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FLOER. HOMOLOGY FOR. CONNECTED SUMS
OF HOMOLOGY 3-SPHERES

Weiping Li

A DISSERTATION

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

DOCTOR. OF PHILOSOPHY

Department of Mathematics

1992

ABSTRACT

FLOER HOMOLOGY
FOR CONNECTED SUMS
OF HOMOLOGY 3-SPHERES

By

Weiping Li

Supervising Professor: Ronald Fintushel

In this thesis, we try to understand a Mayer-Vietoris principle for Floer homology.
Floer homology is defined from a chain complex whose chain groups are roughly gen-
erated from the S U (2)-irreducible representations. And boundary maps depend on the
1-dimensional moduli space of self-dual connections on the (homology 3—sphere)xR. For
Floer homology on connected sums, it relies on understanding the gluing procedure on
noncompact 4-manifolds with almost-harmonic 2-forms in the gluing region. A particu-
lar gluing data and analysis are introduced. The splitting and perturbation effected on
1-dimensional moduli spaces are also considered.

Using this gluing result and the much simpler calculation of the spectral flow of the
Chem-Simona Hessian for the connected sums we are able to calculate Floer homology in

several examples.

To my grandparents.

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my dissertation adviser, Professor Ronald
F intushel, who has been a constant source of inspirational advice for me. I would also like
to thank him for introducing gauge theory to me, for suggesting the problem, and for the
supervisors] guidance which made this work possible with his constant encouragement,
help and support during my graduate study at Michigan State University.

My sincere thanks also goes to Professor Selman Akbulut with whom I went through
all the important papers of S.Donaldson. He taught me how to single out insights from
details. Also I am grateful to Professor T. Parker for his helpful conversations and time
during the course of the work. I also learned much from his courses. I would also like
to thank Professor Tom. Mrowka of California Institute of Technology for help at crucial
time during the preparation of this thesis.

I would also like to thank my dissertation committee members Prof. J. Wolfson, Prof.
T.Y. Li for their time.

Furthermore, I would like to thank former director of Graduate Studies, Professor
Charles Sebeck, who made it possible for me to return to M.S.U. to study topology. My
studies in the United State would not be possible without Professor Shin-Nee Chow’s help.
I also owe a lot of thanks to him.

My sincere appreciation goes to my wife Xiaoli Liu. Her love, caring and understanding
have been a source of encouragement to me during my entire study in mathematics. In

particular, she has always supported me at my hardest times.

ii

Contents

1 Introduction 1
2 Floer homology of homology 3-spheres 3
2.1 Floer homology .................................. 3
2.2 Spectral flow ................................... 7
3 Grafting 11
3.1 Properties of balanced connections ....................... 12
3.2 Smallest eigenvalue on Y x R .......................... 16
3.3 Structure of the trajectory flow on the connected sum ............ 24
4 The Floer homology of Y0#Y1 31
4.1 General properties of the Floer homology of Y0#Y1 .............. 31
4.2 Description of d1 ................................. 33
4.3 Examples ..................................... 39
5 On spectral properties 43

5.1 The Laplacian on a connected sum ....................... 43

iii

Chapter 1

Introduction

Floer homology is a mod 8-graded homology theory for homology three spheres which
relates Donaldson’s polynomial invariants in the relative and absolute cases via a Mayer-
Vietoris principle. It is defined from a chain complex whose chain groups are (roughly)
built from the S U (2)-representations of the fundamental group of the homology sphere.
These can often be straightforward to compute. (They rely “only” on linear analysis.)
The boundary operators, however, depend on nonlinear analysis, namely, the structure of
the 1-dimensional moduli space of self-dual connections on the (homology sphere)xR.
The first calculations of Floer homology were carried out by R.Fintushel and R.Stern
who computed Floer homology for Brieskorn homology spheres and outlined a program for
their calculation for all Seifert fibered homology spheres. A natural question is to ask about
the Floer homology of a connected sum of homology 3-spheres Y0 and Y1. The difficult
point is understanding the structure of the 1-dimensional moduli space of anti-self-dual
connections on the tube (Y0#Y1) x R. This relies on understanding the (Taubes) gluing
procedure on noncompact 4-manifolds with almost-harmonic 2-forms in the gluing region.
Each connection in a connected component of a moduli space M 2Y°#Yi)xn(a#fl,a'#fl')
must limit asymptotically to flat 5 U (2)-connections on Y0#Y1 which in turn correspond

