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5/08 KIPrd/AMPdeIRC/Dateoue.indd

GROUP ACTIONS. COBORDISMS, AND OTIIER ASPECTS OF 4—MANIFOLD
THEORY THROUGH THE EYLS OF FLOER HOMOLOGY

By

Nathan S. Sunukjian

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Mathematics

2010

ABSTRACT

GROUP ACTIONS, COBORDISMS, AND OTHER ASPECTS OF
4-MANIFOLD THEORY THROUGH THE EYES OF FLOER
HOMOLOGY

By

Nathan S. Sunukjian

There are two main divisions of this dissertation. each dealing with a different aspect
of smooth 4—manifold theory. and each employing a different variety of Floer homol-
ogy as the central tool. In the first. we use monopole Floer homology to construct.
families of finite cyclic group actions that. are equivariantly honicomorphic but not.
equivariantly diffeomorphic. In the second main division. we will use Heegaard-Floer
homology to look at the relationship between a simple class of colmrdisms and the
Ozsvz—rth-Szaho 4—manifold invariant. We will prove that the Ozsvath-Szaho invariant
provides a lower hound on the conmlexity of certain cohordisnis. To accomplish this.
we will calculate the Heegaard-Floer homology of the plumbing of two spheres which

have been plumbed zero times algebraically.

DEDICATION

To my parents and my friends at University Reformed Church, East Lansing.

iii

ACKNOWLEDGMENT

\Yithout the help of a number of people. I might still have written a disserta-
tion. but it would not have been this one. and it would have been very. very bad.
Danny Ruberman. Tom Parker. Ron Stern. and Ian Hambleton all offered ideas and
encouragement when things seemed hopeless. Effie Kalfagianni lent her expertise
about. 3—manifolds. and Matt Iledden cheerfully allowed me to pester him with in—
numerable questions about Heegaard-Floer homology. Fellow students. Chris Hays.
Cagri Karakurt. Jeff 'the hipster“ Chapin. and Chris Cornwell all answered questions
for me. and helped me brainstorm. Chris Hays was particularly indulgent in this
respect. Adam Knapp and Inanc Baykur both shared their enthusiasm and answered
questions. Tom Mark shared his entlmsiasm and also bought me ice cream.

I am occasionally asked why I left my ancestral homeland “the golden paradise of
California". \Yhile there is not a single reason for this. most. of the blame can be laid
on my advisor. Ronald Fintushel. During my time here he has undeservedly treated
me like a real mathematician. He answered questions with patience: was quick with
encouragement. motivation. and advice: and endured many the Monday afternoon
rant of a crazed lunatic (namely. me) with patient forbearance and a generosity for

which I will be forever grateful.

iv

TABLE OF CONTENTS

List of Tables

List of Figures

Introduction

Exotic group actions

2.1
00

IO
00

2.4

2.5

History. . . . .

Exotic Constructions . . . . . . . . . .
2.2.1 .A warm up: Exotic 111v olutions on 2/\#(S 92' X S?) .
Se1berr1-\\1tten ill(01‘\ basics. . .

2.3.1 Seibe11g—\\ itten on closed 4—111anifolds

2.3.2 Seil1erg—\\ itten on 3—111a11ifolds . . .
2.3.3 Seil1e1g—\\itten on 4- manifolds with boundary
2.3.4 Seil 1erg—\\'itten invariants of pairs

Knotted surfaces

2.4.1 Twist. spun knots
2.4.2 Rim surgery. .
2.4.3 Examples
Ixnottiug g1oup ac tions
2.5.1 Examples

Complexity of cobordisms via Heegaard-Floer homology

3.1
3.2
3.3
3.4

Basic Definitions . .

Simple co bordisms and surgery. . .
Desciiptions of the sutgety 3- manifold T(n.. m) .
Ileegaa1d diag1ams fo1 4-111anifol l theo1ists .
3.4.1‘Bottom-up handlcbodv descriptions

3.4.2 Inducte (I handle st1u< tu1es (111 8 x81

3.4.3 Gluing handlebodics and constructing T(n. 71:.)
3.4.4 Converting to Ileegaard diagrams . .
3.4.5 Identifying the. generators of Ilngfn. ml)
Background of Ilecgaard Floer homology

3.5.1 Definition of Heegaard-l—Iloer homology

3.5.2 Calculating IIt~1egaard Floer homology from a diagram .

3.5.2.1 Domains. . . . . . . . . .
3.5.2.2 Visualizing m(:r. y) using domains
3.5.2.3 Calculations using domains.

vii

viii

34
35
35
38

44
44
45
4G

3.5.3 Admissili1le Heegaard diagrams .................. 48
3.6 The I-Ieegaard Floer homology of multiply plumbed spheres . . . . . . 49
3.7 Applications ................................ 65

4 Appendix: Idiosyncrasies of the knot surgery formula . . . . . . 68

Bibliography..........................72

vi

LIST OF TABLES

3.1 Algorithm for maximizing (lefizt‘l R1) ................. 55
3.2 Data to calculate ((11153715’2) ..................... 56

vii

to

90

LIST OF FIGURES

Decomposing (03.1) into pieces.
T(11. m) surgery. .

The figure on the left is the pltunbing of two spheres SO and S3 in

. 11 . . p
.X‘ f“). The plctures on the right are the result of surgery 011 each of
these spheres respectively.

The top figure is a decomposition of T(4. 0) into two copies of 5'1 X A”
and the bottom figure is the surface R1.

Crossing a surface with .51

. , . ‘7
Two p1ctures of Ar) x 51. The top has dT : T“ whereas we are more
interested in the bottom picture. where (7+ = (A

One of the tori is shown in the top picture where we have explicitly
drawn a 1- and 2-ha11dle. The other tori are represented more typically
by the bottom pictures. where it is understood that part of each torus
is contained on the 1- and 2-handles which are reprt—1st-1nted only by
their attaching regions.

The top figure shows how to glue IOgether two copies of 51 X A3. The
bottom left shows the standard gluing which gives 5'1 x So while the
alternate gluing on the right gives T(3.1‘). .

The pictures on the left represents a 1-handle attached to a 0-112111dle.
To convert to a Ileegaard diagram. dualize to get a 2-handle (plus an
unpictured 3—handlc) which we attach. in this case. to a torus. Note.
that we represent a l-handle and the surface with the same notation:
a labeled pair of circles.

viii

14

36

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

This is a picture of H1 (shaded) in TM. 0‘). (a) .44 x pf. is shaded. (h)
The two annuli are shaded. (c) The composite. R1. is shaded with a
perforation where .44 is glued to the annuli. ..............

This is a picture of R2 (shaded) inside TM. 0). (a) The two annuli are
shaded. (1)) pt. x A4 is shaded. (c) The composite is shaded with a
perforation where .44 is glued to the annuli. ..............
R1 in pieces I and II ...........................
R1 in piece III ...............................
The top figure has the domain corresponding to T0 shaded while the
bottom picture has T),'_1 shaded. \Ve have not shaded 7191' because it

overlaps with To(-_1. ...........................

The domain corresponding to T‘M—Q is shaded. We have not. shaded
4 271—1 l_)ec.2_ius(—) 1t overlaps w1th T2 11—2“ ................

H2 in pieces I and II of 7‘(2n. 0} .....................
HQ in piece III of TUE/1.0) ........................

The domain Pi ..............................

ix

39

4O

54

‘1

CI

58

.59

60

61

Chapter 1

Introduction

lily soul is an entangled knot.

Upon a liquid nortezr wrmrqlzt‘

By Intellect. in the Unseen residing.
And thine cloth like (I, convict sit.
lli’ltlz nzarln'zsp'ilre unturisting it.
Only to find its knolfn‘icss abiding;
Since all the tools for its tutti/mg

In four-dimension(:(l space are lying.
—— .la mes C l e Tli' illnzrwel l

Ever since Donaldson's lanchnark work in the 1080‘s. gauge theory has played a
central role. in the study of 4—manifolds. However. the invariants arising from gauge
theory are notoriously difficult to compute. Floer homology is an attempt to medi-
ate this difficulty by. in a manner of speaking. breaking the problem up into pieces.
Several versions of Floer homology have been defined. but they all have l;)asically the

same structure: For Y a. 3-manifold. some group (.}y is defined: a 4-manifold X with

boundary Y has an associated relative invariant o X E Cy; and to two four mani-
folds with a homeomorphic boundary. there is a pairing of their relative invariants.
which ideally recovers some gauge theoretic invariant of a closed 4—manifold. Today
there are three main sorts of F loer homologylz Instanton Floer homology poineered
by F loer himself. which recovers Donaldson theory: the monopole Floer homology of
Iironheimer and Mrowka. which is associated to Seiberg-“Htten theory: and Heegaard
Floer homology of Ozsvath and Szabo. which has an associated 4-manifold invariant.
albeit one which lies outside the provenance of gauge theory proper. All three of these
theories are conjectured to be equivalent. but to date the best. and only real evidence

. . . +7
for tlns is that it holds on all known examples".

Various techniques have been developed for computing the Floer homology groups.
In fact. the three varieties of Floer homology are formally similar enough that tech-
niques for computing in one theory very often work in the other two. In particular.
Floer's surgery exact triangle and the. excision theorem have become mainstays.

In due course. some of the differences and relative advantages of the different
varieties of Floer homology will become evident in this dissertation. Since the three
theories are formally so similar. often the advanta—rges of one theory over another will
be manifest in the definitions themselves. In the first section. we will describe a. simple
situation involving monopole Floer homology. Our goal will be to show how monopole
Floer homology can be. used to construct ‘exotic' group actions on 4—manifolds.

In the second section. we will turn our eyes to much broader questions about 4-
manifolds. Specifically. we will define a }_)articular surgery operation on 4-1'nanifolds

that is related to h-cobordisms. As a first step in investigating this surgery. we

 

1For our purposes here we will ignore Floor l'iomology theories such as Lagrangian—
Floer homology and concentrate on primarily on Floer lioniologies that give rise to
3-manifold invariants.

2While. this manuscript was in preparation. Iiutluhan. Lee. and Taubes announced
a proof of the equivelcnce of IIeegaard-Floer and monopole Floer homology in [‘23]

O

calculate the Heegaard—Floer homology of the 3—manifold on which this surgery is
performed. Here. our calculation appeals directly to the definition of Heegz-iard—Floer
homology. It is not clear how one would accomplish this computation in monopole

Floer homology.

0.)

Chapter 2

Exotic group actions

The world of smooth 4-manifolds exhibits a beguiling array of exotic behavior.

1. Erotic mantfolds. There exist 4—manifolds that are homeomorphic but not dif-

feoniorphic.

2. Erotic Surfaces. There exist surfaces 2 and ‘3’ in a 4—manifold X. such that

(X. E) is homeomorphic to (X. 2’) as pairs but not diffeomorphic.

3. Erotic differentorplusms. There exist homeoniorphisnis that are topologically

isotopic but not smoothly isotopic.

In this chapter we are interested in investigating a more rigid version of the third
item. That is. instead of considering a general diffeonnn'phism on a «'l-manifold. we will
look at diffeomorphisms that generate finite group actions. The following question
arises: Do there exist smooth finite group actions on a 4-manifold that are equiv-
ariantly homeomorphic but not equivariantly diffeomorphic? In particular. are there
such actions on irreducible manifolds? In this chapter we will answer this question
in the affirmative by constructing such exotic group actions on 4—manifolds. In sec-
tion 2.5 we will give a full statement of the circmnstances to which our construction

applies.

After briefly reviewing the. history of exotic actions on 4-manifolds in section 2.1.
we will survey a general strategy for producing exotic behavior in section 2.2. As an
example. we‘ll review the technique of knot-surgery for producing exotic manifolds
since our construction of exotic group actions is modeled on it. Once all the necessary
machinery is in place. we will be able to construct exotic actions of finite cyclic groups
on irreducible 4—manifolds. This result originally appeared in [13]. and is joint work
with Ronald Fintushel and Ronald Stern. The. proof presented here is slightly different
from the original: we remove all mention of 'twins and 5'1 actions. \Ve shall end this

chapter with various examples.

2.1 History

It has long been known that the fixed set and orbit data of a group action can tell us
quite a bit about the action itself. In dimension 3. things are particularly rigid. The
classical Smith conjecture from 1939 states that if a finite cyclic group acting 5'3 has
non-trivial fixed set. then that fixed set has to be the unknot. It was finally proved
in 1978 using the cmnbined work of Thurston. i\leeks. Yau. Bass. and Gordon.

