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The Seiberg-witten Theory
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THE SEIBERG-WITTEN THEORY OF HOMOLOGY

3-SPHERES

By

Weimz'n Chen

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1998

ABSTRACT

THE SEIBERG-WITTEN THEORY OF HOMOLOGY 3-SPHERES

By

Wez’mz’n Chen

In this thesis we study the Seiberg-Witten theory of an oriented homology 3-
sphere. The goal is to extract t0pological invariants ——- the Seiberg-Witten invariants
— by counting the solutions to the Seiberg—Witten equations on the manifold. The
first question we consider is whether the Seiberg-Witten invariants depend on the
geometric or analytic data involved in their definition. In the first main result of this
thesis, we completely determine the dependence of the Seiberg—Witten invariants on
the data involved in their definition. In particular, we show that even for the simplest
manifold, the 3-sphere S 3, the Seiberg-Witten invariants take infinitely many different
values.

The rest of this thesis is devoted to understanding the Seiberg—Witten invariants
in a specific geometric setting — the surgery setting. In that context we prove a
gluing formula, which identifies the Seiberg-Witten invariants as certain “homological

intersection numbers”.

To Rong, Xiu and Peter

iii

ACKNOWLEDGMENTS

This thesis grew out of an unsuccessful endeavor searching for a homology bor-
dism invariant lifting of the Rohlin invariant of an oriented homology 3—sphere via the
Seiberg-Witten theory. The existence of such an invariant would imply that there are
no 22 torsion elements with non-zero Rohlin invariant in the 3-dimensional homology
bordism group, which in turn would imply that not every higher dimensional topo-
logical manifold is simplicially triangulable. I am very grateful to my thesis advisor
Professor Selman Akbulut for suggesting that I work on this problem and for sharing
with me his ideas of using gauge theory. I have benefited greatly from numerous
discussions with him in the past three years. I would not have been able to survive
the hard work in these years without his patience, encouragement, and support.

I would also like to thank other members of my Thesis Committee, Professors Ron
Fintushel, Tom Parker, Jon Wolfson and Zhengfang Zhou, as well as other faculty
members in the department for their generosity in educating me, their interest in my
work, their help, and the useful conversations with them. Thanks also go to Professor
Wei-Eihn Kuan, Director of Graduate Studies, for his generosity and support in these
years, and to the Department of Mathematics at Michigan State University for her

financial assistance during my participation in various conferences. While working

iv

on this thesis, I was awarded with a Summer Acceleration Fellowship in 1996 and a
Dissertation Completion Fellowship in 1997 from the Graduate School.

During the years of my graduate study at MSU, I was fortunate to meet sev-
eral outstanding fellow graduate students. I benefited greatly from interactions with
them. I especially wish to thank Liviu Nicolaescu who has taught me my first lessons
in elliptic PDE, and Slava Matveyev who has taught me a great deal of geometric
topology and whose generous help has made my life a lot easier.

Finally thanks to my family and friends, especially my wife Zhaorong, without

whose love, understanding, and support, my Ph.D would not have been possible.

TABLE OF CONTENTS

INTRODUCTION 1

1 Topological Invariance

1.1 Seiberg—Witten theory in dimension 3 .................. 5
1.2 The definition of X and a ........................ 13
1.3 Topological invariance of a ........................ 17
1.4 Perturbations of Dirac operator ..................... 24
1.5 The a-invariant perturbations ...................... 34
2 Seiberg—Witten Equations on Cylindrical End Manifolds 38
2.1 The Fredholm theory ........................... 39
2.2 Perturbation and transversality ..................... 48
2.3 The finite energy monopoles ....................... 53
3 The Gluing Formula 63
3.1 The gluing of moduli spaces ....................... 63
3.2 Geometric limits ............................. 76
3.3 Spectral flow, Maslov index and the gluing formula .......... 85
APPENDIX A 97
APPENDIX B 101
BIBLIOGRAPHY 110

vi

Introduction

Let Y be an oriented homology 3—sphere, i.e. H...(Y) :2 H..(S3). Equip Y with a
Riemannian metric go. The unique spin structure on Y gives rise to a (unique) 5' U (2)
vector bundle W on Y such that the oriented volume form of Y acts on W as identity
by Clifford multiplication. Consider pairs (A,tb) where A is an imaginary valued
1-form on Y and 1,0 is a smooth section of W'. The 3-dimensional Seiberg-Witten

equations for (A, 1/2) read as

Dgow + Aw = 0
*dA + 7(1/2, 11)) = 0.

Here D90 is the Dirac operator on Y associated to the metric go and T(-, ) is a certain
bilinear form on F(W) with values in the space of imaginary valued 1-forms on Y.
The group of gauge transformations 9 (Y) = M ap(Y, S 1) acts on the pairs (A, 2/2) by

the following rule:

8 - (A,¢) = (A — s—lds, 3w) for s E 9(Y).

The Seiberg-Witten moduli space M(Y) is the space of gauge equivalence classes of
solutions to the Seiberg-Witten equations (these solutions are called monopoles). It
is compact and has virtual dimension zero.

The algebraic count of the elements in M (Y) is called the Seiberg—Witten invariant
of Y and is denoted by x(Y) throughout. M(Y) can be regarded as the set of critical
points of the Chern-Simons-Dirac functional and x(Y) its Euler characteristic.

The first question we consider is whether the Seiberg-Witten invariant x(Y) is
independent of the data involved in its definition, such as the Riemannian metric on
Y and the perturbations of the Seiberg-Witten equations. Unfortunately, the answer
to this question turns out to be negative. To be more precise, suppose that the
oriented homology 3-sphere Y bounds a smooth spin 4-manifold X endowed with a

Riemannian metric which is a product near Y. We set
7 - 1 - ,
a(Y) = x(l')—(1ndexDX + §SIgn(A)),

where D X is the Dirac operator on X defined with the APS global boundary condition
([2]) and Sign(X) is the signature of X.

In Chapter 1, we give a rigorous definition of x(Y) and a(Y) and prove the
following theorem.

Theorem A

Let Y be an oriented homology 3-sphere. Then
1. a(Y) is a topological invariant of Y, and a(Y) + a(—Y) = O.

2. 0(Y) E ,a(Y) (mod 2), where ”(Y) is the Rohlin invariant of Y.

The Casson’s invariant satisfies both these properties. Thus this result strongly
supports the recent conjecture of Kronheimer and Mrowka ([18]) that a(Y) equals
Casson’s invariant of Y.

In order to define x(Y), we need to consider the following perturbations of the

Seiberg—Witten equations:

Dg¢+Azl+fw=0

*dA + T(’l,/), 7,0) + [L = 0,

where g is a perturbation of the metric go, f is a real valued smooth function on Y and
a is a small, co—closed, imaginary valued l-form on Y. The topological invariance of
a(Y) is roughly saying that the space of pairs (9, f) has a chamber structure and the
Seiberg—Witten invariant x(Y) depends only on the chamber of the perturbed Dirac
operator D9 + f (assuming the perturbation a is small). In [13], Hitchin studied a
family of metrics on S 3 which shows that the Dirac Operator associated to this family
of metrics has infinitely many different chambers. Using Hitchin’s observation, we
show that even for the simplest 3-manifold, S3, the Seiberg—Witten invariant x(S3)
takes infinitely many different values.

The rest of this thesis is devoted to understanding the Seiberg-Witten invariant
x(Y) in the following geometric setting. Assume that Y is decomposed into a union
of two submanifolds Y1 and Y2 by an embedded torus T 2 where Y2 is diffeomorphic to
D2 x S 1. We put a Riemannian metric on Y such that a collar neighborhood of T2 is
isometric to (—1, 1) x R/27rZ x R/27rZ and Y2 carries a metric whose scalar curvature

is non-negative and somewhere positive. By inserting cylinders [0, 2L + 1] x T2, we

obtain a family of stretched versions YL of Y. Our goal is to express the Seiberg-
W itten invariant x(YL) in terms of Y1 and Y2 when the neck is sufficiently long. We
regard YL as a result of cutting and pasting of two cylindrical end manifolds obtained
by attaching infinite cylinders to Y1 and Y2 (still denoted by Y1 and Y2 for simplicity).
It turns out that the (finite energy) Seiberg-Witten moduli spaces of the cylindrical
end manifolds Y1 and Y2 are generically 1-dimensional manifolds which are immersed
into the space of equivalence classes of flat U (1) connections on T 2 via a map which
sends a finite energy monopole to its limiting value at the infinity of the cylindrical
end. After fixing orientations, these moduli spaces define an “intersection” number
#S(Y1, Y2), which we prove equals to the Seiberg-Witten invariant x(YL) when the
length of the neck is large enough. This result is refered to as the gluing formula of
x.

Theorem B

For large enough L, x(YL) = #S(Y1, Y2).

In Chapter 2, we set up the Fredholm theory for Seiberg—Witten equations on
cylindrical end 3-manifolds. The issue of perturbation and transversality, and analytic
properties of the finite energy monopoles such as exponential decay estimates and
“compactness” are discussed. The gluing formula is proved in Chapter 3.

Two technical results needed in Chapters 2 and 3 are included as Appendice A
and B.

Part of this thesis has appeared in the Proceedings of 5th Gokova Geometry-

Topology Conference (1996) ([6],[7]).

CHAPTER 1

Topological Invariance

1.1 Seiberg-Witten theory in dimension 3

Let Y be an oriented homology 3-Sphere equipped with a Riemannian metric 9 (many
facts stated in this section hold for general 3—manifolds). There exists a unique
SH (2) vector bundle W0 over Y as a Clifford module of the Clifford algebra bundle
Cl (T Y) (293 C such that the oriented volume form on Y acts as identity on W0. Let
W 2 W0 69 L, where L is the trivial complex line bundle over Y. W is a U (2) vector
bundle.

Let (e1, e2, e3) be an oriented local orthonormal basis of T‘Y. This gives rise to a
local unitary basis of W0 and W, within which the Clifford multiplication is given by

the following matrices:

Let i0 = (21,10), ab = (u,v), i0,¢ E W, we define

1 Re(zfi - an?) 2.17 + 211v.
7W), (15) = g

2v + ”LU’U —Re(zfi. — 1027)
It is straightforward to Show

Lemma 1.1.1 ir('z0,q$) = %(Re(zfl—wv)(e1)+Im(zv+v‘m)(e2)+Re(z27+zDu)(e3)),

so r00, d5) 6 A1(Y) <8) iR. Moreover, we have

(i6 - 10, ¢)Re = —2<6a ZW11), 9(9))

for any 6 6 A10”), and WIMP)? = ill/44-

The Levi-Civita connection of the Riemannian metric g lifts to a connection on
W0. Coupled with a U (1) connection A on the complex line bundle L, the Dirac

operator DA: F(W) ———+ I‘ (W) is given in a local frame by

3
DA 2 Zej ' (V3). + ZAJ').

i=1

Let A = C x I‘(W) where C is the space of smooth U (1) connections on L. The
gauge group g = Map(Y,Sl) acts on A by s - (A, 7,0) = (A — s‘lds,sv,0), s E g,
(A, 7,0) 6 A. Note that mg) = H1(Y, Z) = 0. Each element in Q can be written as 6"
with f E F(A°(Y) <8) iR) determined up to a constant 27rik, k E Z. So 9 = K(Z,1).
Let B = A/g. The action of g is free on the subset A* = A\ {20 E 0}, and with

stabilizer 51 on the rest. Hence 8* = A‘/Q is homotopic to CP°°.

We shall work within the context of Sobolev spaces and Banach manifolds. By
fixing a trivialization of L, C can be identified with {21 (Y) ®iR, the space of imaginary
valued 1-forms on Y. Define A2 = L%(A1(Y) ® iR) x L§(W0), 912 2 {L3 maps from

Y to S 1}. For simplicity, we still use the old symbols to denote the Sobolev objects.

Lemma 1.1.2 8* is a Banach manifold whose tangent space at (A, w) is
T316130) = {(a,¢) E A] — d‘a + i(ii0, <0)Re = 0}.

Proof: Standard arguments. The key point is that the operator d*d+ |i0|2 is invertible
if 20 is not identically zero. See [12]. [:1
Remark: A neighborhood of [(A, 0)] in B is diffeomorphic to U /Sl, where U =
{(a, 45) E Ald‘a = 0, ”(0. ¢)|| < 5}-

There is a natural Z4 action a on A given by 0(A, 10) = (—A, J20), where J is the
quaternion structure on W0. The action a descends to an involution on B and acts
freely on 8*.

The Chern-Simons—Dirac functional on A is defined by
1 l
csvm, it) = —5 f} A /\ dA + 5 [yo/i, DAo),Revozg,

which is gauge invariant and descends to B. It is also o-invariant. The gradient of

CSD at (A,1,0) is given by

3(A, it) = (*dA + rel), w), Div)-

It can be regarded as a ‘weak’ tangent vector field on B“ in the sense that it is not in
TB“ but in its L2 completion C, i.e., LOW) = {(a, (0) 6 L2] — d*a + i(i10, (0)36 = 0}.

The covariant derivative V3 is given by

V3(A,1/J) (a, Cb) : (*da + 27111145) _ df(¢)a DA¢ + 0.1,!) + f(¢)bb)

where f(q§) is the unique solution to the equation (d‘d + [w]2)f = i(iDA7,0, (0)38. As

in [29], we have

Lemma 1.1.3 V3(A,w) defines a closed, essentially selfadjoint, Fredholm operator on
£01.10); and its eigenvectors form an L2-complete orthonormal basis for 504.10% The
domain of V304,,” is the Lf—Sobolev space completion of £(A.¢)- The eigenvalues form
a discrete subset of the real line which has no accumulation points, and which is

unbounded in both directions. Each eigenvalue has finite multiplicity.

The 3-dimensional Seiberg-Witten moduli space M is the set of critical points of

CS’D on B, i.e. the equivalence classes of solutions to the Seiberg—Witten equations

mm + ml, 20) = 0
DW = 0.

Let [0] denote the unique reducible solution [(0,0)]. Then the moduli space of irre-

ducible solutions is M“ = M \ [0]. As in [17], we have

Lemma 1.1.4 The moduli space M can be represented by smooth sections and it is

compact.

In order to define the Seiberg-Witten invariant, i.e. the Euler characteristic of

C8D, we need suitable perturbations of C8D.

Definition 1.1.5 A perturbation CSD' of C823 is admissible if:

1. The critical points ofCSD' in 8* are non-degenerate, i.e. VSIKAAOH is invertible

at [(A,10)] E [3* {ls—1(0).
2. The Dirac operator at the reducible [0] is invertible so that [6] is isolated.

Here 3’ is the gradient of CSD' and Vs' is the covariant derivative of s’. The Dirac

operator at [0] will be clear when we specify the perturbation.

An admissible perturbation has only finitely many isolated critical points in B‘.
This is because the reducible [6] is isolated so that M“ is compact. Each irreducible
critical point is assigned a Sign by the mod 2 spectral flow of Vs’. Since 7r1(B*) = 0,
the spectral flow does not depend on the path chosen. See [29].

We will consider two classes of admissible perturbations. The first class is o-
invariant. First we need to perturb the Dirac operator so that it is invertible and still
quaternionic. These perturbations take the form of D9 + f where g stands for the
metric and f is a smooth real valued function on Y. The perturbed Chern-Simons-

Dirac functional takes the form of
1
CSD’(A,10) = CSD(A,1,0) + i [Y flwlgvozg + u,

where u is some functional on B which will be constructed in Section 1.5. The

10

corresponding Dirac operator at the reducible [6] is D9 + f. For convenience, we set
mpg/1,10): CSD(A,)+10 —21—/ f|u|§vozg.

The following proposition is proved in Section 1.4, in which Met stands for the

space of metrics.

Proposition 1.1.6 Let Y be a closed oriented 3—manifold. For a generic pair (9, f) 6
Met x C"(Y), the perturbed Dirac operator D9 + f is invertible. Moreover, any two
such regular pairs (g0, f0) and (g1, f1) can be connected by a generic path (gt, ft) such
that the perturbed Dirac operators D9, + ft are invertible except for t,- 6 (0,1) with
Ker(Dg,i + ft.) = H, i = 1,2,. . . , n. Let At,1,0t be the eigenvalue and eigenvector near
ti, i.e. (D9, + ft)10t = Atwt with At, = 0 and ”104le = 1, we have

Cid/it“): /<%(Dge+ft)(t Wt ),10t,.>3.¢0.

As a corollary, the spectral flow of D9, + ft at t,- is :l:4 for i = 1,2,. . .,n

The next proposition concerning the existence of o-invariant admissible perturba-

tions is proved in Section 1.5.

Proposition 1.1.7 F is: a regular pair (g, f) so that the reducible [6] is isolated. There
erist o-invariant admissible perturbations of C8D] which are supported in the comple-
ment of [6] and the non-degenerate critical points of CS’Df. Any two such admissible

perturbations can be connected by a path supported in the complement of [6].

11

The second class of admissible perturbations of C81? has the form of
CSD’,,(A,10)= csv(.4,a) — [Y A /\ *p.

where ,a is a generic imaginary valued co-closed l-form. The gradient of C817,, at
(A10) is
s',,(A,10) 2 (MA + r(10,10)+ a, 0,110).

CSD'y has a unique reducible critical point [6,,] = [(a,,,0)] where a” is the unique
solution to the equations *dap + u = 0 and d“a,u = 0. The covariant derivative Vs’ u

is given by
Vs’#,(A,¢)(a, (0) 2 (*da + 27(10, <0) — CUM), DA¢ + a10 + “(MW

where f ((0) is the unique solution to the equation

(dltd + [WU], = ifiDAt/J, ¢>Re-

The corresponding Dirac operator at [6”] is D“ = D + 0.“.

Proposition 1.1.8 For a generic 11, C817,, is admissible. Moreover, any two such

regular no and 111 can be connected by a path m, t 6 [0,1], such that
1. 3’“, is transversal to the zero section of the Hilbert bundle C over 8* X [0,1].

2. D,“ is invertible for all but finitely many points t,- 6 (0,1) with KerDmi = C.

Moreover, if At and 1,0, are the eigenvalue and eigenvector of Du, near t,-, i.e.

12
Dm10¢ = /\t10t with ||10t||L2 = 1 and At, = 0, then

d/\t

Kai) : /,,<%(Dm)(ti)¢'t,,10i,)38 7/: O,

In particular, the spectral flow of Dru (as complezr linear operators) at t, is equal

to :tl.

Proof: The “universal” gradient s(u, A, 1,0) = (*dA + 7(10, 10) + u, DA10) is a section
of the Hilbert bundle .C over Ker d“ x A“ which is transversal to the zero section. So
34(0) is a Banach manifold, and so is s‘1(0)/g. The projection P: s‘1(0)/g —-> Ker
d“ is a Fredholm map of index 0. So for a generic 11, Vs’u is invertible at s’;1(0), and
any two such regular #0 and ,ul can be connected by a path pt, t 6 [0,1], such that
3'”, is transversal to the zero section of the Hilbert bundle C over 3* x [0, 1].

Consider the real Hilbert bundle C over Ker d* x (L¥(Wo) \ {0}) given by SW,» =
{d E L2(Wo)|d> is orthogonal to i112}. Then L(a,10) 2 D10 + a1,0 is a section of 8
which is transversal to the zero section. Therefore L_1(0) is a Banach manifold. The
projection II: L’1(0) —> Ker d" is a Fredholm map of index 1. Since Da = D + a is
complex linear, by Sard-Smale theorem, for a generic a E Ker d", H"1(a) is empty,
i.e. D0 is invertible. Two such regular a0 and a1 can be connected by a path at which
is transversal to H. We can take an analytic path at so that for all but finitely many
points ti, Da, is invertible and KerDmi = C by index counting. If D030, 2 Ari/1t with
[I10tIIL2 = 1 and At, = 0, then

dAt d
WU") :f},(a—t(Da,)(ti)t/Jtn1/Jt.)ae-

13

Since at is transversal to the projection II, fy(§;(Da,)(ti)wi,a 10m. # 0. El

Remark: The same conclusions hold if we also allow the metrics to change.

