊗ i − (Recall, the max is between the two coordinates of thefiltration level for afixedi ands.) Whens=i, both the corner (0, s) of and thep (U m r ) lie ony=x+s C s s ⊗ i and multiplication byU translates along the line (Seefigure 4.2). The formula simplifies to: p (U m r )= 0 s ⊗ i � (4.5) ⇔ i2 i m > 2− Theorem 3.3.14 gives the relativefiltration levels betweenU m S and ⊗ i theU m r . We represent their relativefiltration levels pictorially by an ⊗ i approximating right angled isosceles triangle (figure 4.3).U m r lies at the ⊗ i 54 ps (ri) = 0 p U 4 r = 0 ps (U r i) = 0 s ⊗ i � ⊗ p U 2 r = 0 � � s ⊗ i s=3 3 ps �U r i� = 0 2 ⊗ 1 � � 0 1 − s i=0 C Figure 4.2: The unshaded area is the subcomplex s. The exponentm of m 32 3 C m U r must be strictly greater than 3 = − forp (U r )= 0. ⊗ 3 2 3 ⊗ 3 � corner of the right angle, the legs have lengthn i, and the railU m S lies − ⊗ i approximately on the hypotenuse. i2 i i2+i i2 i i2+i − 3, − , 2 − 2 2 2 � � � � p (U m r ) s ⊗ i i2 i i2+i − , 3 2 2 − � � Figure 4.3: A visual representation of the relativefiltration levels between U m r andU m S . The approximating isosceles right triangle is shaded. ⊗ i ⊗ i Using the relativefiltration levels betweenU m S andU m r we derive ⊗ i ⊗ i a non-vanishing condition forp (U m S ) as well. s ⊗ i The idea is to is to adjust the vanishing condition forp (U m r ) by a s ⊗ i quantity proportional to the height of the approximating triangle. First we will look at the case when the index of theU m S andU m r are ⊗ i ⊗ i equal to the index of , namely wheni=s. The calculation then generalizes C s easily by induction to the casei=s. � 55 There are two cases based on the parity ofn i . −| | 1. Ifn i is odd, the center ofU m S is a ‘pushed in corner’ (figure 4.4). −| | ⊗ i p (U m S )= 0 if and only if s ⊗ i � i2 i n i +1 m > − −| | (4.6) 2 − 2 p U m k S s − ⊗ i p U m k r � � s − ⊗ i � � p (U m S ) s ⊗ i p (U m r ) s ⊗ i Figure 4.4: A pushed in middle corner. This happens when the length (n i = 3 here) of the triangle is an even integer −| | 2. Ifn i is even, the center ofU m S is a ‘pushed out corner’ (figure − ⊗ i 4.5).p (U m S )= 0 if and only if s ⊗ i � i2 i n i m > − −| | (4.7) 2 − 2 Now fors=i. We can see this by keepingi constant and shiftings(figure � 4.6). It’s easy to see that shiftings up or down corresponds to increasing 56 m k ps U − S i m k ⊗ ps U − r i � � ⊗ � � p (U m S ) s ⊗ i p (U m r ) s ⊗ i Figure 4.5: A pushed out middle corner. This happens when the length of the triangle is an odd integer or decreasing the length of the projected rail by one. Likewise, changing the exponentm increases or decreases the length of the projected rail. As the vanishing conditions for the projected rail are given in terms of the exponent, We can get the general case ofs=i by subtractings i from the vanishing � − conditions fors=i. The general equations are then: p (U m S ) s ⊗ i m ps (U r i) Cs=i+1 ⊗ Cs=i m Figure 4.6: An example of two projections of holding the indexi ofU S i andU m r constant and shifting the indexs ofp top . ⊗ ⊗ i s=i s=i+1 57 1. i2 i n i (s i) + 1 m > − −| |− − ifn i (s i) is odd. (4.8) 2 − 2 −| |− − 2. i2 i n i (s i) m > − −| |− − ifn i (s i) is even. (4.9) 2 − 2 −| |− − This