In dimension 4. the Smith conjecture is false. In 1906. Giffen constructed infinite
families of finite group actions on S4 with quotient S4 and fixed set a knotted SQ.

\Vhereas in dimension 3 we might say that group actions on S3 are classified by
their fixed set (i.e. there is only one such action with rum-empty fixed set). finite
group actions on S4 which are a subaction of an. 5'1 action can also be classified.
Fintushel showed that 51 actions on S4 are. classified by their orbit data.

The classification of finite cyclic group actions on 5'4 is far from complete. however.
In 1976. Cappell-Shaneson constructed involutirms on homotopy 4-spheres that were
exotic in the sense that they were not equivariantly diffeomorphic to linear actions on

S4. and Akbulut later showed that the homotopy 4—spheres ('(HISU'llCtt-Xl are in fact

S4. [6]. [1]. Through different methods. Fintushel-Stern also constructed examples of
exotic involutions on 84. [10].

The advent of Seiberg-“I'itten theory provided new opportunities for studying
exotic group actions. For one thing. it provided obstructions to the existence of
smooth actions. This is explained in more detail in Section 2.3.1. More constructive
is the result of Ue [42] from 19.98. Ue constructed free actions of finite groups on
simply connected 4-manifolds that are equivariantly homeomorphic but not equiv-
ariantly diffeomorphic. The actions he constructed are distinguished by calculating
the Seiberg—VVitten invariant, of the quotients. In Ue's construction. the 4—Inanifolds

. _ , . .0 0
being acted upon can all be decomposed as a smooth connected sum With 5- x S“,
. . . -3) .0 . . . . .
and it. is this factor of .8“ x .8- that provides the flexrbihty to construct the exotic

actions. ’We will offer an exrnnple modeled on Ue's actions in Section 2.2.1.

2.2 Exotic Constructions
Many constructions of exotic behavior on 4-111anifolds follow the same general pattern:

1. Define some sort of surgery.

.tQ

Check that. the surgery doesn’t change the topological type of whatever behavior

you are studying.
3. Check using gz‘iugc theory that the surgery changes the smooth type.

As an example. we'll describe a nee-classical construction of exotic 4—1‘1’1anifolds
due to Fintushel and Stern. To satisfy step 2. well use the following theorem of

Freedm an.

Theorem 1. [14/ If X and X, are smooth. sin'zply connected 4-Ine-‘Izjfel(l.s. then. they

are homeomorphic if and only if they have 7.9077I.()7‘[)/I‘i(7 (re/remology rings.

(3

To satisfy step 3. we’ll use the Seiberg-XX’itten invariant. This is described in
detail in the next section. For now. it will suffice to know SH’X E Zlf/2(X)] is an

invariant of smooth 4—manifolds.

Knot surgery is a process whereby the neighborhood of a torus is replaced with
something homologically equivalent. but “knotted". Remarkably. this process does
not change the homeomorphisrn type of a 4-manifold. and equally remarkable is the
effect on the Seiberg-XYitten invariant. Specifically. knot surgery is defined by replac-
ing a copy of D x T With b X S \nb(1\ ). If the surgered manifold IS still Simply
connected. then one checks by Freedmans theorem that the homeomorphism type of

the manifold is not changed.

Theorem 2. [1212" Suppose that T is an embedded torus in a 4-mandfold X with [T]:2 =
0. and that Ix" is a knot in 5'3. If X and .X' \ ‘T are simply connected, then X is
homeomorphic to the knot saxrjqered manifold. XK :2 (X \ nth)) U d) (5'1 x (S3 \
nhtlx'))).

Moreover. 2f fl»: is the longitude of K. and o : 081 X (S3\\7’2h(1\')) ——> 8X \ (I)2 X
T21) identifies ( K with, (702. then Sll'X 1' is obtained from Sll'y the multiplication
. x .

by the st/‘Inmctrized Alexander polynomial of K:

SH'XK : Sl—‘l’X ' Alvlngll

It is evident from this theorem that if -X' is a 4-manifold with SI'VX # 0 and
contains a suitable torus. then there exist an infinite number of manifolds that are
homeomorphic but not diffeomorphic to X: apply knot. surgery to X using an infinite
collection of knots with distinct Alexander polynomials. There is an algt-rbraic subtlety

here. but it is minor enough that. we have relegated it to the a ') )endix.

\

2.2.1 A warm up: Exotic Involutions on 2X#(S2 x S?)

Theorem 3. So. ) 2051: that X is a /-mani old to which the theorem 2 a )lies. Then
I I I

. . . . . . J) 7,9
there are an znfinztc another of erotic group actzons on 21X #5“ X 5".

Proof. Let {.X'}\’,} be a collection of non-diffeonior})hic manifolds which all arise as
1

knot surgery on X. If we take the 2-fold branched cover of {Xhh} over a trivially
2

embedded torus we get 2A 5:747:52 x .S“ (for this fact. see [16] or [18]). It has been

2
. . J) . .

shown by Auckly [2]. and independently by Akbulut. [4] that A [(#5- x 52 is dif-

2
~ .' I v9 v9 ’ Ir - -

feomorplnc to .X #5" x 5‘ (see also [a] for a snnphfied proof). Hence. we have an

infinite lannly of involutions on 2.X #5" x .5“ that. are all the same topologlcally since

they came from topologically equivalent branched covers. whereas these actions are

smoothly distinct since their quotients are not diffeomorphic.

2.3 Seiberg-Witten theory basics

2.3.1 Seiberg-Witten on closed 4-manifolds

Let 5 be a Spinc structure on a. 4—manifold X. and let B(X.5) be the set of gauge
equivalence classes of pairs (A. o). where A is a .9me connection and (D is a. spinor
field on X.

For a 4-111anifold with a Spin." structure 5. the Seili‘)erg-VVitten equations are:

oja=0 (an
1/2p(F+ — c...‘+) — (c:>c')*)0 : 0 (2.2,)

4]
4'

where p is the Clifford multiplicz—uion. 0:1. : U‘S'l') ——> I‘(S—) is the Dirac operator.
and (00*)O is the trace free part of the endomorphism 690*. The 2-for1n to is an
arbitrary perturbation.

The Simple Type Conjecture says that. the moduli space of solutions M(X.5) C
B(X.5) to these equations is a zero dimensional manifold for all 4—manifolds with
(ff 2 2 with a generic choice of ax; we will assume this for the remainder of this paper.
In this case. we define the Seiheiy—[latte-n. invariant Sll’X(5). to be an algebraic
count of the points in the moduli space (where signs are assigned via some. choice of
orientation). \X'itten shows that Sll'X (5) depends only on the smooth structure of
X. not. on the choice of metric or perturbation. ([45]. see. ‘28] for a mathematically
rigorous proof). A Spin" structure 5 such that Slvl'Xm) # O is called a basic class.
We encode the information information given from this invariant as an element. of
Z[H2(X)] by defining SH'X :2 Z SXX'(5)(:1(5) where the sum is taken over all S pine
structures on X. In the case that f12(X) has 2-torsio1‘i. we loose information when we

pass to .S'lrl'xr. but since we are primarily concerned with simply connected manifolds.

9

 

this is not a concern.

Note that if (b is a diffeomorphism of X. then 0*(8ll'X) 2 Silk. This provides
a basic obstruction to the existence of certain smooth group actions. For example.
Chen and Ix'wasik [7] use this idea to show how certain actitnis that. exist on K3 cannot
exist on exotic copies of K3.

A seminal result in Seiberg-XVitten theory is the following theorem of Taubcs:

Theorem 4. [41] Suppose (X. we") is a closed symplectic J—manxlfold. Then Sll' X # O.

and specifically .S'll'X(5w-) 2 1.

2.3.2 Seiberg—Witten on 3-manifolds

Let 5 be a Spinc structure on a 3-1nanifold Y. In [132]. Iironheimer and .X'Irmvka defined
the “Monopole F loer homology group invariants. a collection of groups associated to
(ifs). For simplicity. we will restrict our attention to the. circumstance where 5
is torsion. In this case all of the groups defined by Kronheimer and .XIrowka are
equivalent. This invariant is called the reduced monopole F loer homology and we will
denote it by [UN l". 5). It is constructed as follows: Let [3015) be gauge equivalence
classes of pairs (.4. (If) where .4 is a Spinc connection and a") is a spinor field. Then the

chain groups defining 1] ANY. 5) are generated by the elements of B(i"’.5) satisfying:

1’9”]an ,1 — (063*)0 = 0 (2.3)

[)136’) = (l

Ilere we fudge slightly: In fact. it is a suitably perturbed version of these equations
that defines monopole Floer homology. and these perturbations are. the source of much

of the complexity in the. theory. See [22] for the details.

10

\X’e will primarily be concerned in this chapter with the monopole Floer homology
of 81 x 29 where 39 is a surface of genus g. In this case. we have no need to describe

the (.lifferential.

Proposition 5. Let 5'1 x 29 be endowed with a product metric whose restriction to
Eq has constant negative curvature. and let 551—1 be the Spiiiff structure characterized
by (:(51 g_1).$g) = 29 — 2. Then the equations (‘2.3) have a. unique solution, [(10) E

<
8(5'1 X31), and consequently til/(51 X Sg-Sg—ll :

‘-
49—

Proof. A detailed proof can be found in [8). ‘31). and [30). Essentially the proof

L

comes clown to showing that (2.3) is invariant under the obvious 51 action. and in
this case these equations simplify to the abelian vortex equations on Sq. But the

vortex equations can be. solved explicitly. U

2.3.3 Seiberg-VVitten on 4-manifolds With boundary

The full story of Seiberg—Witten equations on 4—manifolds with boundary is a long
one. told in its entirety in {22). In general. if .\ is a -1- manifold with (M: l". then
associated to X we get an element. 'UX E H A] ( l'.5). We will restrict our attention
to the simple case where (7X 2 5'1 X 29.

Specifically. we'll consider the Seilmrg-XX'itten equations on X with an infinite end.
.X'* = X U 5'1 X SKIR‘T: Let. B( .X'*. [00]) be the subset of B('3.X'*) which limit to the
element [(1.0] of Proposition 5 on the end of .X”*. Then we can define 'L'Xfl E [Ll/(5'1 x

59-1) = Z to be the count of isolated solutions to (2.1) in BtX*. led): the count
of elements in the moduli space .ll(.X'*. [00]). that is. Using suitable perturbations.

’*. [00]) along its path

It"); (7 is an invariant. “hat. is more. we can decompose Bt.
components into so called ‘z-paths". B(X*. [00]) = UB3(.X'*.[(10)). and similarly
xiv/(X’k. [(10))—-—U .l/3( X * .[(1())). Moreover. the set of path con‘iponents of B(.X'*. [ooh

1s a prlncipal homogeneous space for 1/"(.X..‘wl x Sq). Hence. if we make some

11

1dent1f1cat1on between H"()\ . .81 X By) and the z-paths. we can define

suxflzz 2: esuuxtpmpheZH(Atflx:n
heHchslxs)

which 1s invarlant 11p to mult1pl1cat1on by an element of H ‘(A . SI X 2).

2.3.4 Seiberg-W’itten invariants of pairs
\X'e can define a smooth invariant of a pair (X. E) as follows.

Definition 6. Let X be a closed 4-manifold containing an embedded surface S. then

we define S'll'(l\,.2) 2 SH,(X\'H()($).8) when [3)? = 0. \X'hen (2)2 2 TI > 0. define

SH' - —Sllv' . ~ .
(3 IS} (,\'#n.C'P2\nb($).0)

 

where E is the total transform of 5.3.

This notation is somewhat non-standard. Typically S H) le) is only defined for
._J
.0 . .
the case that [2)“ = (l. but the extenslon made here makes Sf—EV'Gl'al theorems easler to

state.

Theorem 7. If .X' is a .5;1/'I'/2;1)i("('tl(' manifold and E C .X' is a s;_umplccttc surface with

0 , .
[2]" Z 0.17“?” LS“ Vi: ‘7—'1 0
The proof is standard. but we outline it here for the sake of ccnnpleteness.