1.2 The definition of X and a

Fix an admissible perturbation CSD' of C8D with gradient 3’. Denote the Dirac
operator at the reducible critical point [6] by D’. Let M“ = {[(A,1,0)] E B*|s’(A, 10) =

0}. We define for 63- E M“,

xi: 2 (_1)SF(fi,-,B.)
fliEA/I‘

where SF (13,-, 6,) is the spectral flow between Vs’gj and Vs’m. As in [29], it is easy
to show that [le is independent of the choice of 63-. In order to give a sign to [X3],
we need to fix a sign near the reducible critical point [6].

At (A, 10) E A‘, we have a short exact sequence

0 —> To... dis) TA“ —"—‘+ TB“ —> 0
where d(A,,,',)(f) 2 (—df, f10) and 7r : A“ —> 3*. This enables us to extend any
endomorphism of TB“ to a Q-equivariant one of TA EB TQM. An endomorphism L of

TB“ is extended to
L 0 O

IC’L= 0 0 at“, ,
0 arm 0

an endomorphism of TA {9 Tom 2 TB‘“ 69 Im(d(,w)) GB TQM. IC'L is self-adjoint if and

14

only if L is. For L = Vs’p, we use IC’ for IC’L.

At (A, 10) E A, we define a self-adjoint endomorphism of TA 69 T9"):

[C(A’w)(a, 65, f) 2 (*da + 27-(1/)) 95) — df, DA¢ + (III/2 + flab, —d*(1, + IOU), ¢>Re)

01'

As in [29], we have

Lemma 1.2.1 For smooth (A, 0) E A, [C(Asb) extends to L2(A1(Y) <8) iR EB W0 69
A0(Y) <8) iR) as a closed, essentially selfadjoint, Fredholm operator. It has discrete
spectrum with no accumulation points, and each eigenvalue has finite multiplicity.
The spectrum is unbounded from above and below. The same holds for [Cl/4.10) if
(A,10) E A“. Moreover, one can replace Vs’ by [C for the purpose of computing the

spectral flow.

For any (a, (0) E A“, we need to study the small eigenvalues of ICt(a, d) 2 [Co +

tC(a, (0) as t ——> 0 where

D, 0 0 a ¢. ¢.
Ice: 0 *d -—d ,andC(a,¢)= 2T(¢,.) 0 0
0 ——d* 0 than... 0 0

Here D’ is the Dirac operator at [6] which is invertible. [CO has only one zero eigen-

15

vector which is the constant function i. lCt(a, 90) is expected to have exactly one small

eigenvalue At which is analytic in t as t —> 0. See [14].

Lemma 1.2.2 MO) = 0, MO) = —2 MD'qi, 3),... where is = (D’)_1(igb).

Proof: For simplicity let Kt = lCt(a, (0), C = C(a, (0). Suppose (Kt -— At)ft = 0 where
“ft” = 1, f0 = i. By differentiating the equation, we have

(C — )lt)ft + (Kt - At)ft = 0-

30 A: '-‘ (C(ftlaftla and MO) 2 (C(i),i) 2 (will 2 0- K0(ft(0)) : —C(f0) = ‘W-
Le a3 = (D’)‘1(z‘<b), then X40) = (Comm) + (Ovation = manta}...
Cl

Corollary 1.2.3 For a generic (0, Xt(0) ¢ 0. At ~ /\t2 where /\ = —fY(D’(b,<b)Re
and <0 = (Db-10¢)-

For 63- E M‘“, we define

sign(,6j) = —sign(/Y(D’(Z;, @126) . (_1)SF(fij:¢)

for a generic g0, where SF(l3j,¢) is the spectral flow between [Cg]. and ICt(a,(,/>) for
small t.

Definition 1.2.4 X = sign(6j) ~Xj.

It is easy to see that sign(6j) is independent of (a, g0), and X is independent of 6,-
as in [29].

16

Lemma 1.2.5 X(Y) = —X(—Y), and X E 0 (mod 2) if CSD’ is a o-invariant

admissible perturbation.

Proof: W0 still can serve for —Y if we change the Clifford multiplication by a factor
of —1. Under this change, CSD’(Y) = —CSD’(—Y), Vs’(Y) = —Vs’(—Y), M(Y) =
M(—Y), and fY<D'43,t)Re = — f_Y(D’q3,gb)Re. So X(Y) = —X(—Y). The other
statement is obvious. Cl

Let X be a smooth compact spin 4—manifold with 6X : Y. Equip X with
a Riemannian metric such that a neighborhood of Y is isometric to (—1,0] x Y.
Suppose DX is a perturbed Dirac operator on X which takes the form

c(dt)(% + D’)

near the boundary Y. Here D’ is the Dirac operator at [6] for an admissible pertur-
bation of the Chern-Simons-Dirac functional and takes the form of D9 + f + a where
a is a co—closed imaginary valued l-form, 9 stands for the metric and f is a smooth
real valued function on Y. D’ is invertible. IndexDx is the L2 index if an infinite
cylinder is attached to X, or the index of DX satisfying the APS global boundary

condition.

Lemma 1.2.6 ([2]) Indeer + éSign(X) is independent of X, and
1 1
(IndexD; + gSign(X)) — (InderD§( + gSign(X)) .—_ —SF(D’1, D1),

where D’ = c(dt)(% + D1) near Y. In the case that a = 0 and (g,f) is a regular

pair, Indeer -+- éSign(X) _=_ u(Y) (mod 2) where ,u.(Y) is the Rohlin invariant.

17
Indeer + éSign(X) changes by a factor of —1 if the orientation of Y is reversed.

Definition 1.2.7 For any admissible perturbation, define
1 .
a = X — (Indeer + gSzgn(X)).

Here DX takes the form of c(dt)(% + D’) near Y where D’ is the Dirac operator at

the reducible critical point [6] associated to the admissible perturbation.

1.3 T0pological invariance of a

In this section, we shall prove that a is independent of the choice of the Riemannian
metric and admissible perturbation.

Given any two metrics and admissible perturbations CSD’M, i = —1, 1, we can
connect them by a path CSD’“, t E [—1, 1] for which Proposition 1.1.8 holds. We

only need to consider the following two situations:
1. D“, is invertible for all t.
2. D“, is invertible for all t but t = 0.

Here D“, is the Dirac operator at the reducible point [6“,]. In the first case, IndexDx+

fiSign(X) does not change, neither does X. In fact, we have

Lemma 1.3.1 Suppose two admissible perturbations p0 and #1 are connected by a
path at which provides a partial cobordism Z between part of M5 and part of MI. If
60 G M5 is cobordant to {31 E M; via Z, then SF(1’30,)61) is even. If [30 6 M5 is

18

cobordant to 61 6 M5 via Z, then SF(60,61) is odd. Here SF(60,,61) stands for the

spectral flow between Vs’fio and Vs’gl.

Proof: The lemma follows from the fact that the cobordism Z can be arranged so
that the projection from Z to [0,1] is a Morse function. See [9], p.143. [3

In the second case, IndexDX + %Sign(X) changes by i1. We shall prove that X
also changes by 21:1 which is compatible to the change of IndexDX + fiSign(X) so
that a remains unchanged. This is done by analyzing the Kuranishi model near the
reducible point at t = 0.

Nonlinear F redholm maps between Hilbert spaces admit local reductions to finite
dimensional maps. Suppose \Il: X —> Y is a nonlinear Fredholm map satisfying
\II(0) = 0. Let T = (d\I')0. Then there are splittings X = Ker T EB (Ker T)i,
Y = I mT EB CokerT and a map 1,0 : X ——> CokerT so that ‘11 is equivalent to T + 10
near 0 by a diffeomorphism of X, and 10(0) 2 0, (d10)0 = 0. Moreover, \II“1(O) is
diffeomorphic to {i/JlKei-r = 0} near 0. If there is a group action, the above can be
made equivariant.

The detailed construction goes as follows. Let 77k : X —> K erT, 7rc : Y —>
CokerT be the orthogonal projections. Then X : X —> X given by X : a: —+ 7rk(:1:) +
T“1(1-7rc)(\Il(a:)) is a local diffeomorphism at 0. Define 10(y) : 7rC(\II(X‘1(y))). Then
‘11 o X—1_—_ T + 10, and \II’1(0) = {10lKerT = 0}. See [12].

Suppose two admissible perturbations 11-1 and pl are connected by a path at,
t 6 [—1,1], in the sense of Proposition 1.1.8 and D“, is invertible except for t = 0.

We will study the Kuranishi model near the reducible point at t = 0 of the following

19

family of Seiberg—Witten equations

*tdA + WW), 10) = 0
(Din + AMI} : 0

where A E Ker (1“. Here d* stands for d"‘t at t = 0.

Consider map \II : R Q L1(Ker d“ Q W0) —+ L2(Ker (1* Q I/Vo) given by

‘1’”: A316) : (W(*tdA + ”(ll/)1 76)), (Due + AMP)

where 7r : 91(Y) <8) iR ——> Ker d* is the L2 orthogonal projection. Then Ker (d\II)0 =
R Q Ker D0, Coker(d\Il)0 : Ker D0. Here D0 stands for Dflo. Write 10 = 100 + 101
where 100 E Ker D0 and 101 E (Ker D0)i, then we have a local diffeomorphism

x : R ea L§(Ker d* a W0) —+ RQ Liam d* 69 W0).

Xi(t,A,1/Jo+i/11) ‘2 (t,(*d)_1(7r(*tdA+rt(1/)0+101,100+101))),

100 + D1)—1(1— 77/6000: + AlWO + ‘61)»,

and X-1(t7 0: 160) : (ta/49160 + 161) Where A : 1403160), lit/)1 : $105,100) satiSfy

A + (7r*td)'1(7rn(100 + 101)) = 0
161+ D1)—10‘ 77k)(Dm — D0 + Alfl/JO + 101) = 0-

Lemma 1.3.2 (0,, + Av, 100))(0) + w. a» e Ker Do- If we write

(D0: + AU, 100))(100 + $105,100» = (“0’0 + 151110

20
where a, b are real numbers, then b = 0.

Proof: For simplicity, denote D“, + A(t,1,00) by D. Then b||1,00||2 = fY(ib100, i100)Re =

fnyW’o + 1,191) — 0100, it’x‘olee = fy<D¢13 in)Re = — 00091, Dwolhe = - fY<i¢1a (1100 +

2'wa — Di/Jilae = 0- Cl

Lemma 1.3.3 There exists a constant C so that for smalls, if [[100HL? S s, t S s,
then

llf/jl “17700”ng S C327 and llA(ta wO)HLf S 032-

Proof: We have continuous maps L? x L? —> L2 and (*d)—1,D0_1 : L2 ——> L1. Apply

Banach lemma to the map

B(A,101)=((7r*,d)‘1(7rr,(100 + 101)), D510 — ”U(Dm — D0 + AMI/)0 + 101)),

which maps “MM? 3 Csz, ||101||L§ 3 C32} into itself when t g s and ”100““ _<_ s for
small 3. The lemma follows easily. Ci

Next we examine the finite dimensional reduction ¢IK8,(d\p)o : R Q Ker Do —>

Ker D0. Let 100 E Ker D0, ”100]le =1. We have

¢lKer (d‘ll)o(ta 87.00) = MU)“. + AU, 3¢0))(S’¢0 + 1010, 3700))-

Without loss of generality, we assume that s is real and positive. By Lemma 1.3.2,

alxer (Moe, 3100) = 0 if and only if

/Y<Dm(3¢0 + 10105, 8100)), 8%)Re + /Y(A(t, 3100)(s1,00 + 10103, 3100)), 3100);;e = 0.

21

Lemma 1.3.4 Let wat = Ari/h, /\t(0) = 0, 10,(0) = 100 as in Proposition 1.1.8.

Then
1.
/}"<DM(S¢0 + '¢1(t13¢0)),8‘¢0>11e = 820‘; + 0(8t +132»

as t, s —> 0.

2.
/Y<A(t, 8160x3160 ‘l' 7/11“, 81/10))7 816,0>Re = 284(— /;I<(*d)_1(7_(w0))’ TQM)»
+0(s + t))
as t, s —) 0.

Proof: Let met = Atwt, and 1,0, = atwo + bt10,i where 10,L E (KerD0)L, [[1,0ti||L2 = 1,
at —> 1, b, = 0(t). Then

At : latl2(Dmb/j09w0) + 2lbtl2’\t _ lbtl2(D#tth—iwti)

Since at —> 1, b, = 0(t), we have (D,,,100,100) 2 At + 0(t2).

On the other hand, for any 102 E (K erD0)i, we have
(0,111,622,100) : at—lbt(At(th-iw2) — (Built/131162)) : 0(lllr/12ll ' t)

So

fy(D,,,(s100 + 101 (t, 3100)), 3100);“ = 320‘; + 0(st + t2))

22

ast,s——>0.

For the second assertion, we have

A(t,s100) = —(7t*¢d)_1(7TTt(Sti/’0+"/}1(t33'¢0)))

= _(*d)-1(T(ao))32 + 0(t32 + 33).

So

[1,040, S'l/Jo)(8‘1/10+¢1(ta3'¢’1’0))a5¢0lae "—" 234(—/Y<(*d)_l(7(i’10)),7(’¢90)>

+O(s + t))

ast,s——>0. El

Corollary 1.3.5 The equation dlKer(dq,)o(t, 3100) = 0 has exactly one solution 3 for
and only for those t such that A, and fY((*d)‘1(r(w0)), r(100)) have the same sign, if

fY<(*d)_1(T(t/Jo)), 7000)) # 0. Moreover, we have t ~ 032 as t, s —> 0,

Remark: fy((*d)‘1(r(100)), r(100)) 74 0 is generically true by slightly perturbing ,at
near t = 0, observing that f,,((*d)‘1(r(100)),r(100,(0)) = 0 for any (0 implies that
100 = O, and also observing that at is transversal to the projection II (see Proposition

1.1.8).

Lemma 1.3.6 Let (A,1,0) be the solution to

*tdA + 710/): 11”) z 0
(D,,, + A)10 = 0

23

near the reducible andt = 0, then SF(/C(A,,),),IC,,,,,(0,100)) is odd as t, s —> 0.

Proof: [C(Av) is an analytic perturbation in s = (10,100) of

D0 0 0
K0 = 0 *d —d
0 —d* 0

[Co has three zero eigenvectors E1 = 100, E2 = $0100 + i), E3 = $0100 — i). Let
Kym/0E: = AgE; where E;(0) = E’, A:(0) : 0. Then

Aim) = o, 1":(0) = —8 fyf(*d)—1(T(’l/)0)),TWO», 13(0) = 1, X310) = —1.

So A1, ~ A32, A: ~ 3 and A: ~ —3 where /\ has the same sign with —/\t (see Corollary
1.3.5).

On the other hand, by Lemma 1.2.2, 1C“,,,(0,100) has three small eigenvalues
At,/\t,/\132 as t -—> O and s = o(t) where A1 = -(D,,,1,00,100) and 100 = D;,1(i100).
It is easy to see that /\1 has the same sign with —(D,,,100,100) ~ —At as t —> 0. So

SF(IC(A,¢),IC,,,,S(0,100)) is odd as t, s —> 0. El
Theorem 1.3.7 Let Y be an oriented homology 3—sphere. Then

1. a(Y) is a topological invariant of Y, and a(Y) + a(—Y) = 0.

2. 010/) E 11(Y) (mod 2), where u(Y) is the Rohlin invariant of Y.

Proof: There is a family of irreducible critical points disappearing or being created

when t passes 0. Call it 6,. It is easy to see from Lemma 1.3.6 that sign(6t) = sign/\t.

24

The rest of M1,, provides a cobordism between the rest of MIL, and M1“. The sign
convention fixed near the reducibles does not change since 1C“,,5(0,'l/10) has a spectral
flow equal to 21:1 when t passes 0 (the point is that D”, is complex linear). So we have
x,,_1 — X,,1 = —SF(D,,_,, D01) and a remains unchanged. As for oz(Y) + a(—Y) = 0,
it follows from Lemmas 1.2.5 and 1.2.6.

The second assertion is an easy consequence of the existence of o-invariant admis-
sible perturbations. We will construct them in the next two sections. [:1
Remark: In [13], Hitchin studied a family of Riemannian metrics on S3 which shows
that the second term in the definition of a may take infinitely many different val-

ues. Therefore we prove that even for the simplest manifold, S3, the Seiberg-Witten

invariant X(S3) takes infinitely many different values.

1.4 Perturbations of Dirac Operator

In this section, we show that the perturbed Dirac operators D9 + f are invertible for
generic pairs of (g, f) and they admit a chamber structure.

Throughout this section, we assume that Y is a closed oriented 3-manifold. Given
a metric g on Y, let P50 be the orthonormal tangent frame bundle of Y. Let
H C GL(3,R) be the subset of symmetric matrices with positive eigenvalues, then
C’C(P30 x AdH) which is the set of C"c sections of the associated fiber bundle P50 x AdH
parameterizes the Ck—smooth Riemannian metrics on Y. We use the Ck—norm of
C"(P50 x M H) to topologize it. Let h be a section of P50 x M H, g" be the corre-
sponding metric, and P130 be the orthonormal tangent frame bundle associated to g“.

Let 6 be a given spin structure on Y, 7r : PSpin(£) —> P30, 7r : P”

h
Spin(€) ——> P50 be the

25

Spin(3) bundles correspondent to the metrics g and y”, then we have a lifting h

’3 h
PSpmié) —" Psz’ne)

[1 11

h h

Note that if h is not symmetric, we may not remain in the same spin structure. Let
V = Pspmm xp C2, V "- —ngm(€) x,, C2 be the spinor bundles where p : Spin(3) ——>
SU (2) is the standard representation. We have an isometry h : V ——+ V" given by
Mae) = (h(o),6).

Let D : F(V) x C"(P50 XM H) ——> I‘(V) be the map defined by 17(10, h) =
Dgh -h(10) where 10 E I‘(V) and h E C’“(P50 xAd H). Let a be a local frame of Pspm“)
7r(0) 2 (e1, e2, 63), and (f1, f2, f3) = (e1, e2, e3)h which is the local orthonormal frame

with respect to the metric 9". Write 10 = (a, 6),h = (7r(o), (h,,~)), then

~

DWW) —_— ifl h-( (a), 6)
3
Z ()(7: Cifi(6 $204,110). )c,ckc,- 6))
2'-—-12k<j
3
: (012(Cihsies( _ — :2 wk,( h)CzCkCJ‘6))
i=12Ic<j

where wig-(h) is the Levi-Civita connection 1-forms of the metric g" with respect to

(f11f27f3)9i'e'1v?1fj: fszj(h), and

26

Direct calculation shows that

. 1
“11001) 2 (hkr12 hlihSJ' + hj rlhlkhsi" hi’hvhstWls — wil)
1 1
+ —h;,1h,,—e,(h,,) — 511;} h,,e,(hl,) + ghgslhikedhs.)

1 1
—hi_slh(j8[(hsk) 'l' §h51h3k83(h1j)

"1h 1hsiesUllk)“ 2

2 J"
where Vac,- : ekwfcj, halhfl, = 6,1,. See [19] and [16].

Lemma 1.4.1 D(-,h): I‘(V) —> I‘(V) is smooth in h. Moreover, D(-,h) is self-

adjoint if det(h) = 1 pointwise on Y.

Proof: That D(-, h) is smooth in h follows from the local expressions of D(-, h) and

w,11.(h). For the self-adjointness of ’D(-, h), we have

~

/Y<D(10,h),<b>,volg = / <13- -_,D h(1l), (0)9Volg
= [(0,-(ah h(10 ,W ))g,,voz,.
= fy<h (a) D. (136)) 11/01.».
= [ya/2, ir‘ -D,.. -h((0))gVolg
= f,<w,v(¢,h>>.Vot.

where Vol, 2 VOlgh Since det(h) = 1 pointwise on Y. El

Lemma 1.4.2 Given any metric g on Y, let (e1, e2, e3) be an oriented local orthonor-

mal frame in an open subset A of Y. Let f be a smooth real valued function on Y.