verifies half of the corollary, namely that the subcomplex consists C s of only thosep (U m S ) with exponentsm s ⊗ i ∈K Now for the second part of the corollary. We show that if p (U m r )= 0 s ⊗ i � then 1.p (U m S )= 0 s ⊗ i � 2.p (U m r )= 0 for k < i s ⊗ j � | | | | Figure 4.7 is a pictoral illustration of the complex. r 1 r0 r+1 − S 1 S0 S0+ S1 − − Figure 4.7: A pictoral representation of the subcomplex ofC i containing all of the non-vanishingp(r i). j2 j j2 j j2 j j2 j Notice that if ( − , − ) then ( − , − ) for j < i as 2 2 ∈C s 2 2 ∈C s | | | | 58 well. Since these are thefiltration levels of thep (U m r ), we have that s ⊗ i if (p (U m r )) , then F s ⊗ i ∈C s p (U m r ) for j < i . (4.10) s ⊗ j ∈C s | | | | Since, every element inp s (Si) hasfiltration level less than or equal to the filtration levels of (r ), also contains: i C s p U � S j ( i 1) , ..., i (4.11) s ⊗ j ∈{− | |− | |} � � Putting thep U � S andp U � j together the resulting subcom- s ⊗ j s ⊗ j plex looks as shown� infigure� 4.7. � � The homology is computed analogously to our computation of the homol- � � ogy ofCFK ∞ above. e.g.∂p U r equates the cycles fromp U S , s ⊗ j j ⊗ j p U � S : � � � � j ⊗ j+1 � � U � c = U � c ⊗ ⊗ c ps(S ) c ps(S ) ∈� i ∈� j [OS08] introduce two tricks to simplify the computation ofH (X+(1)), ∗ ‘truncation’ (see lemma 2.0.16) and a condition under which we may identify a mapping cone with the kernel of its chain map. + The chain mapD 1:n is the sum ofv s andh s. We analyze the maps these induce on homology, (vs) ,(hs) . We start with a lemma on the structure of ∗ ∗ + + Bs = CF , whose homology is the range of (vs) ,(hs) . ∼ ∗ ∗ + Lemma 4.0.19 Letπ:CFK ∞ CFK be the quotient map. Then −→ 59 Figure 4.8: A projection ofU m to for a singlem. The subcomplex ⊗R C s generated byp s (Si) and associatedp(r i)= 0 occupies most of the picture (figure 4.7 is a pictoral representation of it).� There is a single rail on the right with zero homology (the right side offigure 3.16). CFK+ is quasi-isomorphic to the subcomplex: m 0 c ≥ ∪ m � c π(U S0) � ∈ �⊗ 3 + + Proof. Since we are dealing with a knot inS ,H (CFK ) = 0 andπ is ∗ T ∗ a surjection. Surjectivity ofπ lets us represent the cycles ofCFK + by a quotient of ∗ the cycles ofCFK ∞: π(U m S ). ⊗ 0 The restriction ofm 0 for the exponent follows from thefiltration on ≥ CFK∞. Proposition 4.0.20 The induced maps on homology: 60 + + (vs) :H (As ) H (Bs ) ∗ ∗ −→ ∗ + + (hs) :H (As ) H (Bs ) ∗ ∗ −→ ∗ k + + are surjective and onU H (As ) = for k >>0, the maps can be identified · ∗ ∼ T s + + with multiplication byU or the unit1 (asH (Bs ) = as well): ∗ ∼ T 1s 0 ≤ (hs) ,(v s) = ∗ − ∗ s U s >0 + Proof. For surjectivity: Letp :CFK ∞ A be the quotient map. Then, s −→ s + + the quotient mapπ:CFK ∞ B = CF factors,π=v p. Thus: −→ s ∼ s ◦ π = (vs p) = (vs) (p) ∗ ◦ ∗ ∗ ◦ ∗ + Sinceπ is a surjection, so is (vs) . wherev s is the map betweenA s and ∗ ∗ + Bs , and thus the induced map (vs) is surjective. Surjectivity of (hs) follows ∗ ∗ similarly (since you can think of it like (vs) by switching the thex andy axis ∗ and then shifting laterally). Under (v ), [p (S )] [π(S )]. Corollary 4.0.19 tells us [π(S )] is the s s 0 �→ 0 0 + homology class with the lowest