1‘) . v . _. , O 0
Proof. Assume {3}" = U. Decompose .X mm /X = /X \ D“ x E and D“ x S and let

X-n define a family of metrics on ,X’ via

.X,, = (To s1 X E x [1). 11)) o (D2 x E u 5'1 x E x ((1.11))

By Taubess result above. 5» 1 is a basic class of .X'. and (5- -

w \ w. E) = ‘29 — 2 by the
adjunction equality. Hence. the basic class 53.3: restricts to 59_1 on 5'1 x 2. By a ba-

. . . . . . . / . “—7 _ , .")
sic (but dlfhcult) hunting argument. .'ll(.Xn .5(,_1) converges to .’ll(.\ . ((10)) x .l1( U“ x

1‘2

V H ~ 1 . ,',,. .'. )1 “.‘_H . . 7"
.4. [00]) as 17 goes to lIlilllll} in some suitable~ eompactihtation of Un€(0.oc) M((An. 59_1)).
But M (Xints) is (algebraically) non-trivial by Taubes’s theorem above.
‘1‘) .
In the case that [Sr 2 71 > 0. the same proof applies because the proper transform

of E blown up n-tinies is still a. sympleetie surface. El

13

U

 

 

 

 

(D4. [#K) (D4. 1) \ nb(a)

 

Figure 2.1: Decomposing (03.!) into pieces.
2.4 Knotted surfaces

We will examine two methods of knotting surfaces in 4-nianifolds: twist spinning a
knot to give a knotted .S" in S (originally defined by Zeeman. [46]). and the closely
related technique of Fintushel—Stern of performing ‘rim surgery" on a. surface in an

arbitrary 4—rnanifold. [11].

2.4.1 Twist spun knots

Heurist‘ically. a spun knot. is e(_)nstruct.ed by removing an annular neighborhood of the
equator of a trivial 5" 1n 54 and replaemg It With .51 times a knotted are. Let us
spell this out. in greater detail. Let n. .5- E .S“ be the north and south pole respectively,
and let I C D3 be a straight segment from 'n to s in D3 C R4.
. , “'4 ' 7 ‘ W1 3 V2 0 . . . .
Decompose .S into .S x D U .S x D- by thinking of it as the boundary of

. r) 0 . . . . . -
D“ x D". The unknot can be seen in this dt—écernposition as

(54.93,) = (31 x 03.51 x I) u <52 x 02- {m} x D2)

Now if K is a knot in S3. we can form the span knot S l\’ in 5'4 by retflacing

51 x 1 in the definition above with 1K = .S‘1 x [#Ix’ (see Figure (-11,):

14

($4.31.» = (5'1 x 03.1,.» o (5'2 x D2.{n.s} x D?) (2.4)

Let us consider an alternate definition that will be easier to generalize. Let a C 03
be a. meridian of I C D3. Then 5‘1 x a is a torus whose neighborhood we shall write

1')
as 5'1 xaxDp.

Then

(51x 193.1,.» 2 (5'1 x 03.51x 1‘)\51 x o x DB

o 81 x (s3 \ nbtlx')‘;

O
The gluing map (2) is characterized by

d)*([5'1l>= [5'11

(.)*([(r]) = [7'I£I\']

0*(fiNJBl) l{[\'i

where mA. is the meridian to [x' C 5'3. and (1" is the longitude. See Figure 2.1.

A geiieraliZt-ttion of this construction is the lit-twist. spun. knot SK 1; C 5'4. \Vhereas
we defined S' - = 1 U {a 5‘} \< D2 we define S' - = I ' U {n 9} x If2 where I ,.

' “I\ k ’ ' ‘* It}; Ink: - ’ ILL“.

is defined similarly to I," aboye. except we use the gluing map (9 characterized by

0* (lS1]) [51} + Aim/f]

0*“ch = [mA-l

C)*(li')f)3l> = [I m

15

 

Heuristically. SK}. spins 1\' around I; times as we go around the 5'1 factor. Notice
that by these definitions. both spun knots and twist spun knots can be constructed
by performing knot surgery on the torus 5'1 x a in S4. This differs from knot surgery
defined in Section 2.2 in an important respect however: In this case knot surgery does
not change the ambient manifold 84: it changes the embedding of a knotted sphere

in 54.

2.4.2 Rim surgery.

Now we explore knotted surfaces in «la—manifolds that are. more complicated than
knotted spheres in 54.

Let E be a surface in an arbitrary 4~manifold X and let C be a simple closed
curve in E that is homologically essential in S. Motivated by the definition of a twist

spun knot:

(54-51(15) = (sl x 03.1,(75) o (s? x 02411.5} x 01’)

one can define lt—t'ur‘z'sf rim surgery as:
(X. 3:. A119) 2 (5'1 x 1>3.1,\-.A.)o(.\' \nb((.').§3\nbt’CN

This definition was originally made by Fintushel and Stern in [11]. for the case
of k. = 0. The A? ,2’ 0 case was explored by Kim and Ruberman in [19] and [20]. In
contrast to twist spun knots in S4. twist rini surgery does not always change the

topological type of the surface:
Theorem 8. Soy X is simply cormected.

o [11/ If 771(X \ E) = 1. then (X. E) is homeomorp/ric to (X. SK“).

16

o [:20] If W1(X \ S) : Zr! and (die) : 1. then. (X2) is harm—Jornorph'ie to

(X. :3 m“).

In (X 2) let (C x D3.C x I) be a tubular neighborhood of C. Define the rim
torus to be H = C x o C C' X [)3 where o is a meridian of I C DB. Note that this
torus is l‘iornological1y trivial in X. but homologically essential in X \ HMS). Since.
we saw that twist spinning a knot was equivalent to doing knot surgery on just such

a torus. the following theorem should be. not too surprising.
Theorem 9. (5/11/- [917. see also [13]) If: C X has positive self intersection. then.

sw) \

r
14 c—J

.. , J =A.[\r(21t)su'(X

“ LI.

 

y)

 

where I? is the rim torus correspo”ding to the. CUT‘W? C where the run—surgery teas

perform ed.

Proof. Rim surgery is accomplished by replacing (.S'1 X [73.31 X I) with (.91 X
D3'1K k). As with twist spun knots. this is equivalent to doing knot. surgery on
on the torus which is 81 times a meridian of I —— the rim torus It). in this case.

Recall SH '( X

v) corresponds to finding solutions of the Seiberg—XVitten equations on

 

X \ 7212(2) (possibly blown up). and I? is a homologically essential torus in this mani-
fold. Fintushel and Stems original proof of the knot surgery theorem [12] applied to
closed manifolds. but the same proof works in this case once one recognizes that one

should substitute z—paths where they originally spoke of Spin“ structures.

Torn Mark has obtained an analogous result in I-leegz—rard-Floer theory that applies

to (my svtn )lectic surface in a svm )lectic manifold l‘f‘f"2‘11‘(ll(-‘SS of self—intersection. ’26 .
i . . 5 l

17

2.4.3 Examples

One can find an elliptic filnation structure on K3 such that a generic fiber is sym-

plectic and has simply connected complement. This was the original example given in

111C l—‘"‘l‘-l"'-‘ l HWll‘ '- f=' l- l)‘ ..
J. .01in tx a ge naic curves a so p10\ 1( c a aige source 0 cxamp es since t 1(} at e

automatically synnflectic. and hence have non—trivial relative invariant by Theorem

_ _ . . . . _ . _ . )0 _ . .

I. So. for example. if I d is a generIc degree-(I curve. 111 C I '. we can apply rim-surgery

to ld as long as 7r1(CP“ \ la’l is finite cyclic. This is true by the Zariski Conjecture
. r) . .

wh1ch says n1(('P‘ \ l (1) = Zd. See [19) for tlns and other examples.

.. . . .0 .0 . . .0 -3)
Similarly. in .S- X .S- the curve ("(1 representing (1([5' X pt] + [pt X b“

]) has
. .5) ,0 . . . . .
7r] (5" X .S' \ f» d) = Zr] by the generalized Zariski conjecture [32]. Therefore we can

also find an infinite family of exotic (761’s.

2.5 Knotting group actions

\X'e finally have all of the necessary machinery in place to construct the promised
exotic actions. Before we do so. let us look at. two model theorems. Our actions will
arise as branched covers over rim surgered surfaces. First we’ll consider the branched
covers of twist spun knots in 54. In all that follows. denote the cannonical (l—fold

d- when I/1(X \ 2) = :3.

branched cover of X over 2 as (4 X 2)
Theorem 10. Let SAX]; C S4 be a h-t'wist span knot. and let (1 E Z be relatively

prime to k. Then (54. Shikld is (l'z‘fleomorphz'c to S4.

Proof. Ciffen showed that such a branched cover is a homotopy ~1-sphere. [15]. Gordon
extended this. and showed it is a horrrotopy 4-sphere that admits an Sl—action. [17].
Pao (using Fintuslrel's classification of 5'1 actions on homotopy 4-spheres) showed

that any homotopy 4—sphere admitting an Sl-action is diffeomorphic to S4. [36].

Corollary 11. Say (7 is a Q—hanrlle attached to 51 X D3 along 5'1 X {pt}. If we write

(s1 x 0% (3. 11(1),)" as (s1 x 03. 11(1).)‘1 o (rm/:1 arherc mt;- t

z are the (l disjoint

lifts of U. then

(s1 x 03.1,(gpdo (*1 = [)4

Proof. We can extend 1K. K C 5'1 X D3 U U to a twist spun knot in S4:

(s4. s,\-_,‘.) = (5'1 x o3 o LIA-‘1‘.) o (134.1)2 o D?)

By Theorem 10. the d—fold branched cover is again just S4:

19

s4 = (s1 x n3 o t.f.11\-.k)d o (1)41)? U 02y!
z (5'1 X D3~l1crld U (D4 U U, D2 o D2)d

= (31 x D3.1K.k)du D4 o U1

This implies that (.S'1 X D3. 11(5).)(1‘ U (.71 is diffeomorphic to D4.
Cl

Now we will construct actions on 4-manifolds that. are locally just like those given

in Corollary 11.

Theorem 12. [15’] Let. 1' he a simply connected 4 —ma.n'1fold with an, embedded surface

satisfying the following conditions:

0 S is of genes 9 2 1

r
u

the pair (3 E) has non-trivial Seibery- It‘l'itten invariant.

o 2 contains a non se )aratinr loo) C which hounds an embedded S—(lisl; whose
l .1

interior lies in, l" \ 53

Let X be the d-fold branched cover of l". Then X admits an infinite family of smooth

Zd actions that are topologically courtraria'n.t, but smoothly distinct.1

Proof. The first three cor’rtlitions irrrply that E is a suitable surface for the rim—surgery

construction. and the fourth tnovides a tool for indentifying the t’liffeomorphisnr types

 

. . . . / . .
1The sarrre proof works where X is the (ll-fold branched cover of l’ where d divides

(1.

2t)

of the branched covers. Let. k be an integer such that. (Is. rt) 2 l and let HEX/{.11.} be
a family of smoothly distinct pairs. where E Ki}: is obtained by k—tw'ist rim surgery
on C using some knot. Kt- Let X?- be the d—fold branched cover over S Art-,5" i.e. X?- =
(Y. 2K2, )d. Then the induced Zd actions on the X1- are all topologically equivariant.
because they came from branched covers of topologically equivalent surfaces: and they
are smoothly distinct because the images of their fixed sets are the surfaces 2 K1,. I;
which are smoothly distinct.

It only remains to show that X!- is diffeomorphic to X. Note that the branched
covers only differ where the rim surgery was performed. Specifically, Xi is obtained
from X by replacing (C x D3. C x I )d with (C x D3. I K. Add . \Ve'll look at. a slightly
larger region. Let U be a regular neighborhood of the disk bounded by C. Then we
can obtain X,- from X by replacing (C x D3. C X 1)d U U1 with (C X D3. 11".]l;ldU(-"1
where (’1 is a lift of (7. By the Corollary 11. (C x D3. 11".A.)(]Ut.71 2 D4. Therefore
X:

I is diffeomorphic to X.

2.5.1 Examples

In Section 12.4.3 we saw a. number of surfaces to which the rim surgery construction a -
O .
plies. If we wish to use these examples to find exotic actions. it remains to check that
the surfaces in these examples contain a suitable curve C that bounds an embedded
disk.
For complex degree-d curves. l d C CP‘. the curve id is the fiber of a pencrl.
Suppose d > '2 Then we can take C to be any loop in lit bounding a vanishing cycle.
. . .0 .5) . . . . o
The. same is true for the curves (d C S‘- >< .9“ described 111 Section _.4.3 above.
By taking the li)ranch(-~d cover over l'd, we get an infinite family of finite cyclic
. . . , 0 ’17—‘32 . .-
exotic group actions on. for example. (, P“#GC l by looking at the 3—fold branched

cover over ll? . Actions on K3 can be constructed by looking at the 4—fold branched
'3 . C5.