27
Suppose 1,0,g0 E Ker (D9 + f). If

55% (DR/1.6“). ¢>gvozg) z 0

at t = 0 for any symmetric matrix function X compactly supported in A satisfying

tr(X) = 0, then in A we have

2
(ejVe,10, (6)9 + (If), ejvej¢)g "—— —§<f1/),¢)g

forj = 1,2,3, and

1
wear, a... + <1. mm. = 7mm 0,)

for any i, ',k such that e, /\e’ /\e;c = 81A62 /\e3.
J J

Remark: The same conclusions hold with the hermitian product (-,-)9 re-
placed by its real part, if (~,-)g is replaced by its real part in the condition
{ft—(fy (D(10,e‘X), (0)9Volg) = 0.

The proof of this lemma is a lengthy calculation which is given at the end of this
section.

Let Meto be the subspace of Met 2 C"(P50 XM H) given by
A’IBLO 3: {h E 1’)’[€t|d€t(h) : 1}.

Every metric in [Wet is conformal to a metric in Meto.

28

The Proof of Proposition 1.1.6:
Consider the real Hilbert bundle E over the Banach manifold B = M eto x C" (Y) x
(L1(V) \ {0}). At (h,f,1,0) E B, E(h,f,¢.) = {d E L2(V)]<f> is orthogonal to i10,j10, 111,0}.

Here i,j, k E H satisfying
ijzk, jkzi, kizj, and 12=j2=k2=—1.

The map L : (h, f, 1,0) -—> 19(10, h) + f10 defines a section of the bundle E over the

Banach manifold B. Suppose that (h,f,10) E L“1(0), then the differential of L at
(h, f , 10) is

Mama F, 1) = val, h) + f\P + 6D(10.-)(h)(H) + F1),

from which it is easy to see that if d E (ImbLH, then (0 E Ker (D(-,h) + f) and
c0 = a1(i10) + a2(j10) + a3(k1,0) for some real functions a1, a2, a3. Moreover, by Lemma

1.4.2,
f, ova, -><h>(H>, (new = 0

for any H implies that
2 ,
(give,¢a ¢)Re +(1/),61Ve,¢)ne : —§<fh/ia @) Re
for i = 1, 2,3, and

l
(ejveflfl, ¢)Re +(lx9iejve.¢)11e = —§€k(<’l/J, )Re)

29

for i,j, k such that e,- /\ e,- /\ ek = e, /\ e2 /\ e3. From this we obtain that

('10, es - (610001)) + 61619010) + ez(a3)(k10)))zze = 0

for any s,l = 1,2,3. Since 1,0 is not identically zero, we have e;(a,~) = 0 for any
l,i = 1, 2, 3. Hence a1, a2, a3 are constant. So L is transversal to the zero section of

E and L‘1(0) is a Banach submanifold in B. The projection
P: 171(0) —> Meto x our)

is a Fredholm map of index 3. Note that L(h, f, ) = D(-, h) + f is quaternionic, so by
Sard-Smale theorem, for a generic pair (h, f) E M eto x C" (Y), P“1(h, f) is empty,
i.e., D(-,h) + f is invertible. Any two such regular pairs (ho, f0) and (h1,f1) can
be connected by an analytic path (h,, f,) which is transversal to the projection P.
The operators D(-, ht) + ft are invertible except for finitely many points t,- 6 (0,1),
i = 1,2,. . . , n. The fact that Ker (D(-, ht,) + ft,) = H follows from index counting.

Suppose that D(10t, ht) + ft10t 2: Ari/1t near t,- with At, = 0 and ||10,||L2 = 1, then

«at

dt (ti) = /Y<%(D(l/1tuht) + ftt/lt,)(ti)a¢t.~)Re-

Since the path (ht, ft) is transversal to the projection P, we have

hfiintu ht) + 610901), 101.):28 79 0.

Suppose h, 6 Met is conformal to h 6 Meta and g"1 r: eng". Let m : V’ll —> V"

30

be the isometry. The Dirac operators are related in the following way (see [13] or

[19]):

D h = e2“ngh,m"le'“.

9

It is easy to see from this that Dgh, +f is invertible if and only if D91. +e“f is. Similar
arguments justify the chamber structure. E1
The Proof of Lemma 1.4.2:

Let 71) Z (010)) 7T(0') Z (61162783), then

19(1), h) = (0, saw) + cgegw) + caegw) — %((w12(h)+ wf‘3(h))016
+ (9)5233“) _ W112(h))c26 _ (wl3(h) + w§3(h))636

+ (“6201) — Wis“) + wia(h))0102039)).

x 0 0
For h : e”, where X = 0 —x 0 , we have
0 0 0
w12(h) + w?3(h) = (0112 + w1133)(1+tx) + (1+tx)2e1(1 —tx) + 0(t2),
1233(1) — wl2(h) = (0:33 — wlgfll —t1)+(1—tx)2e2(1+tx)+ 0(t2),
w]3(h) + 01.33%) = —(1—tx)e3(1+tx) — (1+tx)e3(1—tx)+ 0(t2),
use) — wish) +w%3(h) = 11<1+m2<w13+ wi2)+(1—tsv)2(wi2 — wt.»

+O(t2).

31

So we have

d 1 2 3
d—t(D (16: h()))( ) = (01$0161(9) " $C2€2(9) “ 5((x(w12 + “’13) — 61(113))Ci6

_ ($0633 _ £0112) - 32(1))620 + “Wis + wf3)c102c36)).
If we write 10 = (o, 6), (0 = (0,6), then

xc1e1(6), E) - ($02€2(0): 5)

A

d /' _
f, (309(1), h))(0).¢>lol —- fy(

—-;—(xw12 + xw133— 8 1(3))<0161€>

1

—§(—xw33 x0212 - €2($))<6261€>
+£11.11. +w13)<9,:>>Voz.

Let (e1, e2 ,e3) be the dual to (e1,e2,e3), then

d(x(c16, g) * el) = e1(x)(c16, {)e1 A e2 /\ e3
+ x((c1e1(6),€) + (C16, 81(5)))e1 /\ e2 /\ e3

— x0012 + w13)(cl6,£)e1/\ e2 /\ e3.

Integration by parts, we have

x(w12+w13)( )(016 €)e /\e2 /\e3.

[Y e1(x)(c16, Se1 /\ e2 /\ e3 = —/x( (c1e1(6 +(,c16 e1(§)))e1 /\ e2 /\ e3

32

Similarly, we have

/Y e2(x)(c26,§)e1 /\ e2 /\ e3 = — [Y x((cgeg(6),{)+(626,e2({)))e1 /\ e2 A e3

+/ x0033 — w12)(c26, {)e1 /\ e2 /\ e3.
}/
These give us

figwwnmtavw = 1 j, $(<61Ve110.¢) + (10,81Ve.1>))61 A e M

—— 1 h x(<e2ve.w. <1) + <1, egve.¢>)el A e2 A e3.

Therefore, if

y dt
:2: 0 0

for all h :2 e‘X where X = 0 -3; 0 , we have
0 0 0

(61Ve110,¢>+ (hr/)1 elvelgb) Z <62V62¢1 45) + <16, 82V82¢)'

Similarly, we have

<61V€1w3¢> + (wielvelcs) Z <83v331/I), ¢) + ((6: e3vea¢)'

33

But 11), a5 6 Ker (D9 + f), we have

3
X((ez-Veiw, (b) + (d), caveat» = (Dgw, a5) + (‘9’), D996)
i:l
So we have
2
(fiver-”(Pa ¢) + (Weivefl) = "50¢, (15)
0 a: O
for 2' = 1,2,3. Similar computation with X = 1r 0 0 yields
0 0 0

(€2Ve1¢’,¢) + <¢a€2Ve1¢l + <81Ve2¢a¢> + (w, €1Ve2¢l = 0-
Combined with
((82Ve1¢,¢)+(¢,€2ve1¢>) — (<81Ve21b, <15) + (77/): €1Ve2¢)) : —€3((1/1,¢)),

we have

(ezvm, a6) + (wezvelas) = ——1—e3(<w, a»).

In general, we have

1
(ejVefl/J, Q» + (w: ejVein) Z —§ek(<wa Cb»

for i,j, k such that ei /\ ej A ek = 81 /\ 82 /\ 83.

34
1.5 The o-invariant perturbations

In this section, we give the construction of the o-invariant admissible perturbations
using holonomy along embedded loops. Assume that (g, f) is regular. Let sf denote

the gradient of CSDf and M 1 denote the set of critical points where
1
05791 = C579 + 5 A flfi’lfiVolg. and 8’04. L9) = (*dA + T(¢,1/)),DA¢‘ + fit?)-

The moduli space M f is compact and can be represented by smooth sections.

Definition 1.5.1 A thickened loop is an embedding 7 : S1 x D2 —> Y, together with

a bump function 77(y) on D2 centered at 0 6 D2, with [D2 n(y)dy = 1.

Given a thickened loop /\ = (7, 77), one can define a pair of o-invariant functions

(p,q),\ : B —> [—1, 1] x R+ by
mm = [0, cos<6.>n<y)dy,

where em” is the holonomy of A along the loop 7y 2 S1 x {y}, and
mm») = /D 2.31 WWW

Lemma 1.5.2 The function (p,q) is smooth on A.

Proof: The same arguments as in [29]. It is useful to know that

WWW) = / isinwmyx/ 75am

02 Slx{y}

35

and

dQA|(A,¢)(a3 ¢) = 2 /l)2x51<w, ¢>77(y)dydt

E]

For any set A of finitely many thickened loops, we have a smooth map (DA : 8* —>

HAG/\(l—I’ 1l X R+)A given by
<PA(l(A,ib)l) = ((p,q)i(A,2/2), /\ e A).

The map (PA is a-invariant and continuous on [3.

Lemma 1.5.3 There is a set A of finitely many thickened loops such that
1. Ker st mm Ker (d<p,q>.) = {0} at any [(A, in 6 Mr.

2. (DA is injective up to the 0 action on A4,. Therefore we can identify Mf/U

with a compact subset of Hie/\(l‘l, 1] x R+),\.

Proof: Suppose (An/2) E M *, and (a, (b) E Ker st , , i.e., (a, p) satisfies
f (Am)

DAQl) + f¢ + 01/} = 0
*da + 27(2/2, ¢) = O
—d*a + 2W, ¢>R.-= 0.

Since A is not flat, if (a, (b) E Ker (d(p, q);\) for all thickened loops, then fslx{y} 75a 2
0 for all 7. So da 2 0. da 2 0 implies T(2/J,<b) = 0. So qb = 111/) for some function
u E (20(Y) ® iR wherever w # 0. This implies d’v + a = 0 and fY(|du|2 + |v|2|ib|2) = 0

by plugging into the equations. Since 112 is not identically zero, we have (a, o) = O.

36

So for each [(14,711)] E Mf“, there is a set of finitely many thickened loops such
that the first assertion holds for [(A,zb)]. Then the first assertion follows by the
compactness of M} and the smoothness of the function (p, q).

For the second assertion, suppose [(A1,z,b1)], [(.42,1/12)] E Mf' such that
(p, q)A(A1,w1) = (p,q),\(A2,z/)2) for all loops. Then dAl 2 :lsz2, and |w1|2 = libgl2.
Assume dAl : dAg, then r(w1) = 7(1b2). By writing in a local frame, it is easy
to see that wl = 3% for some 3 E M ap(Y, S 1). Then it is easy to see that
[(A1,i,b1)] : [(A2, 1122)]. In the case of dA1 = —dA2, apply 0.

Now for any [(A1, #4)] 79 [(A2, 2%)] in Mfl/o, there is a thickened loop A separat-
ing them. By the compactness of M f" / o and the smoothness of (p, q), there exists a
set of finitely many loops separating any two points in M f / o with distance greater
then a fixed number. Combining with the first assertion, since each point in M f" / a
has a neighborhood described by a Kuranishi model, the second assertion follows.

C]

For any smooth function h on HA6A([—1, 1] x R+),\, the composition u 2 ho (DA is
a smooth function on A. We will perturb CSDf by adding u, i.e., 0813' = CSD; +u.

Denote the gradient of CS’D’ by s’. The following lemma is standard (see [29]).

Lemma 1.5.4 1. st and Vs' are continuous families of Fredholm operators from

bundle TB" to .C over 8*, and st — Vs’ is compact.
2. M = s'_1(0) can be represented by smooth sections.
3. There exists a constant e > 0 such that when |dh| < e, M is compact.

4. When |dh| —) 0, the distance between Mf and M goes to zero.

37

Next we define a section G of the bundle [I over 8* x V where V is the dual of

the vector space HA€A(R x R)»

C((A. w). (v. wh) = 8’(A, 11‘) + g'rad(/)(<I>A)(Z (vim + U’AQA)))(A’ 1.0)-
AEA

Here the set A of thickened loops satisfies the conditions in Lemma 1.5.3, and p
is a cutoff function on H A6 1\(R x R) ,\ satisfying that p E 0 in a neighborhood of
HA6A([—1,1]x{0}))t and ¢A([(A,1,b)]) where [(A, 2b)] 6 B“ is a non-degenerate critical

point of 68D], and p E 1 in a neighborhood of the rest of <I>A(Mf").

Lemma 1.5.5 There exists 6 > 0 (depending on ,0) such that G is transversal t0 the
zero section of£ when restricted to B“ x B(e), where B(6) is a ball of radius 6 centered

at the origin in V = (HEAR x R)),)*.

Proof: G is transversal to the zero section of L over M f x {O} by the choice of the
set A. By continuity and Lemma 1.5.4 (4), this lemma is proved. E]
The Proof of Proposition 1.1.7:

Apply Sard-Smale theorem to the projection H : G‘1(0) —> 8(6). For a generic

(21, w),\ 6 8(6), the perturbation CSD’ 2 C817; + u is admissible where

U = P(¢A)(;(’UAPA + UNA»

CHAPTER 2

Seiberg—Witten Equations on
Cylindrical End Manifolds

Throughout this chapter, we assume that Y is an oriented 3-manifold with boundary
which is the complement of a tubular neighborhood of a knot in an integral homol-
ogy 3-sphere (many results proved in this chapter hold for general 3-manifolds with
toroidal boundary). Equip Y with a Riemannian metric go such that a neighbor-
hood of BY 2 T 2 is orientedly isometric to (—1,0] x R/27rZ x R/27rZ. We attach
[0, 00) x T2 to Y and still denote it by Y.

Given a spin structure of Y, there is a unique S U (2) vector bundle W over Y such
that the oriented volume form acts on W as identity by the Clifford multiplication.
The spinor bundle W is cylindrical, i.e. on [0, 00) X T2, W is isometric to the pull
back 7r*W0 where 7r : [0, 00) x T2 —> T2 is the projection and W0 is the total spinor

bundle on T2 associated to the spin structure induced from Y.

38

39

2.1 The Fredholm theory

In this section, we set up the Fredholm theory for Seiberg-Witten equations on Y.
Throughout H1(T2) stands for the space of harmonic l-forms on T2. We fix a cut-off

function p on Y which equals to 0 on Y \ [0, 00) x T2 and 1 on [1, 00) x T2.

Definition 2.1.1 For (5 > 0, let

A; = {(A,1/))|A = B + p7r’a,

B e L1‘6(A‘(Y) a iR), a e L1,6(W),a e 7110?) a 2R},

where 7r : [0, 00) x T2 —> T2 is the projection.

Here L2,); denotes the weighted Sobolev spaces with weight 6 ([21])./16 is a Hilbert

space (over real numbers) with the norm

||(A,¢’)||A.s =||(B.1/J)|ltg‘, + Hallu-

Note that the decomposition of A as B + pn‘a is unique. We define a map R : A; —->
H1(T2) ® iR by R(A, 1b) = a.

Definition 2.1.2 The group of gauge transformations is

g. = {s e Liam suns-Ids = g + MW

9 e L3,,(A1(Y) a iR), h e 711(T2) a 2R}.

95 acts on A5 by the formula s- (A, w) = (A — s‘lds, 31b) for s E 95 and (A, w) 6 A5.

40

Lemma 2.1.3 95 is an Abelian Hilbert Lie group acting smoothly on A5 with the Lie
algebra T9555 2 L§,6(A°(Y) ®iR) GaiR. Moreover, for s E 95, ifs“1ds is decomposed
as g+ p7r‘h, then h has zero period along the longitude and periods in 27riZ along the

meridian.

Proof: Suppose that s 6 95 is in the component of identity, then s = ef for some
f E L§,,OC(A°(Y) ® iR). By Definition 2.1.2, df = s‘lds can be decomposed as
g + p7T*h, from which it follows that h = 0 and df E L§,6(A1(Y) <8 iR). By Taubes
inequality (Lemma 5.2 in [28]), there exists an imaginary valued constant f0 on Y

such that

/Y|f_f0|2e26t S C(6)/;lldfl2e26t,

which proves that the Lie algebra T95,“ is L§,6(A°(Y) ® iR) EB iR.
Let 71,72 be the longitude and meridian, and F be the Seifert surface that '71

bounds in Y. For 3 6 Q5, if 3‘1ds is decomposed as g + p7r*h, then we have

/ h=/d(s‘1ds) =0 and f h:/ s'ldse 2mz.
’71 F ’72 ’72

The rest follows easily from the Sobolev theorems for weighted spaces. Cl

Lemma 2.1.4 The de Rham cohomology group H bR(Y) can be represented by the

space of “bounded” harmonic forms

711(Y) : {a E 91(Y)|da = d‘a = 0, Hallow») < m,tlirg10a(%) = 0}.

Moreover, each element a E 711(Y) can be decomposed as b+p7r“a00 with a0o E H1(T2)

41

and b E Lia for some (5 > O. The map R : 711(Y) —> H1(T2) defined by R(a) = a00
represents the embedding H1133(Y) —-) HbR(T2)- As a corollary, for any Is; 6 H1(Y, Z),
there is an 3,, E C°°(Y, 51) such that sgldsx 6 ’H1(Y) ® iR and [s;lds,,] 2 2525. So
«0(a) = H1(Y, Z) = 2.

Proof: The Laplacian df‘pde : L1(A°(T2)) —+ L2(A0(T2)) restricted to (K erdi;.2dT2)l
is invertible. Let G be the inverse. Suppose a closed form A E (21(Y) is written as
A0dt+A1 on [0, 00) x T2 with A1 6 {21(T2). Then f : C(d}2A1) is a smooth function
on [0, 00) X T2. We extend f to the rest of Y and still call it f. Let B = A — df. We
can further modify f by a function of t so that [T2 BO 2 0, where B = Bodt + B1 on

[0, 00) x T2. (B0, B1) satisfies the following equations:

8B

which shows that B1 is in H1(T2) and constant in t and B0 = 0. Since d‘B is
compactly supported, there is a unique solution g E L£,5(A°(Y)) to the equation
d‘B = d‘dg (see Lemma 2.1.7 below). Let C = B — dg, then C E HWY) and the
cohomology classes [A] and [C] are equal in H 11) R(Y).

Suppose a E H1(Y) and a = aodt + al on [0,oo) x T2. Then the pair (a0,a1)

satisfies the following system of equations

(

530,1 —dT2a0 =0

(9
'5‘? '—d:;w2(ll :0

dT2Cll = O.

 

42

0 d 2
The operator L = T is formally self-adjoint and elliptic on KerdéBQO(T2).

*Tzo

By expanding (a1, an) in terms of an orthonormal basis of eigenvectors of L, we see
that a = aodt + a1 can be decomposed as b + p’lr"'aOO where a00 E ’HI(T2) and b 6 L16
for some 6 > 0.