to representations such as a#fl where a is an S U (2)-representation of «1(1’0) and ,6 of

1303). We have proved:

Theorem 1.0.1 : For appropriate metrics on (Yo-#Y1) x R, any I-dimensional anti-self-

dual moduli space takes the form MlYo#Y1)xR(a#fl’a’#fl) (or (a#,6,a#fl')).

In other words, given a anti-self-dual connection A in M hoxlda, a') and a flat connection
B E MiquMvfl) (constant in t), they can be grafted together to produce a self-dual
connection on (Y0#Y1)XR. Furthermore, each l-dimensional moduli space on (Y0#Y1)XR
arises via this construction (perhaps with the role of Y0 and Y1 reversed).

Using this theorem and the (much simpler) calculation of the spectral flow of the
Chem-Simone Hessian for the connected sum we are able to calculate Floer homology in

several examples.

Theorem 1.0.2 1. For the connected sum of Poincaré 3-sphere with itself, the Floer
homology is

HF0=22€BZg, HF1=Z, HF2=Z, HF3=Z€BZ
HF4=Z2®Zg, HF5=0, HF6=O, HF7=ZEBZ

2. For 2(2, 3, 7)#2(2, 3, 7)

HF0=Z2€3Zg, HF1=0, HF2=0, HF3=Z€BZ®Z

HF4=22€BZ2, HF5=0, HF6=0, HF7=Z€BZEBZ
8. For E(2,3,5)#2(2, 3,7)

HF0=Z, HF1=Z®ZEBZ, HF2=Z2€BZg, HF3=0

HF4=Z, HF5=Z€BZ€BZ, HF6=deBZg, HF7=0

At one time it was conjectured that the Floer homology was actually mod 4 rather
than mod 8 graded. Example 1 above shows that this conjecture is false.

Chapter 2

Floer homology of homology
3-spheres

2.1 Floer homology

In this subsection, we will give a brief description of gauge theory on 3-manifolds and

review the definition of Floer homology. For more see [4], [10], [12], and [15].

Let Y be a homology 3—sphere, i.e. an oriented closed 3odimensional smooth manifold
with H1 (Y, Z) = 0, and let P -—+ Y be a smooth principal S U (2)-bundle. (Since c2(P) = 0,
this bundle is trivial.) Fix a trivialization Y x S U (2) of P and let 0 be the associated
trivial connection. Denote the Sobolev L: space of connections on P by .4(P). It has a
natural affine structure with underlying vector space 01(Y, adP) where adP is the ad joint
bundle. .A(P) is acted upon by the gauge group of bundle automorphisms of P which can
be identified with 9 = Aut(P) = Lz+1(Q°(Y,adP)). Here we need I: + 1 > % so that we
can form the quotient space of gauge equivalence classes B(P) = A(P)/G. The irreducible
connections (those for which the stabilizer of the action of G is Z2) form an open and
dense subspace 8"(P) of B(P). The space B‘(P) has the structure of a Banach manifold
with

TaB'(P) E {a 6 L:(QI(Y,adP))| dza = 0}

where d; is the Lz-adjoint of do (covariant derivative on sections of adP) with respect to

some metric on Y.

The Chem-Simone functional cs : A( P) —o R is defined as

cs(a)= %/ytr(al\da+§al\al\a).