‘21

cover over V4 as well as the 2-fold branched cover over l6- The 3—fold branched cover
of (-3 C S“ x 5" 1s again I\3. In fact. only Z9. Z3. and Z4. can act on Ix3 in such a
way that the fixed set is a connected surface — and we have. constructed exotic actions
in each of these cases. This can be shown by a bit. of algebra using the following two
formulas which relate the euler cl'iaracteristic and signature of a manifold to those of

its branched cover.

 

.
ls;

Chapter 3

Complexity of cobordisms Via

Heegaard-Floer homology

\Ve exhibited a variety of exotic l.)eha\.'iors in the last. chapter. In this chapter, we
will shift. perspective slightly. and interpret exotic behavior on 4—manifolds through
5-dhnensional techniques. In particular. we’ll examine the following 2 questions from

a 5-dimensional perspective:

1. How can you tell if homeomorphic 41—manifolds are diffeon'iorophic?

2. Vl'hen is a self-hon‘ieomorphism of a 4-111anifold isotopic. to a self—diffeormophism'.’

These are very difficult questions to answer in general. \Vhat the 5—din'1ensional
persm—‘ctive will give us is a way to quantify how far two manifolds are from being
diffeomorphic. or how far a. self—honieomorphism is from being a self-diffeomorphism.
This is accomplished by measuring the complexity of a cobordism between two 4—
manifolds as follows. Let M1 and .ll-Z be homeomorphic 4—manifolds and let C( A] 1. .I [-2)
be the set. of all cobordisms from Ml to M2 that possess a decomposition with only

2- and 3-lnnidles. Then we. can define

MM 1. HI A12) 2 min{geometric intersection number of the belt sphere of the

2-handles with the attaching spheres of the 3-handles}

where the minimum is taken over all handle decompositions of W using only 2-
and 3-handles. Define i’\1(\i’l111.ll"..l/9) similarly but take the minimum over handle

decompositions of W with only one 2- handle and one 3—handle. \ ow define

/\(All..l12) : fililrn [\(All. l‘l'. Al?)

C

I

A1(Ml. J12) = ‘niiEnC/\1(i\11. u: M2)

Alter/1. A12) 2 min A( M1. w. M2)
' _ ll EQ‘
\\ is an h-cob
h. .- , . .
Al(-l[1..rl12) 2 mm 1\1(1l[1.l‘/l,1\12)
. . ll 66
V\ 1s an 11—001)

The relation between Question 1 and the A invariants is transparent. It is not
hard to show that A} "(M 1. Mg) 2 0 if and only if -l [2 and A12 are diffeomorphie. And
the greater A is. the greater the disparity between the smooth structures of All and
A12.

A and Ah“ have quite different behavior. however. Many families of exotic mani-
folds that have been constructed have A = 2. This is explained in [53.011 the other
hand. A," exhibits more interesting behavior. even for h-cobordisms of a manifold
to itself (so called inertial li-cobordisn‘is). e.g. the. following theorem of Morgan and

Szabo.

Theorem 13. [129/ For all 'n E 23 there exist 4-771amfolds M1, and an inertial h-

cobordism of gun. say 11'”, such that AMI”. Hint. Mn) is mrho-mrded as 77 increases.

24

These invariants also shed light on question 2 above.

In section 3.7 of this chapter we will construct an obstruction to a self homeo-
morphism of a 4-manifold being isotopic to a self diffeomorphism based on the A
invariants. In, particular. associated to c‘). a self homeomorphism of a 4—manifold AI.
we will construct a cobordism (M. ll’d). A!) such that AMI. W99. M) = 0 if and only
if cl) is isotopic a diffeomorphism. The theorem of Morgan and Szabo gives examples
of horneomorphisms that are not realized by diffeomorphisms. Contrast this with the
previous chapter. In the previous chapter (Section 2.3.1) we saw how the Seiberg-
\Vitten invariant provides such an obstruction. A partial motivation for our study
here is to understand how these two obstructions are related.

Our goal in this chapter is to lay a framework for studying the A invariants.
Ultimately. we would like to understand how, for instance. the Seiberg-\Vitten. or
Oszvath-Szabo 4—manifold invariants of cobordant. 4-manifo1ds are related and to
understand the A invariants. A full tmderstanding is. at present. beyond our reach.
As a first step in this direction. however. we will calculate the relevant Heegaard-Floer
homology groups associated to sample cobordisms (defined in Section 3.1).

The main technical content of this chapter is a calculation of the Heegaard Floer
homology for the plumbing of two spheres. At the end of this chapter we will use this
calculation to derive a relationship between the :l—Inanifold invariants of M. and the

A invariants for certain cobordisms.

3.1 Basic Definitions

The purpose of this section is to fix a consistent set of notation and terminology for
basic handlebody theory.

A handle decomposition of a manifold is a thickened version of a cellular complex:
An i-cell is defined to be a copy of Df. \K’e can “attach an n-cell to a space K” using

0.5

a map (91)” —> X. That. is. we attach cells by gluing their boundaries to a space. A
C W complex is defined inductively by attaching cells of increasing dimension. 'We
can construct n—manifolds in a similar way. but everything must be thickened: Define

an I'i-dimensional i-hamlle. denoted h-

z: to be D1 x Dn—l. which is attached to an

n-Inanifold M" via a map a : (01]) x Dn—f -—‘r 0.” n — we glue handles along the
thickened region that we glued cells. We call (01)". ) >< l)""f the attaching region and
we call a the attach/lag map. Additionally. we call 81.")?- x 0 the attaching sphere. and
0 x 01.7"” the belt sphere.

By elementary Morse theory. every n-niai'iifold Al" has a handle decomposition

where handles are attached in increasing index. \Ve will denote this by:

A!”:leO+lel*i—...+:llyn

Also useful will be the dual of a handleliiody decomposition whereby the roles
of the belt spheres and attaching spheres are reversed: an i—handle h. i. is an ‘upside
= h:‘. the attaching region becomes D" x E)l)”—’. etc.

dowrr (Ii-i)-handle h” _,-

Hence. we can also write:

AI" 2 Eli?) +211: + . . . +2117;
ZZI’IL+lell—1+"'+Zh0

\Ye can also define relative handlebodies which are built on an {n-l)-manifold N:
Ll n 2 X \7 h ‘f‘ . . . ; .
l I l ‘l’ E l + flu

In this case. we denote 0—H = {0} X N and (fl—fill = ('Lll ~— il—-ll For the
subhandlebody composed of handles up to index i we will write ill“). By the middle

. . . . . . , _ m
level of a handle decompos1t1on of odd dnnenslon n. we. will mean (Fiji/fl" ll/ "l.

26

For odd dimensional manifolds. it will be ctmvenient to convert the “bottom-up”
handle decon'ipositions described above into "middle-out" handle decompositions by
dualizing the handles below the middle level. For example. we can convert handlebody

decmnposition of a 3-manifo1d ala:

313 = llo'i‘lel +Zl12 +lt3
=126+Zlff~i~($x 1)+Zh2+h3

=li3+ZlI2+(EX l)+:li~2+li3

That is. we attach two sets of Q-handles handles to the middle level. one going
up. the other going down. For a 3—manifold we can actually draw a picture of a
middle-out handlebody decomposition. Such a picture is called a Heegaard diagram
and these are explored further in Section 3.4. l\li(.ldle—out decomposititms will also

arise in our investigation of 5-dimensional cobordisms.

3.2 Simple cobordisms and surgery.

A cobordlsm between two smooth i'z-iiianifolds All and Mg is an (-n -+— l)—dimensional
manifold X with (7X 2 All U .l/Q. If the inclusion of .‘lll (or equivalently M2) is a ho-
motopy etplivalence. then X is called an h-cobordlsm.. The h-cobordism theorem says
that. if two simply comrected manifolds of dimension. greater than 4 are h-colmrdant,
then they are actually diffeomorphic. It is the failure of this theorem in dimension 4
that is the source of the beguiling exotic behavior exhibited by smooth ~1-manifolds.

Consider the simplest class of cobordisms between 4—manifolds: cobordisms that.
have a handlebody decomposition as a single ‘2 handle and 3 handle pair. We will

refer to these as simple cobordisms. In this section we will characterize how a simple

27

 

Figure 3.1: T(n. m) surgery.
cobordism induces a. surgery relationship between its boundary manifolds.

Proposition 14. If Ml and M2 are simple cobordant 4-manifolds. then Ml can be
obtained from Mg by a surgery of the type given in Figure 3.1. We will call this
operation T(n.m)surgery when the ‘oater' J- and Q-handles cross geometrically n

times and algebraically m times.

We shall refer to the 4—manifold given by this Kirby diagram as D(n. m). and the
three manifold which is its boundary as T(n. m).

Notice that a degree—0 log transform is a T(2, 0) surgery using this terminology.

Proof. A simple cobordisrn can be given a "middle—out” decomposition:

)Cz .’\[1X[+l12+ll3

2113 + I X 3(a) +113

Call the attaching regions for these two 3—handles SO. and $5. In Km). the union
SQ U SD» is the neighborhood of two plumbed spheres. given in Kirby calculus by
Figure. 3.2. Then Mg = 0+(Xl2l + h3). which is equal to surgery 011' S3. This
corresponds in Kirby calculus to changing the 0-framed Q-handle that corresponds to

S3 into a dotted l-handle. Similarly. All is obtained by the same operation on the

28

 

 

 

 

 

Figure 3.2: The figure on the left is the plumbing of two spheres 80 and $3 in

X (2). The pictures on the right. are the result of surgery on each of these spheres
respectively.
other 2—handle of Figure 3.2.

El

Corollary 15. If M1 and Mg are simple h-cobordant. then they are related by a,
T(n. l) surgery. Moreover. if All (and Mg) hare indefinite intersection forms. and
All-#52 x 82 is diffeomomhic to Mg#52 x 52, then. A11 is related to A-Ig by both a

T{n.1) surgery and a T(n'.0j surgery.

We remark that for non-spin manifolds. being simple h-cobordant. is equivalent to
Ml#52 X S2 being diffeomorphic to Mg#8'2 x 82. we do not know if this is true

for manifolds which are spin.

Proof. The handles of a simple h-cobordism necessarily intersect. algebraically once.
proving the first part of the corollary. When Ml#S2 x S2 is diffeomorphic to

Mg#.5'2 x 52 we can build a simple cobordism such that the handles intersect al-

29

. .0 -.0 . .
gebralcally once or not at all as follows: Let 5 To and .8 Tb be the obv10us spheres in

, .. 0,") J) . . J) .J) _ .‘r) J)
.\[1#b“ x b“ and snmlarly .850 and b“ 1n .llg#.5‘ x 5"

.. 2b

Let ”"1 be the cobordism from .‘lll#52 x 52 to M1 given by attaching a 2-
handle to STa' Similarly. define Wg by attaching a 2-handle to 5%“. We now form
the cobordism W 2 W1 U65 ll'g where (z) is a dilleomorl.)hism from M 1#S2 x 5'2 to
Jig-71%."? x 52. By a theorem of \\all. we may adjust our diffeomorphism such that

.0 .0 .0 .
0*( [5 ‘1' (1]) = [5.2"] or [.521]. Therefore. the two spheres to wlnch we attach 3-handles

)

to get Ml and .i/g are 0([Slall and .52“ or 5.20 which intersect algtbraitally onte or
not at all.

[:1

Remark 16. The T(n. l) surgeries coming from h-cobordisms are sulnnanifolds of

Akbulut corks (see. for example [21],). and T(n. 0) surgeries correspond to plugs. [Bl

3.3 Descriptions of the surgery 3-manifold T (n m.)

In the previous section. we showed how we could replace the problem of understanding
simple cobordisms with the problem of understanding surgery along T(n.m). A
first step to understz-mding how 4—manifold invariants change under this surgery is to
understand the Floer homology of T(n. 7”). Before we can compute this. it will be
useful to have as many descriptions of Tt n. m) as possilfle. Above we described it as
. .0. ,. .
the boundary of the plumbing of two 5“ s. They are plumbed n-t1mes geometrically
and algebraically iii-times. Three additional descriptions of the manifold T(n. m)

manifold will be useful to us:

1. Surgmy description. If we change the dotted l-handles in Figure 3.1 into surgery
curves with framing 0. then we have a surgery description of the 3-manifold

T01. In).