Assume a1,a2 E H1(Y), if a1 — a2 2 df for a smooth function f on Y, then
df 6 L116, and by Taubes inequality and integration by parts, df = 0. Hence the map

711(Y) —> H [1) R(Y) is also injective. The rest of the lemma follows easily. [:1
Definition 2.1.5 Let B5 = A5/g5 and B; = A3/Q5 where A; = A5 \ {w E 0}.

Lemma 2.1.6 1. B; is a Hilbert manifold with the slice at (An/2) E A; given by

TQM,“ = U x V where

U = (B. at) + {(a, <25) 6 L3,.(A1(Y) a 2R) as L3,.(W)I — d‘a

on, as)... = o. ”(a, bug, < .},

v = R(A,v) + {am e H1(T2) e iR|||a00||L2 < e},
where A is decomposed into B + p7r‘R(A, w). The tangent space of B; at (A, w)

is

maul) = {aneL3,.(A‘<Y>®2R)mam/nu—d*a

How, am = 0} es ’H1(T2) a iR.

43

2. A neighborhood of [(A, 0)] in B5 is diffeomorphic to T(A,0),6/Sl. T(A,0),6 = U x V

and

U = (3.0) + {(a, <15) E 1335(410’) ® 2'3) EB L§,5(W)|d*a = 0, ||(a, (bllng‘, < 6},
V = R(A,1/J)+{aoo 6 WC”) ® z'1-‘illlaoollm < 6},

where A is decomposed into B + pn‘R(A,w). The action of S1 on TWO” is

given by the complex multiplication on the factor (15.

Lemma 2.1.7 Let L1 = d*d and L2 = d*d + lib]2 where w E L§,5(VV). Then there is
60 > 0 such that for k 2 2 and any (5 6 (0,60], L1 : Li,5(A°(Y)) —+ L§_2,6(AO(Y)) is a
Fredholm operator of index ——-1. Ker L1 = 0, and the range of L1 is the Lz-orthogonal
complement of the space of constant functions. L2 : L£,5(A°(Y))®iR —+ Li_2,6(A°(Y))

is isomorphic ifw is not identically zero.

Proof: The operator L1 = d‘d : Lfi,5(A0(Y)) —> Lfi_2,5(A°(Y)) is Fredholm of index
—1 by Theorem 7.4 of [21]. Ker L1 = 0 follows from integration by parts. From
index counting it follows that the range of L1 is the L2-orthogonal complement of
the space of constant functions. For w E L§,6(W), L2 : L2,6(A°(Y)) —-+ Li_2,5(A°(Y))
is a compact perturbation of L1, so it is also a Fredholm Operator of index --1. So
L2 : L£,6(A°(Y)) EB iR —> L§_2,6(A°(Y)) is an isomorphism if 1b is not identically zero,
since Ker L2 = 0 and index L2 = 0. C]

The Proof of Lemma 2.1.6:

1. The construction of a local slice is standard by applying the implicit function

theorem. The key point is the properties of L2 stated in Lemma 2.1.7. To prove

44

that B; is Hausdorff and the local slice is embedded into 8;, the argument in

[12] can be used, combined with Taubes inequality (Lemma 5.2 in [28]).

2. Part 2 of this lemma follows similarly with Lemma 2.1.7 understood.

Definition 2.1.8 For (An/1) 6 A5, we define

5w») = {(a, <25) 6 14150110”) <59 2'R) 69 Li,5(W)| - d‘d + 1(2'2/2, 4%.. = 0}-

£5,(A,5,) is a closed subspace of L1.6(A1(Y) ® iR) EB LiJU/V).

Lemma 2.1.9 £5 = {£5,(A,,5)} is a Hilbert bundle over A; which descends to a Hilbert

bundle over 83 (we still call it £5).

Proof: For any (a, ¢) 6 Lf,6(A1(Y)®iR)EBL1,6(W), we can project (a, gt) into 56,01,110

by solving the following equation

—d'(a - df) + W11). (13 + fwlRe = 0

for f E L§,5(AO(Y) <8) iR) EB iR. By Lemma 2.1.7, the operator L2 = d‘d + lib]2 is an
isomorphism from L§,5(A°(Y) ® iR) EB iR to Lg,6(A°(Y) ® iR) since (A, 2b) 6 A3. So
the above equation has a unique solution f (a, (b) for any (a, (b). If (a, ab) e £5,(,4,,,5,)
with (A1, wl) close enough to (A, it), one can easily show that the projection (a, (t) —->
(a —- df, ¢+ f w) is one to one and onto, again using the invertibility of L2. This proves
the local triviality of £5. The bundle £5 over A; is Q5-equivariant, so it descends to

a Hilbert bundle over 83‘. E]

45

Definition 2.1.10 For (A,1/)) 6 A3, we define

8(A. 7(1) = (*4A + T('/),1/1),DA¢)-

Here DA 2 D90 + A where Dgo is the Dirac operator associated to the metric go. 3 is
a section of £5 over A3, which descends to a section of £5 over 83.

The covariant derivative ofs is a section of End(TB§, £5) over A; which descends

to 83‘, defined by

VS(A,¢)(G, ch) = (*da ‘1' 271$, ¢) _ df(aa (b): DA¢ "l" 0&0 + f(a, ¢)l//')

where f(a, qfi) is the unique solution to the equation

d‘df + flil’l2 = “DMD/Kim.

The map (a, gb) —+ (—df(a, (b), f(a, (bk/2) from TB;,(A,,,,) to Li6(A1(Y) ®iR) EBL¥,6(W)

is compact by the Sobolev theorems for weighted spaces.

Definition 2.1.11 1. For any r > 0, let H(r) = ’H1(T2) <8 iR \ UpeB D(p,r)
where B is the lattice of “bad” points for the induced spin structure on T2 (see

Appendix A) and D(p, r) is the closed disc of radius r centered at p.

2- «46(7) = 34010)); AMT) = AilTlfl/li, 36(7’) = flaw/96 and 33(7) =
A3(r)/g5, where R : A5 —+ H1(T2) ® iR is given by R(A,w) = a for
A=B+mm.

46

Note that for any a E H(r), the twisted Dirac operator D112 = DT2 + a is invertible,

where 0T2 is the Dirac operator on T2 (see Appendix A).

Proposition 2.1.12 For any r > 0, there exists a (5(r) > 0 such that for each
6 E (0,6(r)), Vs : TB; —> £5 is a continuous family of Fredholm operators of index 1
over A3(r). (So 3 is a Fredholm section of £5 over 830)).

At (A, w) E A3, we have a short exact sequence
d x
0 —> TQM ‘1)” TA; 1; Te; —> o

where d( A,,),)( f ) 2 (—df, f 1b) and 7r : A; —> B; is the natural projection. This enables
us to extend V304,,” : TBE,(A,,,,) —> £54,550 to a Q5-equivariant map [Cl/W) (see [29]),

where
VS(A,1,/)) 0 0

(MI) : 0 0 dew)
0 d( 1W) 0
[Cl/W) is from TA; 63 (L§,6(A0(Y) <8) iR) EB iR) to L1,6(A1(Y) ® iR) EB Lf,5(W) 63
2 0 . . 0 dew») . . . .
L116(A (Y) <8) iR). Smce the operator lS 1nvert1ble, V304,,” 1s Fred-
dow) 0
holm if and only if [C] 1W) is, and indexlC’ = indesz.

For (A,rb) 6 A5, we define a map [C(Anb) : A5 69 (L§,6(A°(Y) ® iR) EB iR) —>
Lf,5(A1(Y) a 2R) ea L1,,(W) e L1,,(A°(Y) a 2R) by

K(A,1,b)(a7 Q5, f) : (*da + 27(1/),¢) — (if) D2145 + (11/) + fill, _dlta’ + 104/): ¢>Re)'

47

Then [Cl/150) is a compact perturbation of (c(/W) and Proposition 2.1.12 follows from

Lemma 2.1.13 For any r > 0, there exists a 6(r) > 0 such thatfor each 6 E (0, 6(r)),

IC is a continuous family of Fredholm maps on A3(r) of index 1.

Proof: Consider the following commutative diagram

0 —) VAJJ 69 L195 —> VA“ EB L115 —> 0

T V’Cmu) T Kora) T
0 —) VA5 63 L3“; —> A5 63' (1135 63 2R) —> 7110"?) ® iR€9 iR. —) 0

where L1,, = L1,6(A0(Y) <8) iR) and L3,, = L§,6(A°(Y) ® iR). VA5 is the fiber of map
R : A5 ——> ”H1(T2) <8) iR and VA5,1 is its Lircompletion. VIC( A15) is the restriction of

KM»). We have a long exact sequence (see [24])

CokeerC(A,,,/,) —> CokerlCMM —> 0
0 —> KerVICMM) —> KerlQ/W) —> H1(T2)®iREBiR -—>.

The lemma follows from the claim that for any r > 0, there exists a 6 (r) > 0 such that
for eaCh 6 E (0, 6(7‘)), VK(A,w) i VA5 ® Lg,6(AO(Y) @211) _2 1]./46,1 69 Lg’6(1«O(Y) ® 2R)

is a Fredholm map of index —2 for (A, 1b) 6 A3(r).

48

VIC/W) is a compact perturbation of an operator of form I (% + Ba) on [0, 00) x T2

 

 

 

 

where
( dt 0 0 o) ( 03" 0 0 0)
I 2 0 *T2 0 O and Ba : 0 0 —dT2 *d-Tz
0 0 0 —1 0 —d*T2 0 0
( 0 0 1 0 ) ( 0 —*de 0 0 )

acting on I‘(W0 69 (A1 69 A0 69 A°(T2)) <8) iR). Here We is the total spinor bundle over
T2, and D? is the twisted Dirac operator with a = R(A, w) E H1(T2) <8) iR. For any
r > 0, let 6(r) = min{|u| : u aé O is an eigenvalue of B, for some a E ”H(r)}. Then
VICUW) : VA5 69 L§,6(A°(Y) <8) iR) -—> VA5,1 EB L§,5(A°(Y) <8) iR) is a Fredholm map

of index —2 for any 6 E (0, 6(r)) by Theorem 7.4 of [21]. E]

2.2 Perturbation and transversality

Fix a small r > 0 and a 6 E (O, 6 (r)) for the weight of the Sobolev spaces. For simplic-
ity we omit the subscript 6 in the discussion. Consider the following perturbations of

s over A*(r):
sin/1.2m = (*dA + 7(W) + u, Div + M) for (A. w) 6 AW).

where u is a co—closed imaginary valued l-form and f is a smooth real valued function
on Y, both supported in Y\[0, oo) XT2. The metric 9 being used here is a perturbation

of go supported in Y \ [0, 00) x T2.

49

Definition 2.2.1 Define the Seiberg- Witten moduli spaces

Mai”) = {[(A, '99)] E 3(7‘)l(*dA + 7(4), 10) + It, Dal/J + fit») = 0},
2.1(7‘) = Mai“) 03*-

Let [R] : B(r) ——) ’H(r)/Z be the map induced by R : A —> 711(T2) ® iR.

Proposition 2.2.2 The moduli spaces M“,f(r) and ML,f(r) have the following prop-

erties.

1. For a generic u, M;,f(r) is a collection of 1-dimensional smooth curves, and

the map [R] : ;,f(r) ——i H(r)/Z is an immersion.

2. Given a set S of immersed curves in ’H(r)/Z, for a generic u, the map [R] :

LAT) -> ’H(r)/Z is transversal to S.

3. For a generic (g,f), the L2-closed extension of the perturbed Dirac operator
D9 + f is invertible. Fix such a (g, f), then for any small enough u, there exists
a neighborhood U” of [(au,0)] in B(r) such that U“ flM;,f(r) = (b, where a“ is
the unique solution to *da,, + u = O, d‘a,, = 0 such that R(ap) has zero period

along the meridian.

4. M#,f(7') \ M;,f(r) = {(a,, + iA,0)|A E H1(Y)/’H1(Y,Z)} 2 SI. Note that for

any A E H1(Y), R(A) is a multiple of dy where eiy parameterizes the meridian.

For simplicity we omit the subscript f. Consider the section s of £ over 8*(r) x

Ker d"‘:

‘s‘([(A, 216M) = [(*dA + 5(2/1, v) + u. DAw + fin]-

50

For any ([(A, #2)], p) E §-1(0), we have the following commutative diagram:

0 —> L —> C —> 0
ivvg TVS" To
0 —) VTB*(r)xKer d* —> TB*(r)><Ker (1* d3] H1(T2)®iR —> 0

at ([(A, 112)], p). This gives rise to a long exact sequence (see [24])

Coker(VV§([(A,¢)],,,)) —> Coker(Vs([(A,,5)],,,)) ——>0

dfi]

0 —-) K6T(VV§([(A,w)],,,)) —> K6T(V§([(A,¢)],#)) H1(T2) ® iR. —) .

Lemma 2.2.3 Coker(VV§([(A,.5)],,,)) = O for any ([(A, 1b)], u) E s-1(0).

Proof: First observe that VV§([(A,,5)],,,) is Fredholm as a map from the Liar
completion of VT 8]} AM] to the Laé-completion of £[(A,¢)]. So by regularity, it suffices
to show that the Lad-orthogonal complement of the image of VV§([(A,¢)],,,) is zero
dimensional.

Let (a’, (25’) E £[(A,¢)] be LENS-orthogonal to the range of vvgwym. Set (a, (t) =
e25‘(a’, (6’), then (a, qb) is L2-orthogonal to the range of VV§([(A,¢)],,,) and e‘26t(a, (b) is
in £[(A,¢,)], i.e. —d*(e‘26‘a) + i(iib, e‘26‘¢)Re = 0. Note that (a, ab) is in L14.

Observe that L§,6(A1(Y) ® iR) 6 L§,6(W) = VTB*(r) 6 Im(d[(A,,),)]) (recall the
map dmflp) is defined by f —~> (—-df, fig», and VVQMWW vanishes on Im(d[(A,,5)]).
So if (a, (b) is L2-orthogonal to the range of VV§([(A,¢)],,,), then for any u’ E Ker d*
and (b, 0) E L§,6(A1(Y) <8) iR) 6 L§,6(W), we have

(*db + 27W. 9) + u’, a) + (DAG + f6 + bu, Q5) 2 0.

51

This implies that DA¢ + f (b + aw = 0 and (bi/2,6) = O for any b. Since 2b is not
identically zero, by the unique continuation theorem for Dirac operators, we have
(b :2 ’ll'H/J for a real valued function h. Then DAqS + fab + azb = 0 implies that
idh + a = 0. Hence

d*(e_26tdh) + lie-25mm2 = 0

That (a, qb) E 0 follows from h E 0, which follows by integration by parts from the
claim that e‘5‘|h| is bounded on [0, 00) x T2.

Next we prove that e‘“|h| is bounded on [0,oo) x T2. First of all, idh + a = 0
implies that 1941 E [41,-6 and deh E L'ffls. Let h0(t) be the Lz-orthogonal projection
of h onto Kerdgsde, then ||h — h0(t)||L2(T2) S clldT2h||L2(T2). So h — h0(t) E [43,—6-
On the other hand, [£54 < C||%—1’||L2(T2) so that d—the 6‘ is bounded on [0,oo) x T2.

So
|h0(t) —(0hO 0)|</0|‘:,—h°|dt<(S 9e“.

It follows easily that e"6‘|h| is bounded on [0, 00) x T2. Cl

Lemma 2.2.4 Given any spin structure on Y, for a generic (g,f), the L2-closed

extension of the perturbed Dirac operator D9 + f is invertible.

Proof: For a perturbed metric g of go which is supported in Y \ [0,00) x T2, the
Dirac operator Dg on Y takes the form of dt(g—t + DT ) on the cylindrical end. Note
that DT is invertible (see Appendix A). So the L2-closed extension of D9 + f is an
essentially self-adjoint Fredholm operator (f is vanishing on the cylindrical end). The

argument for the proof of Proposition 1.1.6 can be applied to prove this lemma. E]

Lemma 2. 2. 5 For small enough 6 > 0, the operator *d: Li (A (Y )) flKer d*——>

52

Li_1,5(A1(Y))flKer d“ is Fredholm with dim Ker * d = 0 and dim Coker * d = 1.
Moreover, for any compactly supported co-closed I—form ,u, there exists a unique a” E
Ker d* such that i) *da,‘ + u = 0; ii) a” can be decomposed as b + ,07r"a00 where
b E Li“; and a00 E H1(T2) with zero period along the meridian. Furthermore, 0,,

satisfies the estimate: HbHLfa + law] 3 Cllulng

—1,6'

Proof: The Fredholm property and the index calculation of *d follows from a similar
argument as in Proposition 2.1.12. Ker * d = 0 follows from H1(Y,T2) = 0. Given
u E Ker d* or equivalently *p E Ker d, since H2(Y, R) = 0, there exists an A E
S21 (Y) such that dA+*/x = 0 or equivalently *dA+u = 0. If u is compactly supported,
the argument for Lemma 2.1.4 can be used to modify A with an exact 1-form and a
“bounded” harmonic form, and the resulting l-form a” has the claimed properties.
C]

The Proof of Proposition 2.2.2:

Since Coker(V§([(A,5,)],,,)) = 0 for any ([(A,1,b)],u) E §-1(0), §“1(0) is a Banach
manifold. The projection H : §“1(0) ——> Ker d“ is a Fredholm map of index 1
(Proposition 2.1.12). So by Sard-Smale theorem, for a generic u, M;(r) = II‘1(/i) is
a collection of 1-dimensional smooth curves. In addition, Ker (VVS') fl Ker II = 0
since VV§ is formally self-adjoint on VTB"(r). So d[R] : TM;(r) —> ’HI(T2) ® iR is
injective.

Since Coker(VV§([(A,¢)],,,)) = 0 for any ([(A,w)],p) E §-1(0), the map [R] :
§"1(0) —+ ’H(r)/Z is a submersion. For any set S of immersed curves in ’H(r)/Z,

[R]‘1(S) is a set of immersed submanifolds of co—dimension 1 in 5‘1 (0). If u is a regular

value of the projection H : [R]’1(S) —> Ker (1*, then the map [R] : M;(r) —> H(r)/Z

 

is transversal to S.

Properties 3, 4 follow easily from Lemmas 2.2.4, 2.2.5 and 2.1.4.

2.3 The finite energy monopoles
Fix a perturbation (g, f, p) which is supported in Y \ [0, 00) X T2.

Definition 2.3.1 (A, w) E 01(Y) ® iR6 I‘(W) is said to be a monopole of finite

energy if (A,w) satisfies

0 the Seiberg—Witten equations

*dA+r(1b,ib)+u=0
Dg¢+A¢+f¢=0;

o the finite energy condition
2 1 4
[Wall + —le ) < 00.
Y 2

The exponential decay estimates

Lemma 2.3.2 (Lemma 4 in [17])
Let X be a compact 3—manifold with boundary. Assume that (A,ib) E 91(X) ®
iR 6 NW) satisfies the Seiberg- Witten equations on X. Then there exists a gauge

transformation 3 E C°°(X , S1) such that for any sub-domain X’ with 75’ C intX,

54
s - (A, w) satisfies:
H8 ' (wlllcux') S C(k,X, X')h1(||¢’llu(X))a
“3 ' (AlllC'k(X’) S C(ka‘YaX’lh2(llwllL4(X))

for a constant C(k, X, X’) and polynomials h1,hg with h1(0) = 0.

Corollary 2.3.3 For a finite energy monopole (A, 1b),

llwllc~0(r2)(t) S Cllwllum—qum.

In particular, w —+ O as t ——> 00. Moreover, there exists a constant K depending only

on the geometry on and the norm of (p, f) such that ||w||Co(y) < K.