degree inH (Bs ). Thus (vs) is injective on ∗ ∗ k + + U H (As ) = , k >> 0 for fors> 0. · ∗ ∼ T j k + + More precisely, (vs) ,(hs) is given byU onU H (As ) = , k >> 0, ∗ ∗ · ∗ ∼ T wherej 0 is the largest exponent such thatU j [p (S )]= 0 ≥ · s 0 � 61 + + + Proposition 4.0.21H (X (1)) ∼= H (X (1;n))) ∼= ker D1:n wheren is ∗ ∗ ∗ the index of the cork boundary ∂W . � � − n Proof. First we show that we can takeb=n whereb is from lemma 2.0.16 andn is the index of our cork ∂W . − n This will follow ifCFK ∞ lies between the linesy=x+n andy=x n, − as this would imply (vs) is an isomorphism, fors>n and also (h s) when ∗ ∗ s < n. Let the railS i lie below the liney=x+C i, andS j belowy=x+C j (where Ci anC j are taken to be the smallest integer possible). Then by looking at thefiltration levels of theorem 3.3.14, we getC C =i j. In particular, i − j − Si lies belowy=x+C j ifj i. Thus all the rails lie belowy=x+C n 1. ≥ − Theorem 3.3.14 lets us calculateC n 1 =n. Tofinish off, note that ther i − ∈R i2 i i2+i also lie below this line as (r ) = − , and (n 1) i n 1. F i 2 2 − − ≤ ≤ − � � By the symmetry ofCFK ∞ aroundy=x, nothing lies belowy=x n. − Thisfinishes the argument for truncation. + + + Now we identifyH (X (1;n)) with the kernel ofD 1;n :H (A (n)) ∗ ∗ −→ H (B+(n)). Since the homology of the mapping cone off:X Y is ∗ −→ isomorphic to the kernel off :H (X) H (Y ) iff is surjective, the claim ∗ ∗ −→ ∗ ∗ + + will follow if the chain map D1:n ofX (1;n) is surjective. ∗ RecallD + is the restriction� of� v h to the truncated mapping cone 1:n s s ⊕ s X+(1;n). Seefigure 4.9. � The preceding proposition, 4.0.20, showed the induced maps on homology: + + (vs) :H (As ) H (Bs ) ∗ ∗ −→ ∗ 62 + vn + An Bn hn 1 − + vn 1 + An 1 − Bn 1 − − + v (n 1) + − − A (n 1) B (n 1) − − h n − − − + A n − + + Figure 4.9:D 1:n. Surjectivity of each of thev s,h s implies surjectivity ofD 1:n walking the surjectivity ofh n up to the top by adding thev s h s one at a − time. ⊕ + + (hs) :H (As ) H (Bs ) ∗ ∗ −→ ∗ are surjective. Using linear combinations of the (vs) and (hs) we can bootstrap this into ∗ ∗ + surjectivity of D1:n . ∗ Since � � + 0 ...0 H Bi 0... 0 (n 1) 63 is the map: 0 ... 0 (h n) (v n) ⊕ ⊕ ⊕ − ∗ ⊕ − ∗ =0 ... 0 (h n) ⊕ ⊕ ⊕ − ∗ + This follows asH B (n 1) is the lowest possible index (n 1) and thus ∗ − − − − � � + (v n) = 0 (as its intended target,H B n is truncated. See bottom offigure − ∗ ∗ − 4.9). � � + In the case of whens> n, the restriction of D1:n to the submodule − ∗ � � + + 0 ... 0 H Bs+1 H Bs ... 0 ⊕ ⊕ ⊕ ∗ ⊕ ∗ ⊕ ⊕ � � � � has image: + 0 ... 0 H A n ⊕ ⊕ ⊕ ∗ − � � as (vn) = 0. ∗ � However if the single summand + 0 ... 0 H Bs ... 0 ⊕ ⊕ ⊕ ∗ ⊕ ⊕ � � + is in the image, the linearity of D1:n implies ∗ � � + 0 ... 0 H Bs+1 ... 0 ⊕ ⊕ ⊕ ∗ ⊕ ⊕ � � is also in the image. 64 Since there are afinite number of + 0 ... 0 H Bs ... 0 ⊕ ⊕ ⊕ ∗ ⊕ ⊕ � � we can induct off of + 0 ... 