30

2. Decomposition into pieces. Let An denote $2 with n open disks removed. and
denote the boundary comptmcnts of An by U122} 5.1-. We can decompose the 3-
manifold Ttn. m) as 5'1 x A), U081 x A,” where (f) : 5'1 x 8.4,, ——> 51 X 8.4,), is.
restricted to each boundary torus. just. one of the two orientation reversing maps

exchanging the factors ( use one of the maps on m of the boundary components.

and the other on the remaining (1: —— '71!) boundary components).
3. Heegaard diagram. This will be exhibited in Section 3.4

From description (2) we see a number of surfaces ccmtained in Ttn. m): There are
the n tori given by 51 x S?- for i = O to n — 1. Call these tori Ti- In the case of
T(‘2n. 0) there are two other obvious surfaces of genus n/L’. R1 and Hg: To form R1.
we cap off 0 x An C T(n. 0) by annuli in 5'1 x Air More specifically. let M be arcs in
A,” such that each component of (7.41;, contains exactly one endpoint of the 0,1; Then
we can arrange C) such that. R1 = 0 X An U Q U i191 x 0,1,- is a closed. orientable surface.

The obvious symmetry of T( 211.0) gives us a second such surface. Rg. See Figure 3.3.

- . r . 9 -9
Lemma 17. o Hg(T(2n. 0)) is freely generated by the 2n — 1 ton {Ttkgtl " and

the surfaces R1 and Hg.
. ~ . —‘)
o Hgf’TOL. 1)) is freely generated by tlgt’f n — 1 ton {TIN-:0"

Proof. This is a simple application of the Meyer—Vietoris sequence. [3

3.4 Heegaard diagrams for 4—manifold theorists

A Heegaard diagram is just a representation of a. 'middle-out' handlebody decomposi-
tion of a 3-manifold Y3. \Ve will restrict our interest to dectimpositions with a single
3—handle on each side of the middle level. The middle level of such a decomposition

. . . . (1 ' ,3.-
is a. surface which we will denote by S. Then i = 113 + E: h.)’ +1 x E + Z 140’ +113.

31

Slx

Ue

   

Figure 3.3: The top figure is a. decomposition of T(4. 0) into two copies of S1 X An
and the bottom figure is the surface R1.

32

Since there are no framing issues to deal with in dimension 3. any such decom-
position can be described by a triple (3.0. 3). where o = UfE-J2107- —- the set. of
homologically independent attaching curves for the 2-handles below the middle level
—— and similarly 13 = U§}:1.3,- the ‘2—handles above the middle. Such a picture is called

a Heegaard diagram. Every closed. orientable 3-manifold admits a Heegaard diagram.

and these diagrams are unique up to a certain set of ‘Ileegaard moves". [40].

3.4.1 ‘Bottom-up’ handlebody descriptions

Before we draw a Heegaard diagram of T(n.m). we’ll first construct a standard
bottom—up handle decomposition using 0. 1. 2 and 3 handles. A 0-handle is just
a 3-ball: rcprcsmit its boundary by the plane plus the point at infinity. It is not nec-
essary to draw the 1-handles. It suffices to draw their attaching region in the plane:
two disks. The 2-handles are also represented by their attaching regions. which are
simple closed curves.

\Ve can also describe relative handlebody decompositions this way by attaching
l-handles and ‘2-handles to a surface. or disjoint union of surfaces. Using relative
handlebody diagrams will make it possible to glue handlebodies together. something

that. is cumbersome with. ordinary Heegaard diagrams.

3.4.2 Induced handle structures on ,S' x 5'1

Given a handlebody decomposition of a. surface S (which may or may not be closed).
we can induce a handlebody deconmosition of S. x 5‘1 as follows.
The snnplest case 1s when .8 is a dlsk D", 1.e JUSt a. 2-dnnens10nal O-l‘iandlc. Then
1 O . . . . .. . ,
S x 1)“ 1s a U-l'iandle. and a l-handle. both of (lnnenslon 3. Slnnlarly. fora handlebody
decomposition of a general surface 5. any k-handlc in 5 gives rise to a k—handle and a

A? + I handle in 5'1 X 5. See Figure 3.4. Also. Figure 361) shows the this for S : :13.

Surface S Induced handlebody handle diagram
decomposition of S X S

Figure 3.4: Crossing 21 surface with S 1

More generally. if we begin with a relative handlebody description of S. (i.e. one
where 0—5 # (0). we see that (9—S X S1 is a collection of tori. On these tori,
our procedure builds a relative Heegaard diagram. by again attaching a, A: and a k + 1
handle to these tori for each k—handle in S. In Figure 3.5 we use a relative l'iandlebody
description of the twice-punctured sphere to find a different picture that also describes

5'1 X Ag. See also Figure 3.6a for S1 X A3.

3.4.3 Gluing handlebodies and constructing T(n., m)

Using this technique. we can find two t’lescriptions of S1 X .43. one with UT a collection
of tori. and the other with ('9— a collection of tori. \Ve can now build. for example.
32 X S1 by superimposing the tori from the second description on top of the boundary
‘o‘io ie's- 1e )0 ma" (_)"a'eSIO' '1'1 ' 'er.'.z s ).‘os'r'(
t 1 ftl fn t Tl l in l Iyt 11 i I an 11 11 Figui 36 1nd then 11 e1 1t1)n
1s glven in F 1gure 3. r.

If. on the other hand. we s1.1)erim)ose the diarrrz-uns. but via different homeo—

. n

morphisms of the boundary tori. then we can form. eg. T(3. 1). where we alternate

34

 

 

 

 

 

>\fi

 

 

 

\ x

/

 

 

... ....~.......

1\
l

 

 

 

 

 

 

 

>> >%

 

 

 

 

‘
0-0.” . .......
.L

4 o
oooooo

  

2-handle

Induced handlebody

. . handle diagram
decompos1t10n of S X S1 0

Surface S

. - . . . . 9
Figure 3.0: Two pictures of .42 X .81. The top has ()+ = T” whereas we are more
interested in the bottom picture. where 8+ = Q)
l_)etween right and left twists. It is straightforward to generalize this to get diagrams

for T( n. m). and more general pictures will be given in Section 3.6

3.4.4 Converting to Heegaard diagrams

To convert these standard handlebody diagrams to Heegaard diagram. we dualize the
1-handles to become rf- curves on a surface. This is illustrated from two different
perspectives in Figure 38 with the pictures on the bottom representing how we will
normally draw our diagrams. The top pictures are to illustrate that. in fact. both

diagrams actually represent a genus—1 handlebody.

3.4.5 Identifying the generators of Hg('T(n. 7a))

~"e wi a 'e. ‘ is o ) )oruni v 0 SIM Oenera ors 0 o " . i ' 1 is eev‘aa (
\\' lltk tli ll t t. t 11 lo 1 fl/(FH 0)) 11 th II D rl
diagram. since we will need them later when we calculate Heegaard Floer homology.

Recall by Lemma 17. 112(T(4.())) is generated by thee tori. plus [1’1 and Hg. The

33

 

 

 

Figure 3.6: One of the tori is shown in the top picture where we have explicitly drawn a
1- and 2-handle. The other tori are represented more typically by the bottom pictures,
where it is understood that part of each torus is contained on the 1- and 2-handles
which are represented only by their attaching regions.

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\

 

 

 

 

 

 

 

 

 

 

 

 

n><n>-<=>—<r5——@ee

 

 

 

 

 

 

 

 

\

m
I
H

 

 

Figure 3.7: The top figure shows how to glue together two copies of 31 x A3. The
bottom left shows the standard gluing which gives 5'1 x 22 while the alternate gluing
on the right gives T(3.1').

37

’13

® ® ®

Figure 3.8: The pictures on the left represents a l-handle attached to a O—handle.
To convert to a Heegaard diagram. dualize to get a 2—handle (plus an unpictured
3-handle) which we attach. in this case. to a torus. Note that we represent. a l—handle
and the surface with the. same notation: a labeled pair of circles.

tori are the tori that we glued along. as mentioned above (Figure 3.6‘). The other two
generators are slightly harder to see. It is shown in Figures 3.9 and 3.10 how R1 and
Hg are constructed from the pieces explained in Section 3.3: pf x .44 and two annuli
from Figure 3.3. The annuli are in one copy of 81 x .44 and pt x A .4 is in the other.

Pictures of the R?- in T(‘21‘i.0} for any integer n are given in Section 3.6

3.5 Background of Heegaard Floer homology

Given Y. a closed oriented 3-manifold. Ozsvath and Szabo define a collection of '3-
manifolds invariants. 11F+(l'). [IF—(Y). IlFx‘O'). and l/IFO'). The original ref-
erence is [.33]. Other surveys are [QTj and [.39].

These invariants are modeled on Lagrangian Floer homology: To a symplc—‘ctic
manifold M with two Lz-igrangian submanifolds L1 and L2. a homology theory can
be defined where the. chain groups are freely generated by the points of LI ('1 L2.
and the differentials are defined by counting J—holomorrmic \Vhitney disks in M with

boundary on Ll M Le.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.9: This is a picture of R1 (shaded) in T(4. 0). (a) .44 x pf is shaded. (b) The
two annuli are shaded. (c) The composite. R1. is shaded with a perforation where
.44 is glued to the annuli.

39

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.10: This is a picture of R2 (shaded) inside T(4. O). (a) The two annuli are
shaded. (1)) pt x .44 is shaded. (c) The composite is shaded with a perforation where
A4 is glued to the annuli.

4O

The Heegaard F loer invariants fit into this structure as follows: Say a given three
manifold Y has a genus g Heegaard diagram (:3. a. B. .2) where a and B are the sets
of attaching curves, and 2 is a point in E \ a U 5. Then Syn-9(2) will play the
role of symplectic manifold in this Floer homology. If we think of SymWE) as the
quotient of Ex”. then the images of Ta := 0'1 X . . . X 09 and T3 2: 81 x . . . x 89 are
transversely intersecting subn'ianifolds of Symgfl), playing the role of Lagrangian

submanifolds.

Definition 18. A l—t’hitney disk between points a and b in Ta 1’) T3 is a map:

a) ; {1: E ClO 3 Beta) g 1} ——> SymWE)

such that
lim (j)(.'r) = a
:r—wo ‘

lim d)(.r) = b
.IY—F—OC'

0(1) 6 T0 for Re(:r) = 0

0(1) E T3 for Reta) = 1

~ I

If. additionally. .] is an almost complex structure on Sy‘m-WE). then Lb is called a

J—h.()lomophic l'l’hritncll disk when q)*J = 2'.

Heegaard F loer homology is only defined for suitably perturbed almost complex
structures. Details can be found in [3:3] and we shall make no further mention of this

technical point.

Definition 19. Let f and 37 he points in To (‘1 T3. Then a2(;i’. :17) is defined to be the

set of homotopy classes of \Yhitney disks connecting 5" to 17 in Symg( 3)

~11

Definition 20. For any point w in Z in the complement of the o and [7’ curves. define

by the algebraic intersection number

We) = #o—luwi x Syn-.a-ltzgi‘)

3.5.1 Definition of Heegaard-Floer homology

Let ($11.6. 2) be a pointed Heegaard diagram for a three manifold Y. The chain
A. .1. _ , . . .
groups of H F. H F ' , H F . and H FDC associat ed to this diagram are defined re-

spectively as:

ZIJETQ-f-TTJ
(.'FOC(a.3.;) = EB 69 zap-.3]
C'F_(a. 3. .2) = {B {B 3[.r.f]
__ CFOCTQ. 13. :)

(..'F+(u. 3 s) _

 

(IF-(as. :)

The differentials are defined by:

(3: CFfo. .13. z) ——> C'F(o'. .13- .2)

re» 2 Z #TItoW)‘

376 TO UT3 (DEW? ( 7.17)
‘ ute)=1

4‘2

and

where [1(0) is the l\Iaslov index of a. the expected dimension of the space of J-
holomorpl’iic \Yhitney disks. and #JTTw) is the count of unparameterized holon‘iorphic
Whitney disks (i.e. mod out the set of .l-ln’)lomorphic \Yhitney disks by the obvious
R action) that are homotopic to o. By various energy bounds. it can be shown that
#JTTM) is finite for a generic choice of almost complex structure on SymWS) (see
[33]. Section 3 for details).