Proof: It follows from Lemma 2.3.2, the V'Veitzenbock formula and maximum princi-
ple. C]

Throughout this section, we use a(t) to denote the harmonic component of A1
where A = Aodt + A1 on [0, 00) x T2. After a gauge transformation, any (A, 2;?) takes

the standard form on [0, 00) x T2, i.e. (£15.41 2 0 and ng A0 : 0 (see Lemma 2.1.4).

Lemma 2.3.4 Assume that the finite energy monopole (A, w) is in the standard form.

Then the following holds for a constant c:
0) fT'2(lA0l2 +|d1~2Ao|2) S 0sz I114“;
5) fr2(lA1- a(tll2 +|VT2(41- a(t))l2) S CfT? Mil“;

6) fr2|%(41- a(t))l2 S ch2 W4;

55
01) fT2 |.%Cll(t)|2 S 6sz W,-
e) llfiAolngm) S c||%'§]|,,2(7~2).
Proof: On [0, 00) x T2, the equation *dA + rub, w) : 0 reads as
dT2(A1 — a(t)) + (11(7/1) = 0,

%(Al — a(t)) + £0“) — dT2A0 + (12W) 2 0

for some quadratic forms q1,q2. Observe that %(Al — a(t)), %a(t) and deA0 are L2
orthogonal to each other. The estimates a), b), c), d) follow easily.
For 6), note that d}2d7~2(%A0) = .}2(%q2(w)). So we have

a , a a all
Ila—tAollLflri’) S Clldr2dr2(5€AolllL2_,(r2) S CHEWWDHLHT?) S CllEllLHT?)

since ng A0 = 0 and ||w||Co(y) < K. [:1

Lemma 2.3.5 For any r > 0, there exists a c(r) > 0 such that c(r)fT2 W2 S

sz [Dz21/JI2 for any a in the closure of’H(r) (see Definition 2.1.11 for ’H(r)).

Proof: Observe that both sz I'll‘l2 and [T2 IDZ2¢|2 are gauge invariant, so we can

assume that a is in the compact set ’H(r) / (Z 6 Z). The lemma follows by taking

 

c(r) = min{u2 : u is an eigenvalue of D12 for some a in H(r)/(Z 6 Z)}. C1

The following estimate turns out to be crucial.

Lemma 2.3.6 There exists a constant c1 with the following significance. Let (A,1/))

56

be a finite energy monopole. For any r > 0, if a(t) is in H(r) for T1 < t < T2, then

[T2] |/|2< Cl [T2] (IV /|2+1|w|“)
T1 T21) _c(r) T1 T2 AU 2 '

Proof: Assume that (A, 2b) is in the standard form without loss of generality. Since

a(t) is in ’H(r) for T1 < t < T2, by Lemma 2.3.5, for T1 < t < T2, we have

c(r) T,IwI2(t> </ IDZLMDI as] Ivfam (t)
s fT,(IvAiI2( + |(A1 — a(t)) ®¢|2 >(t).

But [T2 |(A1— a(t)) ® 1b|2 S K2 fT2I(A1— a(t))|2 S CfT2 libl4 by Corollary 2.3.3 and
Lemma 2.3.4 b). So we have

[FT IwI2_ GEM [WI IV+MDI2 -1-ld2|“)

for a constant Cl. C]

Lemma 2.3.7 Let ’y be a loop in T2. Then there exists a constant C(",’) such that for
any (A, 1b) satisfying the Seiberg- Witten equations on the cylindrical end, the following

estimate holds for any t1 < t2:

/ / W s cm] [71W + [:1]me +1lwl“))-

Proof: Note that it suffices to prove the estimate for t2 = t1 + 1. Also note that

both sides of the estimate are gauge invariant. By the embedding L1’(T2) —> L2("/),

we have

57

2 T2 2 ,,2 2 2 ,52
fIwI sC/T,(IV wl +|¢| )SC/T,(IVAI/2l +|A®w| +ItI>.

On the other hand, in U = [t1 — 1, t1 + 2] x T2, A can be decomposed into A = B + h

in a Hodge gauge (Lemma 4 in [17]) such that llBllwa) g C]|dA|]L2(U) g Cllliblliqu)

and h is harmonic with norm bounded by K. Hence

n+1 2 2 n+1 2 2 2
[,1 /T,IA®2I SK /. (TM +IIBII,,(U,.IIIIIIIL.(U,.

The lemma follows easily from these estimates. C]

Lemma 2.3.8 There exists a constant c2 such that the following estimate

t2 t2+l 1
la(t1)-a(t2)l 3 adj“ fT, lz/2|2+/tl_1 fT,(|VAibl2+§lwl4)

t2+1 1 1
“l.-. [fluvmwgwrno

holds for any finite energy monopole (A, 2)).

Proof: Without loss of generality, we assume (A, 1b) is in the standard form. Then

|a(t1) - a(t2)|

|/\

I /\

;(| 71211051) — 71A1(t2)|
+ WUAIUI) — a(t1)| + [24102) — a(t2llll

2

Ed”

i=1 1

7. [(114] + C(llA1(t1) " a(tllllL§(T2)

58

+llA1(t2) — a(t2lllL2(T2)))

'21 C(ff / Ill/42+ +(ftxx2 WW1 + (ftgxp MPH
62/f7‘2wl2+/i:-11/7~2(|VW|2+1M“)
H/t:2+1/7,2(| VAw|2+%I¢I4))%)

holds for any t1 < t2. Here 71, 72 are the longitude and meridian. [3

|/\

|/\

Definition 2.3.9 Choose an increasing function F(r) > 0 satisfying
(—- +1)F(r) + I‘(r )%< cglr.

A finite energy monopole (Art) is said to be “r-good” if there are to and T with
T S to such that a(to) E ’H(2r) and ff: fT2(|VAi/Jl2 +1le4) < F(r).

The “r-good” monopoles of finite energy have the following good property.

Lemma 2.3.10 Let (A,zb) be an “r-good” monopole of finite energy with T as in
the Definition 1.3.9. Then for allt E [T,oo), a(t) is in ’H(r). Moreover, a0o =

limH00 a(t) exists in ’H(r) and the following estimate holds for any t E [T, 00):

Ia(t)—a..I s ((—+1)f:/,((IVAz/2|2+-1-W|4)

+(/,_1)/T,(I IVA¢|2+ 1w )>2).

Proof: It follows easily from the definition of “r-goodness” and Lemmas 2.3.6 and

2.3.8. B

59

Lemma 2.3.11 For any r > 0, there exists a 60(7') > 0 with the following signifi-
cance. For any 6 > 0, there exists an 61 > 0 such that for any “r-good” monopole
(AM/1): when fiooo—l fr2(lVA?(I/‘l2+%lwl4) < 61 for to 5 11,00): we have fzot: f7"? W112???“ < 5

for any 6 g 60(r).

Proof: Without loss of generality, assume that (A, 2b) is in the standard form. By
Lemma 2.3.10, there is a number T > 0 such that for all t E [T, 00), a(t) is in ’H(r)

and a5o = limHoo a(t) exists in ’H(r). Moreover, the estimate

C1 00 2 1 .4
(a(t)—as s c.((m+1)/,_1/T,(Iv.u +§lw|)
+(/,_, [T,(IVAwI2+1lvl4))2).

holds for t E [T,oo). We can further apply a gauge transformation so that a00

 

lies in the compact set (r) / (Z 6 Z). There exists a 60(r) > 0 such that for any

a E (r)/(Z6 Z),

 

IIDZQwIIim-z) Z 450(Tllllblliyr2)
for any 1b E PUT/0). Set u(t) 2 fth2 liblg. Then we have 5—12u(t )— —— 2ftx7~2((5-t2¢ ¢>+
[afiflb’lzl- But

82 8A 8A
f T267222 «2) = / MID f—wI2 IAowl2— (—ii+ —°v i)

f... IDZZwIT— 0(t>IIwII2,(T.,(t)

IV

where 0(t) ——> 0 as ft°flfT2(IVA2/2l2 + 1|wl4) —> 0 by the estimates in Lemma 2.3.4,

Corollary 2.3.3, and Lemma 2.3.10. So when f,:°_1fT2(]VAw|2+1|i/2|4) is small enough,

60

we have
2

bat—Qua) Z 460(r)u(t)

for t Z to. By the maximum principle, we have u(t) _<_ e460(7)(‘°“)u(t0) for t 2 to.

Hence

/°° anemia g C(60(r))u(t0)
2to
holds for any (5 _<_ 60(7) The lemma follows easily. [I

Proposition 2.3.12 Assume that the “r-good” monopole offinite energy (A, w) is in
the standard form. Then (A,2,/)) is in A5(r) for any weight (5 E (O,min(6(r),(50(r))).

Moreover, the following estimate

00

”(A — amawflng‘éflTpflxT?) S 03(5)(/

2 26t l
Tzlwle )2

holds for a constant 03(6) and any T E [l,oo). Here 6(r) is referred to Proposition

2.1.12.

Proof: It follows from Lemma 2.3.11, the estimates in Lemma 2.3.4, Taubes inequal-
ity and standard elliptic estimates. [:1

The convergence of “r-good” monopoles of finite energy

Proposition 2.3.13 Let (Ami/2n) be a sequence of “r-good” monopoles of finite en-

«1

ergy. Then a subsequence of (Amwn) converges in M5(%) to a Er-good” monopole

offinite energy (A0, wo) for any weight 6 E (O, min((5(§), 60(5)».

Proof: The Weitzenbock formula and maximum principle yield an upper bound K

for the C0 norm of the spinors (see Corollary 2.3.3). Then the existence of a local

61

Hodge gauge ([17]) plus elliptic regularity and a patching argument ([30]) imply the
existence of a “weak” limit. More precisely, there exist a finite energy monopole
(Aw/)0), a subsequence of (An,T/)n) (still denoted by (A,,¢»,,)) and a sequence of
gauge transformations 3,, such that 3,, - (An, wn) converges to (A0, $0) in C°° over any
compact subset of Y. We can further assume that 3,, - (An, ([4,) are in the standard
form and therefore the limit (A0, do) is also in the standard form. For simplicity we
still use (An, #2”) to denote sn - (An, 2,19”).

Take To large enough so that £9311 fT2(|VA01/20|2 + élwol“) < Hg) (see Definition
2.3.9). Note that for any finite energy monopole (A, w) the Weitzenbock formula

yields the following equation

00 1 2
[0 /T,(vaI2 + 5W) = (XT,<D§.w,w>.

It follows from the “weak” convergence of (An, wn) to (A0, wo) that there is an N such

that when n > N we have

°° 1
fTo_1/T,(|VA,.%|2 + §|¢n14)< Hg) < F(r)_

Since (Amt/2n) are “r-good”, by Lemma 2.3.10, an(t) is in ’H(r) for any n > N and
t E [T0,oo). From this it follows that (Ami/20) is a “%r-good” monopole of finite
energy, and therefore is in 115(3) for any weight 6 E (0,min(6(g), 60(3)). It is also
easy to see that an,00 —> 00,00-

In order to prove that (Ami/1n) converges to (A0, 2,00) in A5(§) for any given weight

6 E (0, min(6(§), 60(5)), it suffices to prove that given any 6 > 0, there exist to E [1, oo)

62
and N such that when n > N,
”(An — (111,00: w)HL§‘6([2tO+1,oo)xT2) < 6.

This goes as follows. By Lemma 2.3.11, there exists an 61 > 0 such that when

00 1
f / (IV/inlpnl2 + —|?/)nl4) < '31,
to—l T2 2

we have

00 2 26t —1 2
A [T, IwnI e < (c3 (6)6).
0
Now take to large enough so that

00 1 e
2 _ 4 _1
[M /T,<vaoI + 2Ion )< 2.

Then there exists an N such that when n > N we have

°° 1
/ 2 _ 4
/,O_1/T,(|V.4nwnl + 2WI ) < 61.

Therefore we have

“(An _ an,oo: w)|lL§‘6([2to+1,oo)xT2) < 6

by Proposition 2.3.12. Hence the proposition is proved.

CHAPTER 3

The Gluing Formula

3.1 The gluing of moduli spaces

Assume that Y,- (i = 1,2) are oriented cylindrical end 3—manifolds over T,2 where
Y2 is actually diffeomorphic to 02 x S1 carrying a metric whose scalar curvature
is non-negative and somewhere positive. By the Weitzenbiick formula, the moduli
space M(Y2) actually only consists of reducible solutions. Assume that there exists
an orientation reversing isometry h : T12 -—> T 22 which is covered by the corresponding
bundle maps. For any L > 0, we can form an oriented Riemannian 3-manifold YL as

follows:

YL=Y1\[L+1,OO) XT12U1/2\[L+1,OO) XT22,
h

where h: (L,L+1) x T12 ——> (L+1,L) XT22 is given by h(L+t,a:) = (L+1—t,h(3:))
for t E (0,1) and a: E T12. Note that the isometry h : T,2 —> T22 induces an isometry
h : 711(T12) —> H1(T22).

Throughout this section, we fix a small r > O and a small weight 6. For simplicity,

63

64

we omit the dependence of r and 6 in the discussion. We also omit the perturbation
data since it is vanishing on the neck.

Let g~(Y,-) be the normal subgroup of Q(Y,-) which consists of elements in the
component of identity. We define A3106) to be the space of Q~(Y,-)-equivalence classes
of the solutions to the Seiberg—Witten equations on Y,- (i = 1, 2). Then AMY.) is a

Z-fold cover of M(Y,-):

O—+Z—+M(Y,-)—>M(Y,~)——>O.

The irreducible part of M(Y1) is denoted by M*(Y1), which is a Z-fold cover of
M04).

Let 8(Y1, Y2) be the set of pairs (a1,a2) E M*(Y1) x M(Y2) such that there are
smooth representatives (A1,z/)1) and (A2,w2) satisfying hR1(A1,w1) = R2(A2,w2),
where R,- : .A(Y,-) —+ H1(7]2)®iR.

Definition 3.1.1 (01,02) E S(Y1, Y2) is said to be regular if

~

1. the map [R1] :M*(Y1) —> H1(T12) <8) iR is injectiue at al.

2. h[R1](/\;i*(Y1)) and [R2](./\;i(Y2)) intersect transversally at [R2](C¥2).

Note that for a generic perturbation, 8(Y1, Y2) consists of regular pairs (Proposition
2.2.2). We assume that 8(Y1, Y2) is regular throughout this chapter.

For any L > 0, fix a cut-off function pL(t) which equals to one for t < L and
equals to zero for t > L + 1 with |p’L| < 2. Given a regular pair (a1, (12) with smooth

representatives («MM/)1) and (A2,w2), we construct an “almost” monopole (AL,wL)

 

65

on YL as follows:

'11-’14 = mil/’1 + (1 — pL)h"1'rb2 on [L, L +1] x T12
AL = PIX/41“ R1(A1)) + 31(441)

+(1— pL)h‘1(A2 — R2(A2)) on [L, L +1] x T,2
10L = 1/12' on Y; \ [L, 00) X T?
AL 2 A,- on Y; \ [L, 00) x T12.

Note that M, is compactly supported in Y1 \ [L + 1,00) x T2.

Proposition 3.1.2 Assume that ((11,012) is regular. Then for large L, the “almost”
monopole (AL, 1%) can be deformed to a non-degenerate monopole T(AL, wL) satisfy-
ing

[[T(AL:1/)L) _ (AL’h/JLHILflYL) S CL2€_6L.

Moreover, any monopole (A, 11)) on YL which is in the Liz-ball of radius K1L_6 centered
at (AL,wL) is gauge equivalent to T(AL,wL). In particular, there is a well-defined
gluing map T = 8(Y1, Y2) —> M‘(YL) by T(al, 02) = [T(AL, 1121.)]-

The following estimate on (AL, $1.) is straightforward.
Lemma 3.1.3 ”(*dAL + T(7,L’L, ’tlJL), DALlellLZO’L) S 06—61’.

Next we estimate the lowest eigenvalue of AL 2 d’d + [1,/214]” Set

, A 2
AL =1I1f-———-——fh’l Lfl .
”50 fYL flz

 

66

Lemma 3.1.4 Assume that one of 1/11 or 1/22 is not identically zero. For any function

A/(L) = 0(L“4), there exists an L0 > 0 such that when L > L0, we have AL > 7(L).

The basic idea of the proof is the same as in Theorem 4 of Appendix B, but the
argument is more difficult. We postpone the proof to the end of this section. From

now on, we assume that one of 1/11 or 2122 is not identically zero.
Corollary 3.1.5 The norm of Ail : L2(YL) —> L§(YL) is at most L3 for large L.

Proof: In Lemma 3.1.4, take 7(L) = K2L"6 with K to be determined later. There

exists a constant C independent of L such that for any f E L§(YL), we have

llflltgm) S C(HALfHLzm) + Wilma/t))
S C([IALfHL2(YL) + K_1L3”ALfHL2(YL))

S LBIIALfllL2(YI.)

for large L and a suitable choice of K. This proves the lemma. CI
Let TBlAL.wL) be the tangent space of B‘(YL) at (AL,z,bL), ['(Amlm) be the L2
completion of TBZ'AL’W). Then TBEALM) = {(a, (b) E A(YL)| —d"‘a+i(i1,bL,(QR,3 = 0}.

Lemma 3.1.6 There exist constants K1, K2 with the following significance. When

L is sufficiently large, for any (A, w) E A(YL) satisfying
[[(A, Ill) - (Aull’LlllL'f-(YL) S K1176,

there exists a gauge transformation 3 E 9(YL) such that s 0 (A, it) — (AL,?,./)L) E

67

TBEAL va) 071d

lls - (A’s/J) - (ALalellLflYL) S K2L3II(A, 1D) — (Anmhltgm-

Proof: The point of this lemma is to have an estimate on the size of the local slice
at (AL, #21,) if an upper bound for the norm of All is known (Corollary 3.1.5).
Assume that (An/J) is in .A(YL). Set (a, gb) = (A, w) — (AL, I/JL) for simplicity. To

construct the local slice, we look for f E Lg such that
—d*(A — AL - df) + “What/J — wL)Re = 0-
This can be written in terms of ((1, gb) as
(AL + (1.1/M, 1'45)an + C(Cb, f) = dia — “Wt, (files,
where G(¢, f) = i(ii,bL, (ef — f — 1)(¢ + wL))Re satisfying
“0%, f1) — GUI), f2)HL2 S CmaXUIfIHLgaHf2lltglllf1- f2lltg

for some constant C and ¢,f,- satisfying ||cf>||Lzlz < 1 and ||f,-||L§ < 1 for i = 1,2. The

lemma follows by applying Banach lemma to the map

BU) = (AL + (m. i¢>R.)‘1(d*a — mm, (2)3. — Gm, f))

mapping an Lg-ball of radius K L’3 into itself for some constant K. E]

68

Next we deform the “almost” monopole (AL, $1,) to a monopole. Let H be the L2

orthogonal projection onto [hag/M). For any (a, ct) E TBZ'A ”I, 1.), we define

L(a, (l?) = H(*d(AL + a) + TWL + 6b), D(AL+a)(7tDL + ¢))

= (*dAL + TEAL), BALM) + V3(ALfll’L)(a7 <15) + “QM, <15)-
Here VSMML) : TBE‘ALM) —> L(Amt) is given by
V8(ALa1lJL)(a7 (t)) = (*d0 + 270/2, ¢) — df(¢)a DAM + aI/JL + f(cblt/JL)
with f(gb) given by the equation
ALf = i<DALwLai¢)Re-
Q(a, ¢) = (TM), CW5) SfltiSfies
||Q(ait¢1)— Q(a2t (Z52)||L2(YL) S C(||(ait¢1)||1.§ + “(a2t¢2)||L'~;)||(aia¢1) — (a2,¢2)||1.§-

Lemma 3.1.7 There exists a constant K; such that when [[(a, ¢l|lLf(YL) g K3L‘3 for

large enough L, L(a, (b) = 0 implies that

(*dML + a) + TWL + ¢), D(AL+a)(?l’L + W) = 0-

69

L(aa ¢) : H(*d(AL + CL) + T(wL + (b): D(AL+a)(wL + ¢))

: (*d(AL + a) + TWL + (P) _ dg(aa Cb): D(AL+O)(wL + (15) + g(a, gill/1L)

where g(a, 45) satisfies the equation

ALg : Z.<l)(AL+a)(7~.DL + (A), z.Qise-

If L(a, ab) = 0, then D(AL+a)(Tl)L + (b) + g(a, (3)101. = 0. So for large L, we have

”g(aa¢)HL§(YL) S LBHALg(aa¢)HL2(YL)
S L3H<9(aa¢leti¢>ReHL2(YL)

S CLBIlwtllcochlltgmfllgma¢>||Lg<m-

If [[(a, ¢lllL§(Yt) S K3L‘3 for a small enough constant K3, we have g(a, (b) = 0, which

proves the lemma. D

Lemma 3.1.8 Assume that (01,02) is regular. Then Vsmbm) 1 TBZ'ALML) —’

L(ALn/u.) is invertible for large L. Moreover, the norm of (Vs(AL,¢L))‘1 is at most

L2 for large L.