0 H A n ⊕ ⊕ ⊕ ∗ − � � and conclude surjectivity. Theorem 4.0.22 2 + F[U] H (X(n)) = 0 m(s,0) ∗ T ⊕ U [U] 2 2 n Where n i (s i)+1 min (0,(i s)) −| |− − ifn i (s i) is odd m(s, i) = − − 2 −| |− − n i (s i) min (0,(i s)) −| |− − ifn i (s i) is even − − 2 −| |− − (4.12) Note: i , s must be Proof. By theorem 2.0.12,H ( ∂Wn) is given by the homology of the map- ∗ − + + + ping cone ofD 1 :A B . Proposition 4.0.21 tells us this is given by −→ 65 + Red + ker D1:n . SinceH (As ) ker(v s) ker(hs) we may write this as: ∗ ∗ ≤ ∗ ∩ ∗ � � + Red + ker D1:n = H As ∼ ∗ ⊕B ∗ s � � � � � Where is the kernel of B + k + + D1:n : U H As H Bs ∗ · ∗ −→ ∗ n s n (n 1) s n � � − �≤ ≤ � � − −�≤ ≤ � � with k >> 0. Red Red Proposition 4.0.20 shows that for eachs, either (v s) or (hs) is an ∗ ∗ isomorphism. This lets us further truncate our mapping cone down to a single + in the domain and a map into 0. T0 + + Red + Thus,H (X ) = 0 s H (As ). ∗ ∼ T ∗ Red + Red Proposition 4.0.17 gives� us an isomorphism betweenH (As ) andH ( s). ∗ ∗ C Red Corollary 4.0.18, showedH ( s) is given by the homology of a direct sum ∗ C of rails: Red m H ( s) = Hs ps (U S i) ∗ C ∼ ⊗ i m � ∈�I(n,s) For afixedi, the subcomplexes consisting only of railsp (U m C ) will s ⊗ i form a submodule inH Red ( ), F[U] . s U m F[U] ∗ C · The exponentm is calculated by counting the number of rails that have non-trivial homology. A rail is acyclic if and only if it has an even number of generators: 2 # p (U m S ) . (Seefigure 4.10). This happens if and only if: | { s ⊗ i } 0< min [ (U m r ) (0, s)] (4.13) F ⊗ i − 66 otherwise its homology is: H (U m S)) =F ∗ ⊗ m i2 i Figure 4.10: An example of an acyclic railp s (U S i) withm< 2− + min (0, i s). ⊗ − Now equation 4.13 becomes 4.14 0< min [ (U m r ) (0, s)] F ⊗ i − i2 i i2 +i < min − m, m s 2 − 2 − − � � ⇔ i2 i i2 +i m < min − , s 2 2 − � � i2 i m < − + min (0, i s) (4.14) 2 − 67 Thus the homology isomorphic to the underlying chain complex modulo m i2 i the railsp (U S ) withm< − + min (0, i s). s ⊗ i 2 − Red 2 The absolute homological grading is given by adjustingH ( s) bys s. ∗ C − (Theorem 2.0.12 tells us to shiftA + bys 2 s+1 and the boundary map in the s − Red + Red 2 L.E.S. relatingH (As ) andH ( s) drops this to a shift bys s.) The ∗ ∗ C − powers ofU m correspond to changing this grading by two so we must multiply by two before shifting bys 2 s. − For a given s, i, this gives: F[U] m(s,i) U F[U] ((s2 s) (i2 i) min(0,2(i s))) ⊗ − − − − − With n i (s i)+1 min (0,(i s)) −| |− − ifn i (s i) is odd m(s, i) = − − 2 −| |− − n i (s i) min (0,(i s)) −| |− − ifn i (s i) is even − − 2 −| |− − (4.15) Putting all these subcomplexes together and remembering to double count wheni = 0, gives the result. Example 4.0.23HF + ( ∂W ). Plugging into the formula, − 1 2 F[U] + H (X(n)) = 1 0 +1 0 0 0 ∗ m(0,0)= −| | 2−| − | ⊕T �U � �F[U] �0 2 F[U] + = 0 U F[U] ⊕T � · �0 68 Chapter 5 Further Directions 5.1 Four-Manifold Invariant The calculation ofHF + ( ∂W ) is thefirst step in computing the four-manifold − n invariant: + + + 3 F :HF ( ∂Wn) HF S . Wn(K) − −→ � � [OS08] gives the cobordism map + + 3 + F :HF S HF ( ∂Wn) Wn(K) −→ − � � as the map induced on homology by the inclusion: + ι :H (B0) H X (n) . ∗ ∗ −→ ∗ � � 69 One would hope for a similar result for the cobordism map + + + 3 F :HF ( ∂Wn) HF S . Wn(K) − −→ � � Namely, for it to be given as the map induced on homology by the projec- tion: + π :H X (n) H (B0). ∗ ∗ −→ ∗ � � In principal this should be more time consuming than hard. In conversa- tion, Peter Ozsv´athsuggested it could be