Moreover. by Lemma 3.2 of [33]. [1(0) 7% (Z) only when 713(0) 2 0 (see also Propo-
sition 29 below). Therefore ("F‘— and CF+ are sub and quotient complexes respec-
tively of CF36. and hence the differential 0°C defines all three homology groups
HF+(1'). HF-(l'). and IIFx(Y). That this notation makes no reference to the

underlying Heegaard diagram is justified by the following theorei'n:

Theorem 21. [.35] If (2.0.147). :) is a weakly admissible Heegaard dia._(_)7"(1.m. then
[IF+(Y) and HF(1') are invariants of 1': and if it is strongly adrrzjssfblc. than
H F—(Y) and H F DC are. inferir'iav'its of 1' as well (that. is, they do not depend on

the particular If cry/(111771 diagmm (.r/ioscnj.

“hat is meant by 'weak and strong admissibility" will be defined in Sec. 3.3.3
after a few more preliminaries.
As one further refinement. we remark that the chain complexes decompose accord-
ing to Spin"7 structures on Y. So. for example. ('fota. ‘3. z) = 3‘, w - p ,» C'F'xm. :3. 2.5).
- » ~5€Spm (i )
which gives rise to the decomposition 11/7360 ) = \T (3) H FOUO’. 5). There

44565;)le

4'3

are similar decompositions for the other Floer homology groups. An algorithm for

determining this partitioning will be given in Proposition [?]

3.5.2 Calculating Heegaard Floer homology from a diagram

If we wish to do a ‘by hand calculation of these invariants, there are a number
of things we need to find. First. we will divide tip the generators by their Sptnc
structure. Second. we identify the hmnotopy classes of VVl’iitney disks between all
generators within a given Spfnf' structure. Then. we will calculate the Maslov index
of each homotopy class of disks. And finally. if we are lucky. we will be able to
count the number of J-holomorphic disks the homotopy classes with Maslov index 1.
Techniques for accmnplishing each of these steps are explained below. Proofs can be

found in [33). unless otherwise indicated.

3.5.2.1 Domains

Insofar as \Yhitney disks in Symg are rather ii’iconvenient to deal with. we will pro-
gressively simplify the information they contain. first by discussing an equimlence
between \Vhitney disks and maps into S. and then by ‘discretizing' such maps via a

quantity called the domain.

Lemma 22. [35, Lemma 3.6] There is a. one-to-one correspondence between i'l-"lzxzitrzey
. . V . 7' ‘) . .
dzshs 2n SymWE), and maps of surfaces (1) : F " ——> S such. that the following dzagram
comnmtcs.
r) (5’
F-
l l
T) O ’v ' ‘
D“ ——'> .Stj'rltff($)

where f is a. branched covering map. pl is the projection to the first compmrent, and

C) 2])10 (5),.

44

Such maps can be visualized in 2 by looking at a. discretization called the do-
main which is the algebraic representation of such a map. A surprising amount of
information about the homotopy class of a map c5 6 n9( 1:. y) is contained in this

quantity.

Definition 23. Let E — Uni U 531- = U D). a disjoint union of regions. Then we define
a domain to be a formal sum. 2 Oil)". where a,- E Z.

Moreover. choose a point in the. interior of each region. “I“ E 13-. Now. to a

“z 2
\Vhitney disk 0 E rto(;r. y) we can associate a. domain 17(0) = Z '17:?- ((5)073 Note this

depends on the homotopy class of (f) and not on the choice of the :5 s.

2'

3.5.2.2 Visualizing a2(3:.y) using domains

Using the (TOI‘I‘PSI)()1’1(l(,‘11("€ in the last section. we are now in a better position to
understand aid-123;). In particular. we will see how to construct the domains D(o)

for all (3 6 fight 3)). We begin with the simpler case of n2( :r. .r).

Definition 24. A class 0 E fight. 1:) is called a periodic class ifn;(0) = 0 and 13(0)

is called a periodic domain. The set of periodic classes is denoted HI.

Note that the. boundary of a. periodic. domain is a sum of o- and ,3-curves.

To a periodic class (.9 E H;,;. let 0 be a surface F ——> E given by the correspondence
in Lenmia '22. Since the btnmdary of such a disk is a sum of o and h’ curves. we can
define a closed surface in I" by adding the cores of the 2—handles defined by (.1 and 8.
This gives its a map ”H : ITI ——> 110(1') which. because of the normalizing condition

/ n . u . u
rzgmfi) = 0. is actually an isomorplnsm.

\Vhen g > 2. it can be shown that 7tQ(.S’_z/rizf/(Sq)) = Z. Call a generator of this

group [Sl

J .

. . (5—,; .v' , Q 3., ,1 ,. '7'). . g, x ., If;
PropOSItlon 25. [33) Ilhcn g > _. at. hate that 772(.r..z) is l-bOfltOTpfllc t0 Z<3>

112m.

When 7r-2(:r. y) is non-empty, it is a, homogeneous space modeled on 7T2(.7.‘. .T.)

It is also worth pointing out that the l\"lasloy index is additive under this action.
That, is. Moo + 05) = ,utc'vo) + Me).

The group action is given by concatenation of Whitney disks. Proposition 25 tells
us specifically that if we can find a representation of at least one \Vhitucy disk in
n2(:z:, y). we can find representations for all elements of 772(1‘. y) (and hopefully their
corresponding domains) by adding periodic domains.

For example. if do. 6’) E 772021;) and {hill is a. basis for HT. then
Db!” = DMD) + Z (1f1)(ll‘i)+ le] for some of. s E Z

wl‘iere by [S] we mean simply a sum of all the regions of E \ (Ta (’1 T3)

3.5.2.3 Calculations using domains.

Now we’ll see. how domains can help us conveniently partition the generators of our
chain complex into Spin“ structures as well as calculate the l\lasloy index of a \V’hitney

disk.
Definition 26. Define the order measure of a domain D = Z niDZj by
X([)) = Z 1'2,-(,x(1)21) — 1/4(#of corner points of Bi»
and define
Itsum of multiplic1t1es of D 111 the 4 regions bordering mi)

46

Finally. define the quantity
tiff). :3. fl) = \(D) +,u:f(1)) +H§(1))

. . . . 9
It IS worth pointing out that if D corresponds to an embedded surface F “ whose

. . . . _ . 9
boundary maps to a disJonit union of o- and B-curves. then MD) = xtF“). In

this case. the above definition amounts to a convenient way of calculating the Euler

characteristic of a surface by looking at the domain.1

Proposition 27. [:25] T he Master index of a Il’hitney disk 6) E 772(;r.y) can. be
computed via
MO) = 140(0); 11)

Proposition 28. [34. T/it—sore'm 4.9] If C) 6 new then
(clfsg;).7~t(o)) = /.1([)(o).:r.:13)
More generally '1'..ng E agfx.1'). than.
(01(51'l-Hfo» = II(U(C5)-$-4C) — 271.3(0)

In other words. the 8pm“ structures and the Masloy index can be computed
through completely combinatorial means from a Heegaard diagram.
Say /t(a‘)) = 1. In a few lucky cases we can determine #111 to) just based on the

domain 17(0). This is the final step to finding the differential of our chain complexes.

Proposition 29. o ltUUfJ) contains regions of negative multiplicity. then. #37(0) 2

0.

 

1This simplifies finding. cg. \Ylt’q) in Figure 3.15.

47

o If D(d)) is either a bigon or a square with multiplicity 1, then #Tl(d)) = 1.
o If [)(0) represents a disconnected r zgion in S. then. #JTRQ) = 0.

The first two statements are standard. The third statement has a careful proof in
[38] wherein a number of other cases are analyzed as well. In general. knowing the
domain is insufficient to calculate #il] ((3‘)) since this quantity depends on the almost

complex structure on Sying l E).

3.5.3 Admissible Heegaard diagrams

The Floer homology groups defined above are only invariants if we have. what are
called ‘admissible’ Heegaard diagrams. To understand this necessity. consider the
following situation. Say (:20 E n2(;r,y) has #(00) = 1 and o 6 H55 has ale) 2 0.
Then do * no 6 n2lr.y) has Maslov index 1 for all n. E Z. Since the differentials
in the Heegaard-Floer chain complex are defined by counting holomorphic Whitney
disks in 790;. y) with .\la.slov index 1. a priori our differentials might be infinite sums
in this situation. However. if we begin with a Het-igaard diagram which is properly
'admissible'. we can guarantee that only finitely many homotopy classes of Whitney

disks actually contain a holomorphic representative.

Definition 30. A Heegaard diagram is called weakly admissibly for a Spinc structure
5 if every periodic domain D such that (c1('5)7t(l))) 2 0 has regions of both positive
and negative nniltiplicity.

A Heegaard diagram is called strongly admissible for a Spinc structure 5 if every
periodic domain U such that ((:1f5).7-l(l))> :2 2n. 2 0 has a region of multiplicity

greater than n.

Both. of these conditions imply that for a given it E Z. only a. finite number of
o E n2lrr, 3}) such that 713(0) 2 n. will have Dlo) with only positive regions. Hence.

by Proposition 29. only a. finite number have holomorphic representativcs.

48

3.6 The Heegaard Floer homology of multiply plumbed

spheres

The Floer homology of plumbings is completely understood in a few basic situations.
For example. the three manifold associated to a linear plumbing diagrams is just a lens
space. whose Floer l‘iomology is considered in [34]. In [33]. plumbing diagrams which
are trees are considered. The manifolds considered here, T(2n.0). have plumbing

diagrams that are not simply connected.
We will use the notation in Section 3.3 for representing generators of H 2(T(2n. 0))

Say 5’- E Spinf'(T(2n. 0)) is characterized by:

<ct1(SZ-).[Tj]) =0 ft)1‘allj=0.....2n—- 1

(“llfiilislll = 0

<C1f5il» [32]) = '2'

Theorem 31. IIFOO(T(2H,. 0). 50) is isomorphic to HFOCKT3#('II—1)(S'l X52). 50)—
a standard groap— if the use cocfl‘icients in Z2. Similarly. [IFOC(T(n.1).50) is iso-

morphic to HFOC(#(n — 1)(S1 X 32)) with Z2 coefiicients.

Proof. According to [24]. if Y1 and Y2 are 3-1nanifolds. o : H101) —t» H1(Y2)
is an isomorphism that preserves the triple cup product. and 5 is a torsion Spinc
structure on Y1. then HFOOO’LS) is isomorphic to HFOCO'Q. 0(5)) using Z2 coeffi-
cients. The triple cup product of Tf2n. (l) is characterized by (P[)(R1) U PD(R2) U
PD(T,-).[T(n.m)]) = 1. This is evident from Figure 3.3. If the generators of

I11(T3#(7'l — l)( 51 X 52)) are T1. r213 and S?- for i = l to n — .1. then we have

49

an isomorphism
0:111(T(2n.())) —> 111(T3#(n —1)(s1 x 59))

PD(I?1) r——> 7'1
I’Dllfg) i—> 72
PDlTO) 9—) 7'3
PDij) +—+ T3 + S,- for i = l ..... n —l

The proof for HFoclTln. 1).50) is even easier. By Lemma 1?. H1(T(n.1)) has
vanishing triple cup product and bl = n — 1. the same as H l(#( n — l)S'l x S")

D

2(n—1)

2( 71—1 )unclHF-l—(Tan-Ol'sll :

Theorem 32. Suppose n >_> 2. Then 17F(T(2n. (1)51) = Z

2 72—1
7( ( l)

71-1 fori : 2n — 2. Furthermore, both groups are trivial for i > 271 — 2.

Briefly. we will accomplish this calculation by:

0 Identify the periodic domains.

0 Check that we have an admissible I—lcegaard diagram.

0 Find the points in the Heegaard diagram corresponding to 5271—2-

0 Find domains corresponding to all homotopy classes (I) E 7:9(a. h) for all a. h 6
Ta (1 T3 that correspond to 5211—2-

0 Compute Mo) using the domain and compute #illlo) when plea) ; 1.

In Section 3.4 above. we demonstrated a method for constructing Heegaard di—

agrams for Tfn.m). In Figures 3.11 and 3.12 we exhibit a diagram for T(2n..(l).

50

Following the process in Section 3.4. a diagram for T(2n. 0) is constructed by stack—
ing a piece of type I. then (n — 2) pieces of type II. followed by a piece of type III. We
shall refer to the 3 curve which is dual to a given l-handle by. for example. referring
to the dotted curve encircling the 1-handle ‘a" as 3a.- Notice that we have isotoped
some of the 5 curves to ensure that our diagram will be weakly admissilfle. Curve
‘3 Z has been wound (l — 1)-times around for reasons that will become clear in the

computaticm.
Lemma 33. This H eegaard diagram is wee.ka admissible.