70

Proof: First of all, VSMMJL) : TBIALM) —> £01m“) can be extended to an operator
VSMLJPL) 0 0
I —
KU‘LJJ’L) — 0 0 dMLAbL) ’
0 fALttllL)

where d(AL,¢L)(f) = (—df,f1,/2L). Secondly, by Theorem 4 in Appendix B, the lowest
eigenvalue of
DAL Ulla ”(I/)1;
[C(AL,wL) = 27'(1/}L,) *d ——d
“ML, 'lne —d* 0
is at least K L’2 for any constant K when L is sufficiently large. Here we essentially
use the fact that 21);, is identically zero on the Y2 side and not identically zero on the
Y1 side and the assumption that (01,02) is regular so that the regularity and the
transversality conditions in Theorem 4 of Appendix B hold. Finally, the difference
between ICE/4am) and [CHM/”4) can be ignored since the norm of [Cl/11.312) — ’C(AL,¢L)
is bounded from above by CL6[[DAL1/JL[[CO g cL6e"‘5L by Lemma 3.1.3 and Corollary
3.1.5. So the lowest eigenvalue of V3(At,wt) is bounded from below by K L“2 for any
constant K when L is large enough. The lemma follows easily. [I]
The Proof of Proposition 3.1.2:
In order to deform the “almost” monopole (AL,1,bL) to a monopole, we need to
solve the equation L(a, o) = 0 for (a, ct) E TBZ'ALM). This equation can be written

as

(a, 45) = _(VS(ALt¢’Ll)—1((*d‘4L + 7W1), DAL‘J’L) + “QM, 625))-

71

Assuming that ((11, 02) is regular, it then follows from Lemmas 3.1.3 and 3.1.8 that

the map

B(av ¢) = —(V8(Attwt))_l((*dAL + 717/4), DALwL) + HQ(av (bl)

maps an L2—ball of radius KL’2 in TB“ into itself and satisfies
1 (ALJZ’L)

||B(a1,gb1) — 3(02,¢2)HL§(YL) < |l(a1,qb1)— (a2,¢2)HL’f(YL)

for large enough L and some small enough constant K. Therefore the equation
L(a, (t) = O has a unique solution (aL, ¢L) in the Lf-ball of radius K L”. By Lemma

3.1.7, the resulting monopole is T(AL, in) 2 (AL, 1121,) + (aL, $1,), and
”Twat/1L) — (ALfll’LlllLfo/L) S 01:26—61“-

Suppose that monopole (A, w) is in an Li-ball of radius K 1L‘6 centered at (A1,, tbL).
By Lemma 3.1.6, (A,2b) is gauge equivalent to a monopole in the local slice with
distance from (AL, tbL) less than KgKlL’3, and it must be T(AL, 1111.) by the unique-
ness of the solution (aL, 451,) to the equation L(a, gt) 2 0. In particular, [T (AL,’l/)L)]
depends only on the homotopy class of (AL, #11,), which implies that there is a well-

defined gluing map T : 8(Y1, Y2) —+ M*(YL). By Lemma 3.1.8 and the estimate

HTML, IPL) " (Atttb‘Lllltfo/L) S 01426—61“,

[T(AL, #4)] is non-degenerate. Therefore the proposition is proved. D

72

The Proof of Lemma 3.1.4:

Suppose that there is a sequence of L, —> 00 such that

/\L,. S ”7(Ln) = 0(L—4)1

n

then there exists a sequence of cn > 0, f,, 75 0 such that ALnfn = Cifn and on =

Claim: There exist constants Ad and L0 with the following significance. The f,’,s

can be chosen such that [[fn|[00(yLn) S M, and one of the following is true:
1. either meO) [nt2 or meo) |f,,|2 is equal to one;
2. either anHL2(T3)(L0) or [lntIL2(T22)(LO) is greater than or equal to one.
Here Y,(L0) = Y; \ (L0, 00) x T12, i = 1,2.

Assuming the Claim, Lemma 3.1.4 is proved as follows. By elliptic estimates, we
can select a subsequence of fn which is convergent in C°° to f,- on Y,- (i = 1,2) on any

compact subset. Moreover, f1, f2 satisfy the following conditions:
a) d‘df, + [wilzfi = 0 011 Y." i = 1,2;
b) Hftllcom) S M, i=1,2;
c) one of f1 or f2 is not identically zero.

Lemma 3.1.4 is proved if we show that conditions a) and b) contradict condition c).

In fact, by condition a), for any t, we have

0 : d*di [2'21'71'
(M f+|wl f f)

i

73

'2 I.2 . _12 .
)0de + IeI If.|2) , a; [1,, If.|2)(t)-

- l

I ‘ I

Since {,(t) : ng |f,|2(t) is bounded by b), there exists a sequence of t,, such that
%%(t,) —+ O as tn —> 00. Therefore in(|alf,-|2 + |w,|2|f,|2) = 0, i = 1,2. So f1 and f2
are constant functions and one of them is zero, since one of 1P1 or $2 is not identically

zero. But in the proof of the Claim, it is easy to see that f1 2 f2. So both of f1 and

f2 are zero, contradicting c).

The Proof of the Claim:
For simplicity, we omit the subscript Ln or n if no confusion is caused. Write
f = 91 + 92 where gl E lCer ($2de and g2 E (lCer d}2dT2)L.

Pick L0 > 0 large enough, there are two possibilities:

o On [L0, 2L + 1 — L0], max [g1| s max ||g2|[1,2(T12). In this case, by the maximum
principle, for large L0, ||g2[|L2(T12) can not reach its maximum in the interior of

[L0, 2L + 1 — L0]. The Claim follows easily in this case.

0 On [L0, 2L +1 — L0], max [[92[[L2(T12) s max [g1]. In this case, we need to show
that for large enough L either |g1] reaches its maximum at the endpoints of
[L0, 2L + 1 — L0], from which the Claim follows, or either max [91] g K|g1(L0)|

or max |g1| g Klgl(2L + 1 — L0)| holds for a constant K independent of L.
Assume that we are in the second case. On [L0, 2L + 1 — L0], we have

-;9—t§91(t) + h(t) = c2910)

 

 

where h(t) is the LQ-projection of w], 2f into ICer (1}grlT2. c = 0(L‘1) and h(t)

74

satisfies

[h(t)] 3 K64“ max|g1[ on [L0,L +1)

and

[h(t)] S Ke—26(2L+l_t) max [91] on (L, 2L + 1 — L0].

Set g3(t) = c‘1%91(t), then we have

$91“) = 09305)

%93(t) Z ‘—Cgl(t) + C_1h(t).

These equations can be written equivalently as

8 Ct 910) Ct 0
E e = e
g3(t) c’1h(t)
0 —c cos ct — sin ct
where C = . Note that eC‘ :
c 0 sin ct cos ct

Since 0 = 0(L‘1), we have cos ct > % and
t t
| L sin cs - c"1h(s)ds| S K/L |sh(s)[ds S Ke‘aL0 maxlgll
0 0

for large L and any t E [L0, L + 1). On the other hand,

t
gl(t) cos ct — 93(t) sin ct = —/ sin cs - c”1h(s)ds + g1(L0) cos cLo — g3(L0) sin CLO.
Lo

So if |g1(t)| reaches its maximum in the interior of [L0, 2L + 1 — L0], without loss of

75

generality, assuming that it is in (L0, L + 1), then
émax [g1] S Ke‘“0 max |g1[ + [91(L0) cos cLO - g3(L0) sin cL0|.
Hence for large L0, we have
max |g1| g 4|g1(L0) cos CLO — g3(L0) sin cLOI.
On the other hand,

g3(t) = /: cos c(s -— t) - c’1h(s)ds + g3(L0) cos c(Lo -— t) + g1(L0) sin c(Lo — t).

Assume that [g1(t)| has its maximum at to E (L0, L + 1). Then g3(t0) = 0 and

[91(L0) Sill C(LO — to) + 93(L0) COS C(LO — to)[

to
/ [c'1h(s)|ds
L0

< Kile—26L0 max |g1[

|/\

S 4Kc_le_26L°|g1(L0) cos cLo — g3(L0) sin cL0|.

Since c = 0(L‘1) as L —> 00 and L062“0 2 0(1) as L0 —> 00, we have

l93(Lo)| s 10Kc-‘e‘25L0Igt(Lo)I + Igt(Lo)I.

76

Hence

maxlgll < 4|g1(L0)coscL0—g3(L0)sincL0|

|/\

4|91(L0)| + “0193—16—2“0 [91(L0)[ + [91(L0)|)| Sin CLO]

|/\

5[91(L0)[+10KL0€_26L0[91(L0)[

|/\

K1l91(Lo)l

with K1 independent of L when L0 is sufficiently large. This proves the Claim. (Ob-
serve that |(g1(t0) cos cto — 93(t0) sin cto) — (gl(L0) cos CLO — g3(L0) sin cL0)| _<_ Ke‘“0
and |(g1(t0) cos cto —g3(t0) Sin cto) -(g1(2L+1—L0) cos CLO —g3(2L+1 — L0) sin cL0)| g

Ke"‘5L0 where to = L + a from which one sees f1 2 f2).

3.2 Geometric limits

Let Y be an oriented integral homology 3-sphere decomposed into Y 2 Y1 UTz Y2,
where Y1 is the complement of a tubular neighborhood of a knot and Y2 is diffeomor-
phic to D2 x S 1. Equip Y with a Riemannian metric such that a collar neighborhood of
T2 is orientedly isometric to (—1, 1) x R/27rZ x R/27rZ with (—1,0) x R/27rZ x R/27rZ
in Y1 and Y2 carries a non-negative, somewhere positive scalar curvature metric. We
insert cylinders of lengths 2L + 1 and obtain a family of stretched versions YL of
Y. We also use Y1 and Y2 to denote the corresponding cylindrical end manifolds.
Note that the finite energy monopoles on Y2 are reducible by Weitzenbéick formula,
so the moduli space M(Y2) is identified with the line H1(Y2) ®iR of imaginary valued

“bounded” harmonic 1-forms on Y2 which is embedded into H1(T2) ® iR by the map

77

R2 (Lemma 2.1.4). With a small perturbation, we assume that MOE?) misses the
lattice of “bad” points for the spin structure on T2 where the twisted Dirac operators
are not invertible.

Let (An, wn) be a sequence of monopoles on YLn, Ln ——> oo. Weitzenbock formula
and the maximum principle yield an upper bound K for the C0 norm of the spinors,
which depends only on the scalar curvature of the manifolds. Then the existence of a
local Hodge gauge ([17]) plus elliptic regularity and a patching argument ( [30]) imply

the existence of geometric limits as we stretch the neck.

Lemma 3.2.1 There exists an r > 0 with the following significance. Let (Amwn)
be a sequence of monopoles on YLn, Ln —> 00. Then there exist a sequence of gauge
transformations 5,, and a pair offinite energy monopoles (A,, 712,-) on Y, (i = 1, 2) such
that a subsequence of 3,, - (A,,, w”) converges in C°° to (A,, 1b,) on any compact subset
on, (i = 1,2). Moreover, (A1,w1) and (Ag,1b-2) are “r-good” monopoles and have

the same limiting value, i.e. R1(A1, 1121) = R2(A«2, 1192).

Proof: Exhaust Y,- by a sequence of compact subsets Ki," such that Kim C Km“,
i = 1,2. There exist a subsequence of (Ami/in) (still labeled by n), a sequence of
gauge transformations 3,,” defined on Kim, and monopoles (A,, #2,) on Y,- such that
3,," - (Amwn) converges to (44.3%) in C°° on any compact subset of Y,. Note that
(A,, 112,-) are of finite energy by Weitzenbock formula.

First we show that there is an r > 0 such that the geometric limits (A,,i/2,-)
(i = 1, 2) are “r-good”. Let (1 be the distance between the lattice of “bad” points and
the line R2(M(Y2)) in H1(T2) ® iR. We simply take r z: 7615. Note that w E 0 and

a2(t) = (12,00 for all t E [0, 00), so (A2, $2) is “r-good”. On the other hand, there are

78

t0 and N such that |(D§f‘ltp1,v,)(t0)| < %I‘(r) and [(Djififlnfll‘nxtofl < F(r) when
n > N (see Definition 2.3.9 for F(r)). For large n, 032,".(An)(2Ln) is in ”H(4r), so is
a..(2L..). By Weitzenback formula, hummer + emu“) < I(D£:,.wn, extol <
F(r) when n > N, where Y2(t0) = Y2\(2Ln+1—t0,oo) X T2. So for large n, a.,,(to+1)
is in ’H(3r) (Lemma 2.3.10) and so is asl,,.-(/I,.)(to +1). So a1(t0 + 1) is in ’H(2r). It
follows easily that (A1, 1,01) is “r-good”.

Next we show that

1. The sgms can be chosen so that as n —> oo, 31"},sz . Szanz is in the component

of identity of M ap(T 2, S 1). As a consequence, s1”, and 32," can be extended to

an 3,, E Q(Y1,n).

2' R1(A17,¢)1)= Fifi/42,7192).

Given any 6 > 0, pick L0 large enough so that
”(Aifll‘tXl/o) — RI(At,¢t)|[Ck(r2) < 6, 75:1)?-
For large enough n, we have
H81,n'(Am¢’n)(L0) —(Al,’l/11)(L0)Hck(r2) < 6

and

“San ' (An,1l/'n)(2Ln +1 — L0) — (A2, VJ2)(L0)HC'¢(T2) < €-

79

Let 7 be a generator of H1(T2). Then we have

I flame.) - autism

g 30t+I/(s...-A (Lt)—s...-A.(2L..+1—Lo))l

3 306+] L f (1.4,, +2m[s;},|T. . 32,,IT21I,
0 '7

where [SI—,idfl '82,,1IT2] is the degree of the map 31"},[T2 - sg,n|T2 : 7 —> 31. On the other

hand, we have estimates

I A:""+‘”L°fdA.I s 0]?"+1 LO/Iw I2
g 0]?” °/ (Ivmnl +—Iwn“l)
g 01(|(D§:,wn,wn)(Lo — 1)|

+I(D§:,w.. 2mm. + 2 — LoII

by Lemmas 2.3.6, 2.3.7 and Weitzenbock formula, from which it follows that
R1(A1,ib1) = R2(A2,1b2) in ”H1(T2) ® iR/(Z EB Z). It follows that (Am/2,) can be
modified by an element in 9(12) so that R1(A1,1/11) = R2(A2,w2) in H1(T2) ® iR
(due to the fact that Y is a homology 3—sphere) and 31”},sz - 32,,sz is in the identity
component of M ap(T2, S 1) for large n. I]

In the following discussion, we fix the number r in Lemma 3.2.1, a weight 6 small
and a generic perturbation (g, f, u) with a sufficiently small.

Recall the set 8(Y1,Y2) of pairs (011,02) E M*(Y1) x M(Y2) such that there are
smooth representatives (A1, $1) and (A2, wg) satisfying R1(A1,w1) = R2(A2,7,b2). By

80

Proposition 3.1.2 and Lemma 3.2.1, each pair (01,012) in 8(Y1, Y2) is “r-good”. There-
fore by Proposition 2.3.13 ( convergence of “r-good” monopoles) and Proposition 2.2.2

(3), 8(Y1, Y2) is compact and hence consists of finitely many points.

Proposition 3.2.2 For large enough L, the gluing map T : 8(Y1,Y2) —> M‘(YL)

given by T(a1, a2) = [T(AL, 1%)] is one to one and onto.

Assume that a sequence of irreducible monopoles (Amwn) on Yr... converges to
geometric limits (A,, 112,-) on Y,- (i = 1, 2). Note that ’l/Jg E 0 and (A1, 1121) is irreducible
since the (perturbed) Dirac operator at the reducible point on Y1,” is invertible for
large n and the norm is uniformly bounded from below (Theorem B in [3]). Our next
goal is to show that for large enough n, (An, 21)“) is in the image of the gluing map T.
This is done by showing that up to a gauge transformation the L? distance between
(An, 21)”) and the “almost” monopole (ALn, ibLn) is less than K 1L; 6 (see Proposition
3.1.2).

For simplicity we omit the subscript n in the notation if no confusion is caused.
As in Lemma 2.3.11, there exists an L0 > 0 such that for any t E [L0, 2L + 1 — L0],

we have

[T, We) 3 ewe—t) [T, Ii>I2(Lo) + emu-”Mr [T, [4426314 +1— L.)

For each L > 0, fix a cut-off function pL which equals to one for t g L and equals to

zero for t 2 L + 1. We construct an “almost” monopole (A1, 1,51) on Y1 as follows:

A1 = pL(A—a(L+1))+a(L+1) on Y1\[L+1,oo)xT2

81

A1 = a(L +1) on [L + 1,00) x T2
151 = pLQ/J 0n Y1 \ [L +1,00) X T2
ibl = O on [L + 1,00) x T”.

(Note that we have omitted the subscript n in the notation; here A = An and L = L").
Here a(t) is the harmonic component of A[{t}xr2-

The following estimate is straightforward.
Lemma 3.2.3 [|(*dA1+ T('l,l;1), DAlifillllLffiYi) S CC—6L h0ld3 for (14.1, @131) on 1/1.

Recall from Definition 2.1.10 that for (A, 1b) E A", Vsmm : TBZ‘AM —> Low) is

given by
V3(A,w)(at <25) = (*da + 2T(149,4?) - df(a, <15), DA¢ + all) + f(a, can)
where f(a, (15) is the unique solution to the equation
d*df + fit/2)? = i<DA1llti¢>Re

Lemma 3.2.4 For all sufficiently large L, VSM-mii) : TBZ‘A-l ‘51) —> £011,151) is surjec-

tive. So there exists a bounded right inverse P : £041,151) —> T868451) satisfying

||P(a. <f>)ll.4 s KIIIa. alum.)

for a constant K independent ofL (see Definition 2.1.1 for the norm H “Al-

82

Proof: Let H be the L2-orthogonal projection onto £04131”), 7r be the L2-orthogonal

projection onto T B; A} #31) and I be the right inverse of Vsmmm (1 exists by the

assumption that 8(Y1, Y2) is regular). For (a, gt) E £(A-mlit)’ we have

VSMIJIJIWIWG. <15) = (a, 4)) + 0(1)(a, ¢)

as L —+ 00. Here the key point is that d*d + [151? is invertible and the norm of the
inverse is bounded uniformly in L (Lemma 2.1.7). E]

Next we deform the “almost” monopoles (A1, ibl) to monopoles. Let Ill be the L2

*

(AH 151), we define

orthogonal projection onto [’(A'min' For any (a, 45) E TB

L(a, <15) : H1(*d(.41+ a) + r(i,b~1+ 45), D(A'1+a)(1/:1 + ¢))

= (*dA't + Twit). Den/31) + V3(.il,u§1)(aa¢) + mew, <25)
where Q(a, <25) = (Tu), 61¢) satisfying
||Q(at, $1) — Q(a2, ¢2)Iltg‘, S C(l|(a1, ¢I)HA + “((12, $2)l|.4)|l(a1, 451) — (a2,¢2)||A-
Lemma 3.2.5 L(a, a) = 0 implies that
(*d(41+ a) + 7(le + as), D(,cil+a)(¢~l + ¢)) = 0

when ||(a, (t))[IA is sufi‘iciently small.