It satisfies cs(g - a) = cs(a) + 21rdeg(g) for gauge transformations 9 : Y -» S U (2) Thus
cs is well-defined on 3(P) = .A(P)/{g E 0 : deg(g) = 0} and it descends to a function

cs : 8(P) -+ R/2tZ
which plays the role of a Morse function in defining Floer homology. Its differential is

dcs(a)(a) = [Y tr(Fa A a),

and so its critical set consists of the flat connections R(B(P)) = {a E B(P)| F. =
0}. (Here I“.I is the curvature 2-form on Y.) It is well-known that ‘R(B(P)) is in 1-1
correspondence with 12(Y) = H om(n(Y),S U (2))/adSU (2), the S U (2)-representations
of 11(Y) mod conjugacy. Given any metric on Y, the Hodge star operator applied to
the curvature Fa gives a vector field f(a) = ti}, 6 L§(01(Y,adP)) . In fact because
f(g-a) = g- f(a)-g"1, f(a) is a section of the bundle with fiber TaB‘(P). A representation
a e ’R(Y) is called nondegenemte if the twisted cohomology H1(Y; ado) = 0. This is the
same as requiring that ker df(a) = kertda = 0 where *d.. is the Hessian of the Chern-

Simons functional.

Note that a l-parameter family {a(t)| t E R} of connections on P gives rise to
a connection A with vanishing t-component on the trivial S U (2) bundle over Y x R.
Floer’s crucial observation is that trajectories of the vector field f, i.e. the flow lines of
%% + f(a(t)) = 0 or 13% = *F(a(t)), can be identified with instantons A on Y x R and
Alyxu} = a(t). A trajectory flow “connects” two flat connections on Y if and only if the
Yang-Mills energy of the trajectory (as a connection on Y x R with trivial component
in the R direction) is finite. One needs that all zeros of f are nondegenerate and that
their stable and unstable manifolds intersect transversally in smooth finite dimensional
manifolds. Floer has shown that one can perturb the Chern-Simons functional to make

the trajectory flow “MorseSmale” type (see [15] Lemma 2c.l, Proposition 2c.l and 2c.2

4

). These perturbations are based on Wilson loop functions. For the rest of this paper,
we assume that the Chem-Simon functional has been so perturbed. Then all irreducible
representations are isolated and nondegenerate. Since ‘R,(Y) is compact, it is then also

finite.

Fix a Riemannian metric on Y. For any connection a in the trivial real 3-plane bundle

over Y, define the elliptic operator
D, : (I11 613 0°)(Y, adSU(2)) -§ (01 @ (1°)(Y,adSU(2))

by D.(a,fl) = (*d.a —- dad, —d;a). For a nondegenerate representation a E R(Y) the
Floer grading 13(0) 6 Z; is defined to be the spectral flow SF(a,0) of the family of
operators D... with the asymptotic values limp.-.” at = a, and limg_.+°° a¢ = a, the
element of R(B(P)) corresponding to a. (We also denote ac, by a.) The grading a(a)
is well-defined mod 8 on B( P) independent of the choice of path a.. Define the weighted
Sobolev space LE5 on sections £ of a bundle over Y x R to be the L]: Sobolev space of e; ~£
where e5(y,t) = e5." for |t| Z 1. For 6 sufficiently small (we will be more precise in §3)
and any S U (2) connection A on trivial bundle over Y x R, the anti-self-duality operator

d; 69 d; : Lz+1'5(fll(Y x R, adP)) —-> L]:’5((Ilo 63 Q1)(Y x R, adP))
is Fredholm. We say that A is regular if d; ® d} is surjective. In terms of the complex:
4.
L; +1.,(n°(y x R,adP)) 3A» Lz,,(n1(y x R,adP)) it L;_,,,(oi(y x R,adP))

A is regular means that H 0 = 0 (irreducible) and H} = 0 (generic). For a nondegenerate
critical point a of cs, the spectral flow is S F(a,0) = I nde:r(dj4 69 d: )(a,0), the Atiyah-
Patodi-Singer index of the anti-self-duality operator over Y x R. So

”(01) E I ndez(d; 63 d;)(a,0) mod 8

where A is any family of connections {a(t)} E 8(P) over Y with a(+oo) = 0,a(—oo) = ac,
(see [15] or [12]). Floer’s chain group CJ-(Y) is defined to be the free module generated by

irreducible flat connections a with a(a) = j mod8.