Proof. If we can show that every nontrivial class 01 E Hg; such that. Mair) = 0
has regions of both positive and negative multiplicity. then we have achieved weak
admissibility. Equivalently. assume that for OT, 6 Hg: we have that DEBT) has only
all positive or all negative regions. We will show that [)(o) = 0. Our diagrams have
a number of 'test domains labeled that we will use to accomplish this.

A basis of the periodic domains is given by R1 —l[$]. R2. and T2- for 27 = 0. . . . , 2n. —
2. Therefore we can write 0(051') = 7‘1(R1 — (El) + r2132 + 2:12:62 1.1T]; Then
[40%) = r1/_L(H1)+ r2)i(1?2)+ 2 M73) 2 rl(2n — 2). Since 72 _>_ 2. this can be 0 only
when r1 = 0. It is also true that r2 = 0 for the following reason: The multiplicity
in 13(0) of region I)??? is 7'2 while the multiplicity of region 0%2 is —r2 (see Figure
3.15). N ow our ass1m‘1ption that D((;'>) has only regions of positive multiplicity implies
7'2 2 0.

The reason that the t2- vanish is slightly more intricate. Refer in the following
to Figure 3.13. The multiplicity of region DO in I)((,D) is to. while the multiplicity
of region DC is —{.0. Hence to = O. The multiplicity of region D31 in 12(0) is
tj_1 — tj. Because the multiplicity of this region is non-negative. inch-letively 0 =
t0 2 "'t2n—1 2 t2,,_2. But the n'iultiplicity of D? is ’7' which implies t1- 2 0.
= 0. Therefore the only periodic domain with all nmi-negative regions

Therefore I i

51

is the trivial domain.

C]

We will see that on each i3 curve. there will be at most two intersection points
which will be. used in any generator corresptmding to 597,4). For tidiness. these are
the only intersections labeled on the figures. To refer to specific 1", we introduce

the following shorthand. Order the elements in f in the order given by. e.g. if =

.a a .1)

,1
{.rZ.a.Bl.aCl.....rtB

.123} which we shorthand as a:(1. a. a ...... b. a).
n.—

Lemma 34. The interesction points that correspond to 5212—2 are of the form x(i. . . .)

where i = 1 or 2 and where, there are ('n-J) 12‘s in the string.

Proof. \Ve. claim that these are the intersection points that maximize (('1(5J;). R1).
and we will show this using Proposition 28 and Figures 3.11. Specifically. we will use a
so—called ‘greedy' algorithm: Choose a 8 curve and decide which intersections on this
curve contribute the most to (c1(51~,). R1). Move to the next 5 curve. If the intersec-
tion point that. would contribute most to the sum is still available. then. we will use it.
Then repeat for the rest of the 13 curves each time choosing the intersections that will
n'iaxin‘iize the sum. If for some reason the maximally contributing intersection point
is unavailable (that is. if we have previously used its a curve). then we need to check
that if we reset our greedy algoritlnn. and began the entire process with this D curve.
then we would not maximize (c1(5.;;). R1) — the best choice on this )3 curve does not
maximize the sum. In this case. we choose the next. best. intersections avz-rilable. This
algorithm is carried out in Table 3.1 referencing Figures 3.11 and 3.12.

Conveniently. all such its actually correspond to points in To (‘1 To‘ (a priori such
a string might “double use. an a curve). There are 2”+1 such generators.

—0

Since ;\(l?1) = 1 — 3a. and /I.( R1417. .r) is found by adding up the entries in Table
3.1. by Proposition 28. any such 5 will have ( [1’1 . 5.1-) = 2n —2 if we place the basepoint

:. in region DZ (Figure 3.11).

 

 

 

 

 

 

 

 

 

 

f

.“\ f"

9
.th

fi‘
-

 

Q

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.11: R1 in pieces I and II
3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.12: R1 in piece III

 

 

 

 

 

 

 

Step ,[3- curve possible contribution to maximizes?
intersections MR1. :13. :r)

1 B Z xlz or 1:22 41 — 4 yes

2 8A 1A 2 yes

3 3a

Bbi
BC?- 1' star 2(times (2n — 1)) yes
bid
(.36-
4 3C], 23%, or $127 1 No. There is a point that con-
’ 7 tributes 2 to MR1, x. 1:). How-
ever. this point is on the same
a curve as :rA. therefore if we
chose the maximizing point on
307. we would have to choose a
different point on 3A. the only
other intersection points on 8
either a) are on the same a
curve as 2712 and 3:2. and for
big enough 1. any 5? not includ-
ing those points won’t maxi-
mize; or b) contribute nothing
to the sum, and again one can
check that this creates a deficit
which cannot be overcome.

5 51) at?) or 1:2) 1 No. But they are second best,
and should be used, for reasons
identical to the Sci case.

6 13’ B 2732' or fl}; () None of the intersection points

 

 

 

 

contribute to the sum. These
are the only two intersections
whose 0 curves have not al-
ready been used in previous
steps.

Table 3.1: Algorithm for maximizing ((‘1l5r)- R1)

6- curve intersections contribution to MR0. $.13)

 

 

» T .2
6.4 IA 1
.3... .31.)?” .39. . 13d. 3. 2:5th 1x (27:. _ 1)
, ”v (I .1)
13(1). mcvi . ICV’ 0.2
”a .b
9 .(l ..f)

Table 3.2: Data to calculate (cl(sg;).11’2)

In Figures 3.13 and 3.14 we label the tori T0. . . . .T 271—2 (only some of the cor-
responding domains are shaded because. e.g. the domains To and T1 overlap). One
checks. again using Proposition 28. that (Effigy) = 0. This is necessary. by the
adjunction inequality. for the Floer groups to be non-trivial.

Now we will identify the subset of the :r(. . ) such that (c1(5;1;).1?2) = 0. The
data compiled in Table 3.2 (calculated using Proposition 28 and Figures 3.15 and
3.16) shows that intersections lalmled with superscript. ‘b’ contribute 2 to ya: while
those with ‘a' contribute nothing. If we say I) is the number of intersections with
superscript ‘b" in a given intersection point, then since y-(RQ) = 2 — 4n. we get
that (01131“)- RQ) = 2 — 2n + 21). Therefore the f E Spinf'(52n_2) have exactly
7: —- 1 intersections with superscript b. This implies that in 617(2. a, 6. $52,142) is

2(‘71—1)

Generated by 2( . ) elements.
0 ~ 71—1

[3

Finally we are in position to calculate the Floer homology. \V'e have identified
2(2(“—1)) O'enerators for 677(2) o 13 ~ 6 ) all of the form e 0' r(2 a b a )
72—1 b C. L . . n .l‘ .~.~"?_2 . . z-o. «."_. q . .....
where there are n — 1 Us in this string. If we can show that 0 = 0. then we are
done. To accomplish this. well find the general form of a domain [)(0) for any map

0‘) E 7T2(g1‘(—)..’E(~)).

Consider the domain 1) Z“ We remarked in Proposition 29 that such a domain

56

 

c/

 

 

 

 

 

 

(wit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.13: The top figure has the domain corresponding to T0 shaded while the
bottom picture has T2,-_1 shaded. We have not shaded T2) because it overlaps with

T2'f—l‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U

Figure 3.14: The domain corresponding to T2n—2 is shaded. We have not shaded
T27'z.—1 because it overlaps with T2,,_2.

 

 

 

 

 

 

 

 

 

a
R2
\9\ A
i“. Y
\
\.
\
\
‘1 a 13*

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.15: R2 in pieces I and II of T(2n. 0)

59

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

uc-uuoououonoounufl'

 

 

Figure 3.16: R2 in piece III of T(2n. 0)

60

 

 

 

 

 

 

Figure 3.17: The domain I}-

corresponds to a holomorphic disk. and further we see that this Whitney disk is in
7r2(."c(1. —). 1(2. —)). Similarly the domain —D Z corresponds to a. Whitney disk in
7r2(:r(2. —). .r(1. —)). however it has no holomorphic representative.

Furthermore. consider the annular domain F7- in Figure 3.17. This corresponds to
a. Whitney disk in 7rQ(:r(i. . . . .a. b. . . .). :r(i. . . . ,b. a ..... )) where i = 1 or 2. a \Nhitney'
disk. that is. that leaves the net number of b’s constant. By summing the Whitney
disks corresponding to the. F i- and to DZ we can construct. a Whitney disk between
any two generators of EFKo'. B. 3. 5271—2):

Hence. for (I) E 7rQ(:r(—).at(~)). the domain will have the general form D((_z>) =
r1R1+ r2 H2 + Z fiT-i + Zea-F.) + 6DZ + s[2] where (i is either 1. -1. or 0.

Using the additivity of the Maslov index. and Proposition 27. we have that
Mo) = 7'1(2l+2n —2)+6+2s (3.1)

Furthermore. observe that

713(0) = r11 + 6 + s. (3.2)

We will use these two equations to glean information about i) and 0+ in the

61

following series of claims.
Claim 1. Ifft((.b)=1, then 6 = 1 or —1.

Proof. The only option to rule out is the case 6 = 0. This cannot happen; if 6 is 0.

then by (3.1) we would have that Mei) is even. Cl

Claim 2. If cf) is 0. Whitney dish such. that [1(0) 2 1. and a admits a holomorphic
representative. then 0 is a member of either 7T2(.’L‘(1. —). x(2. ~)) or 7T2(17('2.—)..T(1.~

))-

Proof. This follows immediately from Claim 1. Notice in particular that if 6 = 1. then

to is in 772(r(1. ——)..r(2. ~)): whereas if 6 = —1. then C) is in 7rr2(.r(2. —-)..r(1.~)) Cl

Claim 3. Suppose we have C) E 772(.r(1. —-). .r(2. ~)) such that [1(0) 2 1 and n;(d)) =

0. Then a cannot admit a holomorphic representative.

Proof. Since a is in 7:2(:r(1. —).;z:(2. ~)). we have that 6 = 1. Additionally, Md) —
2715(6)) = 1 implies. by Equations (3.1) and (3.2). that r1(n — 1) = 1. Since n is
positive. this can only happen if n : 2 and r1 = 1.

Suppose this is the case. Equation (3.2) now implies l + 1 + s = O. \Vhich
implies that s is negative. However. out of the regions under consideration. only [3]

conntains the point at. 00. Therefore n r O) = s. which cannot be negative

(pt at oc1}(

by Proposition 2.9 if cf) is to admit a. holomorphic representative. Cl

Claim 4. Suppose we have (I) E r3(:r(1. —). :L‘(2. ~)) such that /.1.(c)) : 1 and 7?;(0) =

1. Then [)(0) = DZ and #TTto) = 1 or —1. That is. c') is in r2($(1. —).;r(2.—))

and admits a unique holomorphic representati've.

Proof. Since 0 is in 772(.r(1. —).:r(2. ~)). we have that 6 = 1. Additionally. ,u.((D') —
2n;(¢)) = -—1 implies by Equations (3.1) and (3.2) that r1(n—1) = 0. Since n > 0. this

im )lies that r = 0. and bv E( uation 3.2 . we can now say that s = (l as well. Hence
1 . .

62

0(a)) is of the form r2 R2 + Z tl-T/ + Z “riff +192. However. if there is a holomorphic
Whitney disk corresponding to this domain. then according to Proposition 29 the
domain .1)(o) must be connected. 1) Z is disconnected rom H1. the P2“ and the Ti-
Therefore r2. the ti’s. and the 75's must. be 0 as well. Hence, 0(0) 2 DZ. and by

Proposition 29 again. we have that #AHO) = 1 or —1 since DZ is a bigon. [:1

Claim 5. For all intersect/rm, points at 2. —) and 117(1. ~). we have that

Z #fita) = 0

Proof. Here we will use the fact that 0+2 = 0. Assume 1' >> 0. Then by Claims 2., 3.

and 4, we have that

0+[T(1—}I]= #TROHHQ ~)1]

a2 ~> ()EW2(:T(1-—) re ~>>
Mel—:1
113(O)=(l

+ Z #JTT(0)[:I:('2 N) (-1]
r(‘2 ~) OEn2(;r(l.—) ”13(2 ~l)
/_L((D\)=1
713(0)=1

+ lower order terms

2 0 + [:r(2. —).1' — 1] + lower order terms

63

Therefore.

E)+2];zt(1. —). 2'] 2: 0+(].r(2. —). 1' — 1] + lower order terms)

#TT(cb)[a'(1. ~).2' - 1] + lower order terms

1(1 ~)C)E7t2(t(2 — .a:(1.~))
[.l.((f))=l
n;(0)=0

- . o
The claim now follows from the fact that (Th = 0.