done by redoing [OS08] using the CFK∞ + CFK∞ subcomplex CFK i<0 or j j=s j=s Figure 5.1: CFK∞ on the left instead ofA + = CFK∞ on CFK∞ i<0 or j 5.2 Cork Twist Cobordism The interest in corks arises from the difference that arise between attaching it via the identity map or via the twistτ. A natural way to approach this it to compute map induced by the cobor- dism between a cork and its twist byτ . ∗ This approach has already had some significant successes. In [AD05] they 70 looked at the Mazur cork ∂W n and showed thatτ , acts non-trivially on its − ∗ Heegaard Floer homology. In [AK11]τ was shown to act non-trivially on the contact invariant. ∗ The information carried by the cork twistτ has thus not been lost by pass- ing to Floer homology. Further, becauseτ ’s action on the contact invariant ∗ is non-trivial, there is reason to hope the Heegaard Floer invariant may be picking up an interesting aspect of the corks. With the Floer homology of the corks computed, one could hope that the cobordism maps could be computed directly from the diagram. The hope being that the differential on the Heegaard diagram could be computed explicitly by reasoning backwards from what we know the Floer homology must be. 5.3 New Exotic Manifolds Even with a full understanding of the Heegaard invariants of corks construction or detection of new exotic manifolds would pose difficulties. To compute the Heegaard Floer invariant of a four-manifoldX, you need information not only about the corkW , but alsoX W. − To sidestep this, one idea is to modify manifolds, that work well with some of the Mazur corks, to produce manifolds which work well with other Mazur corks. LetB denote the cobordism representing the boundary map in the surgery exact sequence shown in Figure 5.2. (There is an analogous sequence for Positron corks.) By computing the Heegaard Floer invariant ofB we could ask: ifW is a cork in a smooth four-manifoldW=M W , isW � was also a ∪ 71 cork inM � =M B W �? ∪ ∪ One would hope tofind a large number of new exotic manifolds by iterating this process of ‘stacking’ corks. 0 0 0 0 0 0 n 1 n 1 n 1 − − − − − − K 0 +1 n n n − − − 0 0 n − n+1 − Figure 5.2: +1 surgery along the knotK gives a surgery triangle containing ∂Wn and ∂W n 1. − − − 72 5.4 More corks This approach to the computation of the invariant is extendible to other corks as well. Even if a closed formula is not obtainable, one should be able obtain an algorithm. As an example, the computation of thefirst cork of the Positron family is sketched. Following the procedure for reducing the surgery to a knot, we get Figure 5.4 from the two bridge link in Figure 5.3. We build the complex by inspection + then computeH (As ). ∗ From Figure 5.7 we get: + H (A 1) =F (2) ∗ ± and + H (A 0) =F (0) F (0) ∗ ± ⊕ So + + HF ∂W1 = 0 F(0) F (0) F(2) F (2) − T ⊕ ⊕ ⊕ ⊕ � � � � � � 73 Figure 5.3: Kirby Diagram of ∂ W − 1 Figure 5.4: Heegaard Diagram forCFK ∞ of the knot associated to ∂ W − 1 74 c 3 r� r r 3 2 1 r� a b c a 2 1b Figure 5.5: Detail of the Heegaard diagram with labeling corresponding to the chain complex below 3 rc a r� ∗ b a 2 1 c b ∗ 1 ∗ r� ∗ ∗ 2 3 ∗ ∗ 1 Figure 5.6: whereCFK ∞ =F[U, U − ] . 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