Proof: A similar argument as in the proof of Lemma 3.1.7. The key point is that

d*d+ [1131]” is invertible and the norm of the inverse is bounded uniformly in L (Lemma

83
2.1.7). I:

Lemma 3.2.6 The “almost” monopole (Abi/il) can be deformed to a monopole
~I ~I ~ I ~I ~ ~ ~ I ~ I ~ ~
(A1 MP1) SUCh that (A1 31/11) — (AMT/11) E TBfA'th) and [[(A1I¢1)—(A1,1/J1)HA S

(76—6L

Proof: A similar argument as in the proof of Proposition 3.1.2. The fact that
d*d+ lib-1|” is invertible and the norm of the inverse is bounded uniformly in L (Lemma
2.1.7) is also used here to get an estimate ||H1Q(a, fluid; 3 c||Q(a, ¢)||L§'6. E]
The Proof of Proposition 3.2.2:

We need an estimate on the restriction of (A, it) on the Y2 side. Similarly we

construct “almost” monopoles ($124.02) on Y2:

A2 = (l—pL)(A-—a(L))+a(L) on Y2\[L+1,oo)xT2

A2 = a(L) on [L + 1,00) x T2
1/32 = (1 — pLW on Y2 \ [L +1,oo) x T2
I32 = 0 on [L+1,oo)><T2.

By Weitzenbock formula and the exponential decay estimate for the spinor, we have
1 _
/ (Iva)? + -I¢l4)SI(D§2.1I.i/J)(L)ls Ce 6‘”
Y2(L+1) 2 ~
for a small (5 > 0 where Y2(L +1) 2 Y2 \ (L + 1,00) x T2. It then follows that

H * dAglng S C8—6L and IltljzllLf’é S Ce’6L.

84

Therefore the distance between R2(A2,0) and [R2](./\;l(Y2)) is controlled by Ce”‘”‘
(Lemmas 2.1.4, 2.2.5). On the other hand, the distance between R2(A2,0) and
R1(A1, 2,51) which is given by |a(L+1)—a(L)| is also controlled by Ce‘“ (Lemma 2.3.4
((1) and the exponential decay estimate for the spinor it). So is the distance between
R1(A1’, 1131') and [R2](M(Y2)) by Lemma 3.2.6. By the assumption of transversality

(Definition 3.1.1 (2)), we have

R1(TM*(Y1)(A1,¢1)) n R2(TM(Y2)(A2.¢2)) : {0}

Then it follows that the distance between R1(A1’, 1131') and R1(A1, 1121) is controlled by
Ce‘aL. Since [R1] : M*(Y1) —> H1(T2) <29 iR is an immersion at [(A1, 421)] (Definition
3.1.1 (1)), the distance between [(Al’nfil’fl and [(A1,i/)1)] is controlled by Ce‘”.
So is the distance between [(441,151)] and [(A1,1/21)] by Lemma 3.2.6. The distance
between [(A2,w2)] and [(A2, 2/22)] is also controlled by Ce‘M‘. Now it is easy to see
that up to a gauge transformation (Amit'n) is within an L? ball of radius Ce'fL"
centered at the “almost” monopole (ALm’l/Jlm) for large enough n. By Proposition
3.1.2, (Ami/2n) is in the image of the gluing map T. On the other hand, it follows
from the “weak” convergence of the gauge transformations that the gluing map T :

8(Y1, Y2) -—> M*(YL) is also one to one. Hence the proposition is proved.

85
3.3 Spectral flow, Maslov index and the gluing for-
mula

First we recall the basic relation between Maslov index and the spectral flow of a
one-parameter family of first-order, self-adjoint, elliptic differential Operators of APS
type on a stretched manifold. The basic references are [3] and [4].

Let M be a closed, oriented, smooth manifold that is decomposed into the union

of two submanifolds Ml, Mg by a co-dimension 1, compact oriented submanifold E,
A! = All UA/Ig, 2 = M1 “A12 2 61% = 611/12.

Equip M with a Riemannian metric such that the hypersurface 2 has a collar neigh-
borhood isometric to (-—1,1) x 2, and 2 = O x 2, (—1,0) x )3 C Ml. We stretch
M by inserting cylinders [0, 2L] x Z and obtain a family of manifolds M (L). Let
M1(oo), Mg(oo) be the cylindrical end manifolds obtained by attaching [0, 00) x Z to
MI, and (-oo,0] x E to M2.

Let D : I‘(E) —> I‘(E) be a first-order, self-adjoint, elliptic differential Operator
acting on the space of smooth sections of a real Riemannian vector bundle E —) M
which is of the APS type near 2. More precisely, on (—1, 1) x 2, E is isometric to

the pull-back bundle 7r*E0 and D can be written as

a
D : 0(a + D0):

where 7r : (—1, 1) x Z ——) E is the projection, E0 —> 2 is a Riemannian vector bundle

86

on E, o : E0 —> E0 is a bundle isometry, and D0 is a first-order, self-adjoint, elliptic
operator acting on I‘(E0). Then E and D naturally extend to a vector bundle E (L)
and an operator D(L) on the stretched manifold M (L), and to Ej(oo) and Dj(oo) on
the cylindrical end manifold Mj(oo), j : 1, 2.

Let t,- be the space of limiting values of the extended L2—solutions of Dj(oo) = 0

over Mj(oo). Denote Kler Do by ’H. Then we have (see [3])

Lemma 3.3.1 1. ’H is a symplectic vector space with the preferred symplectic form
{23.31} = /18 (33,031)-

2. l1, l2 are Lagrangian subspaces in ’H.

We call I,- the Lagrangian subspace associated to Dj(oo).

Let E1, E2 be the restriction of the vector bundle E and D1, D2 be the restriction
of the operator D on the submanifolds M1 and M2 of M. For any pair of Lagrangian
subspaces l1,12 of the symplectic vector space ”H = lCer Do, we have a pair of self-

adjoint Fredholm operators D1(l1), D2(lg) defined with global boundary conditions:

D101) I L%(E1, P+ 6 l1) —) L2(E1)
D202) I L¥(E2, P. 6} l2) —) L2(E2)

where Pi are the subspaces of L2(E0) spanned by the eigenvectors of positive / negative
eigenvalues of Do, and the space Lf(E1,P+ 65 ll) is the Lf-Sobolev completion of
smooth sections of bundle E1 whose restrictions on 2 lie in the space P+ EB l1 and

similarly is the other space Lf(E2, P_ 69 lg) understood.

87

Each homotOpy class (with fixed ends) of one-parameter families of pairs of La-
grangian subspaces (11(3), 12(3)) : a S s S b is associated with an integer which is
called the Maslov index of (11(3), l2(s)) and denoted by Ma8{(11(8),lg(3)) : a S s S b}
(868 [4145])-

The (61, €2)-spectral flow is defined as follows. Let D(s) : a S s g b be a family
of real self-adjoint operators such that for some fixed 6 > 0 the total spectrum of
D(s) in the range of eigenvalues A with IA] < 6 is finite-dimensional and has no
essential spectrum. Furthermore, after taking into consideration of multiplicities,
these eigenvalues /\ with IAI < 6 vary continuously with respect to 3. Let 61,62 be
real numbers with |€1l < 6, |62| < 6, such that 61 is not an eigenvalue of D(a) and 62 is
not an eigenvalue of D(b). Then the (£1, €2)-spectral flow of D(s) : a S s g b is equal
to the number of times the eigenvalues /\ of D(s) in the range |/\| < 6 cross the line
joining ((1,61) and (b, 62) from below, minus the number of times crossing from above
(see [4] for details). The (e, e)-spectral flow will be called briefly as e-spectral flow.

Let D(s) : a S s _<_ b be a one-parameter family of first-order, self-adjoint, elliptic
differential operators on M which are of the APS type, i.e. in the collar neigh-
borhood (—1,1) x E, D(s) = a(% + Do(s)). Furthermore, there exists a 6 > 0
such that there are no eigenvalues of D0(s) in the range (—6,0) and (0,6), and
H = ICer D0(s) is a fixed symplectic vector space for a g s S b. A one-parameter
family of pairs of Lagrangian subspaces (11(3), l2(s)) : a g s S b in ’H is said to satisfy
the endpoint condition if (l1(8),12(8)) is the pair of Lagrangian subspaces associated
to (D1(oo)(s), Dg(00)(3)) at the endpoints s = a, b.

The basic relation between Maslov index and spectral flow is given by the following

88

Theorem 3.3.2 ( Theorem C in [4])
There exists an L0 > 0 such that for any choice of smoothly varying pairs of
Lagrangian subspaces (l1(8),l2(8)) : a S s S b satisfying the endpoint condition, for

all L > L0, the (L’2)-spectralflow of D(s)(L) on [M(L) for a S s g b equals to
2
ZSFE{DJ-(s)(lj(s)) : a S s g b} + [Was{(l1(s),l2(s)) : a S s S b}
j=1

where SF€{DJ-(s)(lj(s)) : a g s g b} is the c-spectralflow of Dj(s)(lj(s)) : a g s S b.
Here 6 > 0 is chosen so that the eigenvalues of Dj(s)(lJ-(s)) in the range [—6, e] consist

of at most zero eigenvalues for the endpoints s = a, b.

Now let’s go back to our own problem. Suppose that Y is an oriented inte-
gral homology 3-sphere that is decomposed as Y 2 Y1 U72 Y2 with Y1 being the
complement of a tubular neighborhood of a knot and Y2 = D2 x S 1. Y carries a
Riemannian metric such that a collar neighborhood of T2 is orientedly isometric to
(—1,1)xR/27erR/27rZ with (—1,0) xR/27erR/27rZ C Y1, where we assume that
the first and second factors in R/27rZ x R/27rZ represent the longitude and meridian
respectively ([1]), and the metric on Y2 has non-negative, somewhere positive scalar
curvature. By inserting cylinders [0, 2L + 1] x T 2, we obtain a family of stretched
versions YL of Y. We also use Y1 and Y2 to denote the corresponding cylindrical end
manifolds if no confusion occurs.

The basic result we’ve obtained so far (Proposition 3.2.2) is that for a large enough
L, the irreducible Seiberg-Witten moduli space M*(YL) of YL is identified via the
gluing map T with the set of “intersection points” 8(Y1, Y2). Here 8(Y1, Y2) consists

of the pairs (01,012) E M*(Y1) x M(Yg) such that there are smooth representatives

89

(A1,i,b1) and (A2,zb2) having the same limiting value, i.e. R1(A1,1/J1) = R2(A2,ib2).
Our next goal is to orient M*(Y1) and M(Yg) appropriately so that their “intersection
number” #S(Y1, Y2) equals to the Seiberg-Witten invariant x(YL) as the oriented sum
of the points in the moduli space M*(YL). This is referred to as the gluing formula
of x.

Fix a generic perturbation (g, f, u) compactly supported on the Y1 side according
to Proposition 2.2.2 and thereafter omit it in the discussion for simplicity. Assume
that L0 is large enough so that Proposition 3.2.2 holds for YLO. Pick a smooth section
d) of the spinor bundle W —> YLO which is compactly supported in Y1\[0,oo) x T2 and
satisfies ((D9 + f)"1(igb), (i¢)) < 0. Then by Lemma 1.2.2, for small enough t > 0,

the self-adjoint operator (on YLO)

D9 + f 0 0 0 (15- (fi-
lC(t,¢) = 0 *d —d +t 27(4), -) O 0
0 —d* 0 i(i¢, )3, 0 0

acting on P (W 69 (A1 EB A0) <8) iR) is invertible and has one small eigenvalue

At ~ -((Dg + f)_1(i¢). (icbllt2 > 0-
According to Definition 1.2.4, the Euler characteristic x(YL0) is defined by

X(YLo) = Z Signfl, where signfi = (—1)SF(’Cfiv’C(t.¢)l
fiEM‘(YLO)

for small t > 0 (SF denotes the spectral flow). Here if fl is represented by (A, 1b),

90

then
DA 2/1' 1/1'
(C3 = (c(/w) = 27W), -) *d —d
“W, '>Re —d* 0
Let (AL0,1,/)L0) be the “almost” monopole being deformed to (A, it) under the gluing
map T (note that wLo is compactly supported in Y1 \ (Lo + 1,00) x T2). It is obvious
that (CB can be replaced by [Q A L0 #140) for the purpose of spectral flow calculation. For
any L > 0, we insert cylinders of lengths 2L into YLO and obtain a family of manifolds

YLmL and operators K(.4L0.wro)(L) on YLOJJ from IC(,4L0,¢L0) in the obvious way.

Lemma 3.3.3 For large enough L0, [C(ALO,¢L0)(L) are invertible for any L > 0. In

particular, the spectral flow between [C(ALofll’Lo) and K(AL0.wLO)(L) is zero for any L > 0.

Proof: For large enough L0 > 0, the operators (C(ALO.¢LO)(L) are invertible for all
0 < L < L0 by Theorem 4 in Appendix B. Suppose that [C(ALO,¢L0)(L) has a non-zero
kernel for some L 2 L0, i.e. there is an :1: 79 0 such that K(ALO,¢L0)(L)$ = 0. On the

inserted cylinder, (c(/41.0 ,¢L0)(L) has the form I (g + B) where

 

 

 

 

(dt 0 0 0) (D? 0 0 0)
I = 0 *T2 0 0 and B = O 0 —dT2 *de
0 0 0 —1 0 —d:;. 0 0

( 0 0 1 0) ( 0 —*de 0 0)

acting on F(W0$(A169A063A0(T2))®iR). Here Wo is the total spinor bundle over T2,
and 0212 is an invertible twisted Dirac operator. It follows that a: can be decomposed

as a: = 2:0 + 13+ + :13- with x0 6 Ker B constant in t and pi have exponential decay to

91

the right / left. Take a cut-off function '7 in the middle of the inserted cylinder, define

yi on the cylindrical end manifolds Y1/Y2 by:
y+ : 7(1: -— $0) + 1:0, y- = (1 —— ’7)(.’E - 1:0) + 270.
Then it follows that

ll’C(ALO,wL0).1(OO)y+HL§ S 08—6L(|ly+ — SFOHLg + ”y— — IEOHLg)

and

ll’C(AL0,wLO).2(OO)y—||L§ S 66—6L(||3/+ - Itolng + ”31— - Ivolng)

for some small (5 > 0 and a constant c. Here [C(ALO,.),LO),J-(oo) is the corresponding
operator on the cylindrical end manifold Y], j = 1, 2. On the other hand, observe
that y+ and y- have the same limiting value 11:0 and for all large enough Lo, the
Lagrangian subspaces associated to (c(/41.0 Judd-(co) (j = 1, 2) are transversal to each
other with angles larger than a fixed number ( due to the fact that 8(Y1,Y2) is
regular). Then the above estimates yield

6L(

“170” S cle“ ”31+ — IOHLg + H3!— - CCOHLgl-

Since both of K(AL0,¢L0),1(00) and [C(ALO,¢LO),2(OO) have no L2 kernels, we have esti-

mates

6L(

”ll/i — 130'ng S 626— ”9+ — 950'ng + H3!- — 330HL§)

which imply that for large L0 (therefore L 2 L0 large) yi vanish identically, contra-

92

dicting the assumption that :1: ¢ 0. Therefore the lemma is proved. C]
The operators considered here have the APS form I (g + B) on the inserted

cylinder where

 

 

 

 

(dt 0 0 0) (193’ 0 0 0)
I z 0 *T2 0 0 and B = 0 0 _dT2 *dT2
0 0 0 —1 0 —d}2 0 0

K 0 0 1 0) K O —*dT2 0 0}

The symplectic vector space ”H = Ker B is H1(T2) <8) iR EB iR EB iR. Let’s fix the
notation about ’H first. Recall that T2 is orientedly isometric to R/27rZ x R/27rZ (lon-
gitude, meridian). Let (:13, y) be the oriented coordinates, then we orient ’l-tl(T2) <8) iR
by idx /\ idy. Furthermore, the 3rd component of 71 corresponds to the dt-component
of the 1-forms and the 4th one is from the Lie algebra of the gauge group.

Let 1C0 (acting on F (W 69 (A1 EB A0) <8 iR)) be the Operator at the reducible point

(0,0):
D9 + f 0 0
[Co = 0 *d —d
0 —d" 0

The corresponding operators 1C0 g(oo) on the cylindrical end manifolds Y,- have no
L2 kernels and the associated Lagrangian subspaces of [€0,1(oo) and [€0,2(oo) are
spanned by (idy, (0,0, 0,1)) and ([R2](TM(Y2)), (0,0,0,1)) respectively. Note that
[R2](TM(Y2)) is transversal to idy since Y is a homology 3-sphere.

Now we are ready to orient the moduli spaces M*(Y1) and M(Yg). Assume that

93

a1 6 M*(Yl) is represented by (A, it). For any vector V E H’(T2) ®iR with positive
ids-component which is not in R1(TM(Y1)(A,¢)), let v E TM(Y1)(A,,),) such that
VAR1(v) = idrrAidy. Pick an L0 > 0 and cut down (A, it) at L0+1. Denote the result
by (ALO, the). We assume that L0 is large enough so that the Lagrangian subspace
associated to [C(ALO,¢LO),1(oo) is transversal to the Lagrangian subspace spanned by
(V, (0,0, 0, 1)). Connect (AmeLo) to the reducible point (0,0) by a path (Ag/2),
which is constant in t on [L0 + 1,00) x T2. Choose a smooth path of Lagrangian
subspaces [1(3) which equals to the Lagrangian subspace (idy, (0,0,0,1)) associated
to 1C0,1(oo) or the Lagrangian subspace associated to [Q A L0"), L0),1(oo) at the endpoints

of the path.

Definition 3.3.4 1. The orientation of M*(Y1) at al : [(A,w)] is determined
by the tangent vector (—1)"’v. Here m is the sum of the (e)-spectral flow of
operators [C(A,¢,)a,1(L0 + 1)(ll(s)) (for a small 6 > 0) and the Maslov index

Mas{(l1(s), lv)}, where [V is the Lagrangian subspace spanned by (V, (0,0, 0, 1)).

2. The orientation of M(Yg) is determined so that the positive direction of
[R2](TJ\;1 (Y2)) has positive idx-component. Note that [R2](T/\;l(Y2)) is transver-

sal to idy-axis since Y is a homology 3-sphere.

Lemma 3.3.5 The orientation on M*(Yl) is well-defined, which induces an orien-

tation on M’(Y1) via the Z-fold covering map M*(Y1) ——> M*(Y1).

Proof: We need to prove that the orientation of M*(Yl) is independent of the choice
of a1 (and its representatives (A, 112)), the vector V E H1(T2) <8) iR, the cut-off point

L0, the path (A, w), and the path of Lagrangian subspaces l1(s).