Now we are able to show that 5 =2 0. Claim 2 implies that any holomorphic
a must be in either 77r_)(:r(1. —)..’r(2.~)) or 7rQ(;r(f2. —).a:(1.~)). For 5. we are only
interested in the Whitney disks where 113(0) 2 0. so by Claims 3 and 5 we have that
#THQS) = 0. Hence (,7 = 0. and the. statement about. HP follows.

Now we can turn our attention to HF+. Filter (7F+(a. 3. $.52.“_2) by defining

One can find the induced differential on the assocrated graded complex T—’—. again
(—1
by looking at domains.

By Claim 2. the only hmnotopy classes of maps that have Maslov index equal
to 1 are in 772(ar(1. —).:r(f2. ~)) and 7m].r(‘2. —).a‘(l. ~)). However. in the associated
graded complex this simplifies. Notice. in fact. that the only non-trivial (:lifferentials
here will correspond to maps in 7r.)(":1:(1. —). 517(2. ~)) where 17.;(95) = 1. By Claim 4.

. . . F,- . . [ Q .
the only non-zero differential induced on 735—— 1s [17(1. —).2] +—-> [a]... —).z — 1] or
(—1
[1(1. —).i] +—> —[.r(2. —).i — 1] given by DZ (the sign depends on how the moduli

space is oriented. but either choice will give the same result). Consequently.

64

F..- 0 when 2' > 0
Z 71-1 when t = 0

 

Associated to the short. exact sequence

'2'

 

 

0 —> Fi—l ——> F,- —> —> 0
i—l
is a long exact sequence
5
->H(F’f_1)—)[”I(Fl')—>H(F )—)
't—l
Using (*). this exact. sequence. and the fact that H (F_1) = O, we arrive at

(201—1))
11F+(T(2n,.0).52n_2):Mutri)=z 72—1 .

3.7 Applications

\Ve offer two calculations of the A invaria-uits. The first is merely to offer some per-
spective on how the relationship between diffeomorphisms and h-cobordisms can be
exploited. The second applies our F loer homology calculation.

Let 65* be an automorphism of H *(M ). Construct the h-cobordism ll '0) as follows:

. . 7 . 7 , J] .2 O . _ .
Let H 1 and H 2 be cobordlsms from M to 11177—5 X 5‘ built. out of a smgle 2-handle.
v . . . o I 1‘.) ‘I2

“all s theorem shows that there is a self-d1ffeomorplnsm q) of .ll#b' X S that
._ .* i. . ,... . -. 2 2 -
induces to on I] (M) and is the identity on the cohomology of S x S . Then define
ll"5 2 W1 U0 H72. By a theorem of Quinn ([37]) and standard surgery theory. it'd)

is determined up to difieomorphism by 0*

Theorem 35. An aatomorphtsm (3* is induced from a dzfieomorphtsm of .«ll if and

only 22f1\(.'\l.ll"‘9..l'\l) = 0

Proof. If 11(111. W“), M) = 0. then W0 is smoothly a product. That is, there is a.
(,liffeomorphism from ( M. I‘ll x I. M ) to ( M. W0. M) which. when restricted to M X 1.
- . * ~* .
induces d on H (M).

Conversely. if a)* is induced from a diffeomorphism of M . then we can arrange the
diffeomorphism a5 : tll#52 x 52 ——> ill-#32 X 82 such that. it is just the identity on
52 x 82. But by construction. this forces the handles of H“9 to cancel.

Cl
In the languaee of we have been usinO‘ here. More‘an and Szabo rove the following
0 O O b O

- , —.——‘) .
Theorem 36. [29/ Say An = CPQfitniI/‘P" where in = (2n + 1)2 + 1. Then there
erist automorphisms 6);: of H *( Xn) such that A(X7-,_. ll"®’7.Xn) is unbounded as n

increases.

From the perspective adopted here. this means that. there exist homeomorphisms
that are arbitrarily far from being diffeon‘iorphisms.

As a. second application of the. A invariants. we derive the. following elementary
relationship between the 4—manifold invariant of Ozsvath-Szaliio. and the complexity

of certain cobordisms.

Theorem 37. Say a simply connected spin 4 -mani fold Ml has a H eegaard-F loer basic
class [h] E H2(X) with dioisibility d. and self intersection 0.
. .r) .0 . . . . , , .9 .0 . .
Say M 1#S“ x .5“ is diffcom(.rrphic to Algae-b" x 5". Then there eatists a cobordism

H” such that AIM/1. ll-'. 21/2) 2 d -+- 2

Proof. A theorem of \Nall says that if two elements of Hg have the same self inter—
section, divisibility. and are either both cln-n'acteristic or regular (not characteristic7
that is). then there is an automorphism of the cohomology ring taking one of these
elements to the other. [43]. \Vall also showed that this automorl)hism can be realized

. . . . . . .0 .0
by a self diffeomorl‘nusm 1f the manifold splits as a smooth connect sum of b" x 5.“

66

and an indefinite manifold. [44]. Note that (l/d)[h] is a. regular homology class. be-
cause it has a dual 2 such that ((1,/d)[h]. S) = l. which cannot happen if (l/d)[h] is
characteristic, since X has an even intersection form.

Hence. there is a diffeomorphism c) : ill2#52 x 5'2 ——> zl‘11#52 x .92 such that
0*([52 >< pt]) 2 (1 / (1)[h]. As in Proposition 1.3. we can construct a simple cobordism
W using this diffeomorphism. Hence. if we take a handlebody decomposition of ll'
where the belt sphere of the 2—handle intersects the attaching sphere of the 3-handle
277 times, then fill and M2 are related by a T(2n.0) surgery in M 1. Moreover. by
construction we have ((l/d)[h]) = H2(D(2n.0)) C 112(.l[1) (recall that D(n.m) is
the 4-manifold corresponding to T(n. m) surgery).

This implies that Floer homology maps that define the Oszvath-Szabo 4-manifold
invariant factor through H F +(T(2n. 0).5d). For this to be non-trivial. the compu-
tation in the previous section requires at 3 2n — 2.

[:1

Examples of manifolds satisfying the hypotheses of this theorem exist. in abun-
dance. Note also that this inequality has no dependence on the manifold Mr). and

that the cobordisms constructed here are never l’i-cobordisms.

67

Chapter 4

Appendix: Idiosyncrasies of the

knot surgery formula

Recall the knot surgery formula. Sl'l'XK = S'll'X - A 1"(2[T]).

It is evident from this result that one can construct infinite families of exotic
smooth manifolds. \Yhat is not prima facie evident is that knots with two different
Alexander polynomials will a/zizays give non-equivalent knot surgeries. The purpose

of this appendix is to clarify and resolve this issue.

Theorem 38. If K1 and K2 are knots with different Alexander p()h_/nomials. then

A K1 and A Kr) cannot be dificomorphic.

This subtlety arises because of the somewhat imprecise way we have described
Sl’l’X E Z[H2(X)] as an invariant of X. It should really be though of as an invariant
up to automorphisms of Z[.1-12( X)].. Here is why: The Seiberg-Witten invariant. is
typically defined as a map Sll' : Spin‘iX) —> Z. We encode this information as
an element of Z[Hg(X’)] by defining S'll'X :2 Z Sll'(5)PD(c1(5)) where the stun is
taken over all Spinc structures on X. \Vhen we do knot surgery on X to produce
XK. our new Seiberg-W'itten invariant Hll'XA, is an element. of lelgt'i‘i'h'fl. \Ve

can think of this as an element in Z[Ilg(.\')] —— which is what we do implicitly in

(38

the knot surgery formula because HQ(X) is isomorphic to H2(XK). In fact, this
isomorphism is cai‘ionical, but only with. respect to the surgery. Different. knot surgeries.
even surgeries that give diffeomorphic manifolds. will induce different isomorphisms
of H2, and hence might manifest the resulting Seiberg—XVitten invariants as different
elements of ZZ[H2( X)]

Consider the following illustrative example: Say X is a 4-manifold containing
two tori T1 and T2 representing different homology classes such that there is a self-
diffeomorphism of X taking T1 to T2. For a single knot K. do knot surgery on T1
and T2 forming X1 and X2. Clearly knot surgery can be performed in such a way
that these manifolds are diffeomorphic. but note that their Seiberg—VX’itten invariants.
as elements in Z[H2( X )J will be different. According to the knot surgery formula,
if c1(5) is a basic class of X. then on X1 we get. new basic classes of the form
c1(5) + n[T1]. whereas our new basic classes on X2 are of the form c1(5) + n [T2].
However. the diffeomorphism of X1 to XQ induces an automorphism of H2(X) that,
takes [T1] to [T2] (and consequently takes Sll-'X1 to Sl-I'XQ).

In the case at hand. where Al‘rl aé A102 and we want to show X K 1 is not
diffeomorphic to X K2. we will need to associate to each element. of Z[HQ(X)] a

quantity that is not affected by automorphisms of H2( X)

Definition 39. Suppose H 2f. X) is torsion free. a is an irreducible element. of Z[HQ(X)]

and a5 is an automorphism of Zlf12(.X.')]. Define a map Fae) : Z[Hg(.\')] —> Z by

.r H :1 of elements of {o'nlofln E Z} that can be factored

out. of x counting multiplicity

'
a

This map is well defined because ZUIQLX )] is a UFD. Moreover. F has the follow-

ing basic properties:

69

q

 

Proposition 40. (i) For (1.. b E Z[H2(X)]. we have T ab) 2 Food”) + 110.0(1))

a.c‘)(
(m; moon/(mi) = I‘MdeQTD) when a e Zl<lTl>l

Proof. Only the third property deserves further comment. Suppose Pa,.@(Ak) >

F

. - (A ). This means that On 0' can be factored out of A . for some integer n
(1.2(l~ l, A 0
such that a" (a) aé a. We will show there can be no such factor.
Since A]; can be factored into irreducibles that are in Z[<[T]>]. we have that
ucb"(a) E Z[({T]>] for some unit. u E Z[I-12(.X')]. Since a E ZMTD]. we can write
nont’o) = u E a;o‘>"([T]f) for a. E Z. This sun'n'nation must. have more than one
term since otherwise g6"(o) would be a unit. Therefore. since m"(a) E Z[<[T])].
.- y 4 .1 , . . .

we have that uon([T]’) = [T]J. and uon([T]’ ) = [T]] for some i # i’ and J # _)’.
_.-l _1 _,," , . . . "- " in i—i’

Therefore. [T] J = u 0([T] ). and this implies that [TV 9 = o ([T] ).

Since of)" must. preserve degree. we get that that o’)([T]) = [T]. and hence 077(0) 2 o.

C]
These properties are sufficient to prove theorem 38.

Proof. 38 Assume A K1 71 AK? but that X K1 is ('liffeomorphic to X K2. We will
derive a contradiction. According to the knot surgery formula. the Seiberg-XVitten
invariants of X K1 and X532 are Sll’X - A K1 and S ”TX - Ali"). respectively. A
diffeomorphism o : X 11‘" 1 —> X K2 induces an automorphism 0* : Z[I—l2(XI\'1)] —>
Z[H2(X;\'gll where 03*(S'll’X - A K1) = s11 \ . AK?

Since A [1'1 74- A 1(2‘ we can choose a to be. an irreducible element of ZZ[([T])]
that divides A K1 with a greater multiplicity than it. divides A KO' In other words
FNMA/{1) > [surfing-

In. fact. Via property (111) we have Fo.c)*(Al\'1) > FOGJAKQ).

70

To the equality 0*(SII'X - 331(1) 2 Sll’X - A113) we apply Fae)... (here short-

handed as F) and use properties (i) and (ii) above:

I“(o'*(SH’X AK.» = I‘tSH'X AK.)
r(’a..(sng\r)) + Nanak/1)) = mswx) + HAN.)
MK.) = m 1.22)

This. however. contradicts our choice of a.

Cl

Remark 41. An essential l'iypothesis of this theorem was that H 2(X ) be torsion free:
Otherwise Z[HQ(X)] is not a UFD and we cannot define F. Note. however. that in
the case H2(X) has torsion. the same proof can be carried out as long as the image
of SWX in ZN! 2(X _) / tor] is non-trivial. Simply replace every instance of Z[H2(.X')]
above with Z[}l2(X)/tor].

Remark 42. The above proof can also be applied to rim surgery to show that any two

knots with different Alexander polynomials will give rise to inetniivalent rim-surgeries.

I9]

[101

I11]

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