94

First of all, the independence on the choice of cut-off point L0, the path (A, it),
and the path of Lagrangian subspaces l1(s) follows easily from Theorem 3.3.2. Sec-
ondly, suppose that two monopoles (A1,z/)1) and (A2, 2/12) are in the same component.
Join them by a path of monopoles (A3, 11),) and then cut down the path at Lo + 1 for
sufficiently large L0 (still denote the path by (A3, 7%)). Let l(s) be the Lagrangian sub-
space associated to [C(As,¢,),1(oo). Then the (e)-spectral flow of lC(A,,¢,),1(Lo +1)(l(s))
is zero because M*(Y1) is immersed into ’H’(T2) ® iR so that [C(Aa,¢s),1(oo) have no
Lz-kernels for large enough L0. On the other hand, since (A,,w,) is irreducible so
that the 3rd component of l(s) is non-zero, Mas{(l(s), (V, (0, 0, 0, 1))} (mod 2) equals
to the sign change of VAR1(vs) where v, is a smooth tangent vector field in TM*(Y1)
along the path (As, its). So the orientation at (A1, 1,121) and the orientation at (A2, wg)
are compatible. Finally, suppose that V1, V2 6 ’H’(T2) ® iR are two different vectors
used in the definition. Then Mas{(ll(s), (V1, (0,0,0,1))}—Mas{(ll(s), (V2,(0,0,0,1))}
(mod 2) equals to the sign change from V1 /\ R1(v) to V2 /\ R1(v) for any v E
TM(Y1)(A,,/,), which implies that the orientation of M*(Y.-) at [(A,i/2)] is indepen-
dent of the choice of the vector V. Therefore we have proved that the orientation of
M*(Yl) is well-defined.

Next we prove that the Z-fold covering map M*(Yl) —> M*(Y1) induces an ori-
entation on M*(Y1). Suppose that (A1,i,b1) and (A2, 2112) are gauge equivalent by a
gauge transformation 31 not in the identity component of Q(Y1). Pick an L0 large
enough and cut down (A1, 2,121) at L0 +1 and still denote it by (A1, $1) (we can assume
that 31 is constant in t on [L0 + 1,00) x T2). Connect (Abwl) with the reducible

point (0, 0) by a path (A,, 1b,) (so 31 - (A3, 1b,) is a path joining 31 -(A1,w1) = (A2, 1122)

95

with (—sf1dsl,0)). Then the (e)-spectral flow of [C(As,¢3),1(Lo + 1)(ll(s)) equals to
that of K31.(A,,¢,),1(Lo +1)(l1(s)) where M3) is a path of Lagrangian subspaces which
equals to the associated Lagrangian subspace of IC(A,,1,/;s),1(00) at the endpoints. On
the other hand, the (e)-spectral flow of K:(—usl’1d31,0),1(L0 + 1)(l3) : 0 S u g 1 is even
(Dirac operators are complex linear) where the Lagrangian subspace Z3 is spanned by
(idy, (0,0,0,1)). Now it is easy to see that the orientation at (A1,w1) and (A2,z,b2)
are compatible. So the lemma is proved. Cl

Now we are ready to define the “intersection number” #8(Y1, Y2) and prove the

gluing formula.

Definition 3.3.6 1. For any (01,02) 6 8(Y1, Y2), let ej be the positively oriented
tangent vector of M(Yj) at 09 (j = 1,2). Then the sign of (01,062) is the sign

of [R1]e1 A [R2]€2 with respect to ids: /\ idy.

2. #S(Y1, Y2) = 2(01,a2)€5(Y1,Y2) 8897?,(01, 02).

Theorem 3.3.7 (Gluing Formula)
x(YL) = #S(Y1, Y2) for sufiiciently large L > 0.

Proof: Let (AL0,’(/}L0) be the “almost” monopole being deformed to fl 6 M*(YL0).
By Lemma 3.3.3, signfi = (—-1)’"1+1 where m1 is the L’2-spectral flow between

(C(AL0,¢LO)(L) and [Co for sufficiently large L. By Theorem 3.3.2, m1 is equal to

Z SF.{IC(A,,-,),,,(L0 +1)(lj(5))}+ A/Ias{(11(s).12(s))}

]—1

for any choice of (A,1/J), joining (AmeLO) with the reducible point (0,0) and any

choice of a path of Lagrangian subspaces (11(3), l2(s)) satisfying the endpoint condi-

96

tion. Here SF€{IC(A,¢)MJ-(L0 +1)(lJ-(s))} is the e-spectral flow of E(A,¢)hj(L0 +1)(lj(3))
for some small 6 > 0. We choose (Aft/2).; such that w, is identically zero on
the Y2 side, and choose [2(5) 2 lg to be the Lagrangian subspace spanned by
([R2](TM(Y2)), (0,0,0,1)) Then the e-spectral flow of [C(A,¢),,2(Lo + 1)(l2) is even.
On the other hand, suppose 6 = T(oz1, a2). Let ej be the positively oriented tangent
vector of M(Yj) at 03' (j = 1,2). Then by taking V = [R2]62 in Definition 3.3.4,
we have sign(al,oz«2) equals to the sign of [R1]e1 /\ [R2]e2 = (-1)"’[R1]v /\ [R2]62 =
(—1)"’+1id:r /\ idy = (—1)m+1. Here m and v are referred to Definition 3.3.4. The

theorem follows from the relation m E m1 mod 2.

APPENDICES

APPENDIX A

The purpose of this appendix is to find out for what a E H1(T2) ® iR the twisted
Dirac operator Da =2 D + a is not invertible. Here D is the Dirac operator on T 2
associated to a given spin structure and the flat metric.

First of all, let’s recall some basic facts about the spin structures on the torus
T2. There are two equivalent descriptions of spin structures. Topologically, a spin
structure on T2 is a framing of its stabilized tangent bundle TT"2 ED 6 (a homotopy
equivalence class of trivializations). There are four different spin structures on T 2
which are parameterized by H1(T2,Zg) = Z2 EB Z2. It is well-known that among
these four different spin structures, three of them are spin boundaries, i.e. spin
structures induced from a spin 3-manifold bounded by the torus. The only one left
which is not a spin boundary is usually called the Lie group framing. Assume that
T2 = R/27rZ x R/27rZ and let (52,5513!) be the tangent vectors of the circles. For

(k, l) = (0, 0), (0, 1), (1,0), (1, 1), the following formula defines four different framings

50,,” of the tangent bundle TT2 which induce all the spin structures on T 2 (framings

97

98

of TT2 ED 6):
cos k2: + ly — Sin k1: + ly i
f(k,z)($, y) = ( ) ( ) 6x , (3:,y) 6 T2.
sin(lca: + 1y) cos(ka: + ly) 6%

The Lie group framing is {(09). See [15] for details.

The geometric aspect of spin structures is related to the groups Spin(n). The
groups Spin(n) sit inside the n-dimensional Clifford algebras Cl (n) and double cover
the groups SO(n). Let it : Spin(n) —+ SO(n) be the double covering map. Equip
the torus T2 with a Riemannian metric, assuming that it is the product metric for
simplicity. Let P509) be the 80(2) principal bundle to which the tangent frame
bundle of T2 is reduced. A spin structure on T2 is then defined to be an equivalence
class of liftings of the principal bundle P50(2) to a S pin(2) principal bundle ngm(2),
i.e. PSpin(2) 1+ 1050(2) such that 7r restricts to the double covering map on each fiber.
Two liftings PéL’mm 13 Pgom and Péiinfl) 32) 1350(2) are said to be equivalent if and

only if there is a bundle isomorphism i such that the following diagram commutes:

i

(1) (2)
PSpin(2) —> PSpin(2)
i 7T1 i 7T2

identity

1330(2) P50(2)

For each spin structure PSpin(2) 1) P509), there is a canonically associated spinor
bundle W = W+ EB W“ on T2, where Wit 2 PSpin(2) X9: C. The representations
pi : Spin(2) —-> U(1) are distinguished by the conditions gi(e1e2) 2 SH for any

orthonormal basis (e1, eg) of R2.

99

The t0pological and geometrical descriptions of spin structures on T2 are related
in the following way. The spin structure induced by the trivialization 50%!) (k, l = 0, 1)
corresponds to the unique equivalence class of liftings Pézflm —> 1050(2) for which the
trivialization a“) of P50(2) can be lifted to a trivialization 601,1) of Péifim, which
further induces trivializations for the spinor bundles Wi and W.

Theorem:

Assume that T2 = R/27rZ x R/27rZ carries the product metric and (397:3 %) is
the oriented orthonormal basis. For (k,l) = (0,0), (0, 1), (1,0), (1,1), define trivial-

izations f(ky) of TT2 by the following formula:

cos(ka: + ly) — sin(k;t + ly)

{yeah/U?!) = , (cc, y) 6 T2.

g

63

sin(k:z: + ly) cos(ka: + ly) 3%
Then within the induced trivialization 50;,” of the spinor bundles associated to the spin

structure given by Q“), the Dirac operator D0”) is given by the following formula

3 k 0 u

u . .
D(k”) =da: 5+; +dy 51+; ,
v a: 0 —k v y 0 —l v

where u, v are complex valued functions on T2. As a consequence, for a E ’H’(T2)®iR,
the twisted Dirac operator ng”) 2 D0”) + a is invertible unless a = £09611” + ldy) +
sida: + tidy for some integers s and t, and dimc Ker D5,“) = 2 if a 2: g(kdx + ldy) +

sidII: + tidy for some integers s and t.

100

Remark: The lattice
Bu.” 2 {ala = g(kda: + ldy) + side: + tidy, s,t E Z}

is called the lattice of “bad” points for the spin structure g(kJ).

Proof: In general, if (e1,e2, ..., en) is an oriented local orthonormal frame, then
within the induced trivialization of the spinor bundles, the induced connection is
given by —% ZKJ- wijeiej, where wij is given by the formula Vej = eiwij (see [19] for
details). Back to our case of the torus, let E(k,l)(:c,y) = (e1,e2), and V62 = elwlg,

then am 2 kda: + ldy by direct calculation. The theorem follows easily from this.

APPENDIX B

The purpose of this appendix is to give an estimate on the lowest eigenvalue of
certain self-adjoint elliptic operators on a manifold containing long necks, a technical
result needed in Chapter 3. See [7].

Let X be an oriented Riemannian manifold with a cylindrical end modeled on Y,
i.e. there exists a compact subset K such that X \ K is isometric to (—1,oo) x Y.
Let E be a cylindrical Riemannian vector bundle over X. By definition, there is
a Riemannian vector bundle E0 over Y such that E is isometric to it"Eo on the
cylindrical end (—1,oo) x Y, where 7r : (—1,oo) x Y -> Y is the natural projection.
Assume that D : I’ (E) —+ f’(E) is a first order formally self-adjoint elliptic operator

on X, which takes the following form on the cylindrical end (—1,oo) x Y

a
D_ua+A)

where I is a bundle automorphism of E0 which preserves its inner product, and

A : F(E0) —-> I‘(E0) is an elliptic operator on Y independent of t. The self-adjointness

101

102

of D implies that I and A satisfy the following conditions:
I2 = —1,1* = —I, A‘“ 2A, IA+AI:0.

Note that the spectrum of A is symmetric about the origin and the automorphism I
maps EA to E_,\ where EA is the eigenspace correspondent to eigenvalue A. See [26].
We assume that Ker A 76 0. Then the automorphism I defines a complex structure
on Ker A which induces a symplectic structure on it. In particular, the dimension of

Ker A is even. The operator D as described will be said cylindrical compatible.

Definition 1
An exponentially small perturbation of a cylindrical compatible operator D is a first

order formally self-adjoint elliptic operator D’ satisfying the following conditions:

a) D’ is a zero order perturbation of D,

b) on the cylindrical end (—1,oo) x Y, D’ = D+P(t) where P(t) : P(EO) —> P(Eo)
is a smooth family of zero order self-adjoint operators satisfying the following
exponential decay conditions: there exist a small 6 > 0, some To > 0 and a

constant C such that when t > To,

_ _ 0P _ _
“P(tlt/JHLW) S Ce 6“ TO’WHLW) and HEY/4mm S Ce 6“ TO’WHLW)

f0?" If} E L2(E0).

Let D’ be an exponentially small perturbation of a cylindrical compatible operator.

103

The space of “bounded” harmonic sections of D’ is denoted by H g(D’), i.e.
Ham) = {w e P(EilD'w = 0. “imam < oo}.
The space of L2 harmonic sections of D’ is denoted by H L2(D’), i.e.
H.417) = {'1/2 e L’(E)|D"w = 0}.

Let 6 be a fixed cut-off function which is equal to one at 00, and 7r : (—1, 00) x Y —> Y
be the natural projection.

Lemma 2

There exists a small 61 > 0 such that for any 11) E HB(D’), there exists a unique

limiting value r(u’2) E Ker A such that
”w - fiW’Tf‘L’lHLglw) < 00-
In particular, 1b 6 HL2(D’) if and only if r('z,b) = 0. Moreover,

dimHB(D’) — dimHL2(D’) 2 édimKer A.

Now consider a pair of triples (X,, E,, D:) for i = 1, 2. Suppose that there is an
orientation reversing isometry h : Y1 —> Y2 which is covered by corresponding bundle

maps which identify A1 with A2 in a suitable way so that for any L > 0, we can

form a triple (XL,EL, D’L) where XL 2 X1 \ [L + 1,00) X Y1UhX2\ [L + 1,00) x Y2

104

with h : (L,L +1) x Y1 —> (L +1,L) x Y2 given by h(L + t,y) = (L +1 —t,h(y)),
EL 2 E1U,,E2, DL 2 D1Uh D2 and PL 2 flLPl + (1 — ,BL)h*P2 for some cut-off

function ,6], supported in (L,L +1) >< Y1 with IVE] S 2, and D’ 2 DL + PL. Set

D’ a 2
AL = inf f—XL—l—Ez—l.
Wm fo WI

The purpose of this appendix is to investigate the behavior of AL as L —> 00.
Definition 3
Suppose D’, D’l and D’2 are exponentially small perturbations of cylindrical com-

patible operators.
a) D’ is said to be regular if HL2(D’) = 0.

b) (D’1,D’2) is said to be a transversal pair if

T(HB(D’1))flh‘(T(HB(Dé))) = {0}-

Here is the main result.

Theorem 4

1))\L=0(#)OSL—>OO,

2) if (D’1,D§) is a regular transversal pair, then for any function “/(L) = 0( L2) as

L —+ 00, there exists L0 > 0 such that when L > L0, we have

/\L > 7(L).

105

In particular, D’L is invertible for large L.

We first introduce some notation. Let A,, i E Z denote the eigenvalues of the
Operator A, and u,- denote the corresponding eigensections. Set a : inf meg Mi]. For
simplicity, we omit the subscript L if no confusion is caused.

Lemma 5

There exist L0 > 0 and M > 1 with the following significance. Assume that w
and c satisfy D’i/J 2 Cd) with w 75 0 and |c| < 6(p) for some small (5(p), then it can

be rescaled so that Ill/’HC‘KXL) < WI and one of the following conditions holds:

|2 is equal to one,

' either fX1(Lo) WIQ 07" fX2(Lo) I’ll)
0 either ||w||L2(Yl)(L0) or ||w||L2(y2)(L0) is greater than or equal to one.
HCTB X1(L0) = Xi \ (L0,00) X K, 221,2.

Proof: Let H1, H2 be the L2-orthogonal projection onto Ker A and (Ker A)l'.
On the cylindrical neck of X L, write it = f1+f2 where f1 6 Ker A and f2 6 (Ker A)i.
Set 60) = Iv |f2I2-

Direct computation shows that

 

(9

3f;— = IfIle—cl(f1)

3f2 _

E _ —Af2 + Ingpw — cI(f2)

32f2 6P an

at2 : (A2—C2)f2+IAH2PW+IH2EW+IH2PE+CH2Plp

For any 6 > 0, there exists Lo > 0 such that on the neck [L0, 2L + 1 — L0] x Y1 we

106

have

825 32f2

IV

K(u2|lf2lligm - €||f2llL§(Y)(||f1lltzm+l|f2||L2(Y)))

for some constant K. Here [cl < 6(a) for some small 6 (u) If f (t) reaches its maximum

in an interior point to 6 (L0, 2L + 1 — L0), then on the neck, we have

2

6
max l|f2l|L2(Y)'

 

u
maxllflllmo') Z llf1||L2(Y)(t0) 2

Otherwise, 6 t 2 f2 22 reaches its maximum at the end points.
L (Y)

On the other hand, we have on the neck that

98% +cI(f1) = 111le
w — C(f1)= —H1P¢.

0 c
SBt C = , then
—c 0
f1 (t) 2 e—C’t /t 803 IH1P1/J (13 + e—C(t—Lo) f1(L0)
Ifl Lo —H1P1/) If1(L0)

This implies that on the interval [L0, 2L + 1 — L0]

||f1||L2(Y)(t) S 01€_6(L°_T°’(max||f1||L2(Y)+ max Hf2HL2(Y)) + Hf1(L0)||L2(Y)-

107

If llf2HL2(y) reaches its maximum in the interior, then

max Hf1HL2(Y) S 2Hf1(LolHL2(Y)

for large enough L0. If ||f2[|L2(y) reaches its maximum at the end points, assuming

that it is the left end point without loss of generality, we have

max(||f1||L2(y) + Hf2|lL2(Y)) S 2(Hf1(LO)HL2(Y) + Hf2(L0lHL2(Y))

for large enough L0. Lemma 5 follows easily from these estimates. [:1
The Proof of Theorem 4:

1). Pick (f) E Ker A with H¢HL2(Y) = 1. Let pL be a cut-off function which equals
to one on [%, % + % +1] and equals to zero outside [a %+L+ 1] with [Vle = 0%).

Then

1 L
I 2 < 2 2 2 : _ 2 > _.
[him/3m _ [XLIVpLI |<z>| +/XLIPL<pL¢>I 0<Li.and/XL|pL¢| -10

SoAL=O(Z’§)asL—>00.

2). Suppose that there exists a sequence of L, -—> 00 such that AL" _<_ 7(Ln).
Then there exist it”, c,, such that D’ann = cnwn with cf, : AL". By Lemma 5, there
exist 1121 E H 3(D’1), 2.02 E H g(Dg) such that a subsequence of 1b,, converges to 2,01 over

X1 and if); over X2 in C"C norm on any compact subset. Note that one of wl and 1122

is nonzero. Part 2 of Theorem 4 follows if we show that r(y’)1) = h*r(i,/22). But this

108

follows from the fact that if we write it = fl + f2 as in Lemma 5,

||f1(t) — f1(2L +1—tlllL2(Y) S C((3—6t + ICOS(C(2L +1— 2t)) — 1|

+I sin(c(2L +1 — 2t))|),

for large enough t and L. C is some constant independent of t and L.
The Proof of Lemma 2:

Suppose 1,!) E P(E) and D’w = 0. On the cylindrical end (To, 00) x Y, write it =
Z,- fiu, where u,- are the eigensections of the operator A corresponding to eigenvalues

A,, and f,- are smooth functions in t. Then we have

3ft
0t

 

+ Aif, : (IP(t)i/),u,-).
3913 91' = (IPUWWi), then 2;“ 91? = “Pll’llifioq and
t
h(t) Z fr GIM’fl’gzlsldS + fi(T0)€-A’(t—T°’-

Now assume that if} 6 L37 for any small enough 7 > 0. Assume that 61 < min(g—, ff)

where [1. = iIlfAfiéO |/\i|-

o For A, = 0, we have for any t’ > t,

t’ 6 1
e"’lf.-(t')-f.(t)l sc<f Le‘fi’lwleolx/dsfl

so f,(00) = limHo‘> f,(t) exists and f, — f,(00) 6 L31.

109

o For A,- > 0, we have for some constant C (n) that
(32516 mm s cm) [T 626”(ng(s))ds + (Zf.-2(To))62"’”°-
o For x\,- < 0. First of all, we have

W) = -—e’*" f°° e*‘:g.(s)ds

since w 6 L37 for any small enough 7 > 0. On the other hand, for some

constant C (,u), we have
errant» s cm) [me2618<zgf(s>)ds.
i t i
Take r(i,b) = Z,- f,-(0o)u,- where u,- E Ker A, then

”'12 — 57F*T(1/’)IIL§I(E) < 00

where 6 is a fixed cut-off function which is equal to one at 00, and 7r : (—1, 00) x Y —> Y
is the natural projection. As for dimHB(D’) —— dimHL2(D’) : %dimK er A, it follows

from Theorem 7.4 in [21].

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