Note: Changing the orientation of Y switches the sign of cs and hence the spectrum of

the Hessian reverses, so —p-y(a) = 3—(-py(a)) mod 8. Le. p-y(a) = 5-py(a) mod 8.

Define My“; to be the moduli space of anti-self-dual connections on Y x R and
let M(a,fi) be the subspace of those A such that limp-” A = a, lim¢_.+°o A = 8 for
fixed flat connections a and 8. It is a smooth canonically oriented manifold which has
dimension congruent to [1(a) - a(fl) (mod 8). The moduli space M(a,fi) has finitely
many connected components each of which admits a proper, free R-action arising from
translations in Y x R. If a(a) - a(fl) = 1 (mod 8), let M1(a,fl) be the union of 1-
dimensional components of M(a, It). Further perturbations make all the M ‘(a, ,6) regular.
Then M1(a,[3)/R will be a compact oriented O-manifold, i.e. it is a finite set of signed
points. The differential 0 : C,- -> Cj-1 of Floer’s chain complex is defined by

60: Z #M(aafl)fl

560;.)

where M(a, ,8) = M1(a, fi)/R and #M(a, fl) is the algebraic number of points. The sign
in this formula can be counted by transporting the orientation on the normal bundle of
the unstable manifold of or along the trajectory flow into the stable manifold of ,6. If this
agrees with the natural orientation on the stable manifold of 5, the trajectory gets the sign
+1, otherwise —1. Floer has shown that 02 = 0 . Hence {C130} jez. is a chain complex
graded by Zg. The homology of this complex is Floer homology, denoted by HP}. Floer

has shown that it is independent of the choice of metric on Y and of perturbations (see

[4]. [101.[15D-

The connected sum Y = Y0#Y1 of two homology 3—spheres is again a homology 3-
sphere. Its fundamental group r1(Yo#Yl) is the free product of «1(Yo) and 11306). There
are four types of S U (2) representations of 11(Y0#Y1):

(1) 9 = 90#91, (2) 90#01, (3) 0015401, (4) 010#01

where the a.- are irreducible representations of x1(Y,-) and 0,- is the trivial representation
of 1r;(Y,-),i = 0, 1. These four types of representations correspond to equivalence classes
of flat connections glued together by the clutching map which forms the trivial S U (2)
bundle over the connected sum from the bundles on the punctured summands. In each
case we have a family ao#a1 of flat connections parametrized by a copy of S U (2), which

can be identified with the automorphisms of a fiber over a point in the gluing region.

Two elements of this family corresponding to automorphisms p0, p; are gauge equivalent
if and only if pop;1 extends to an element of the isotropy group 1‘“, or I‘,,. Thus the
corresponding family of gauge equivalence classes is S U (2)/ I‘M x Fa; . Since 1‘9 5 S U (2)
and 1‘. = Z; for a irreducible, the first three types of representations gives rise to a unique
gauge equivalence class, whereas the last type of representation gives a copy of S 0(3)
for each pair of irreducible representations. In §3 we show that all trajectories between
these representations are obtained by grafting together existing trajectories from each side.
Thus one needs to compute the spectral flow along such trajectories. This is done in the

next subsection.

2.2 Spectral flow

Consider irreducible representations 0, fl 6 R(Y) and let {a,} be a l-parameter family
of S U (2)-connections on Y joining a to ,6. Let A be the corresponding connection over
Y x R. Recall that the spectral flow SF(a,fi) is (modulo 8) the index of the Fredholm
operator DA = d? 69 d: on the weighted Sobolev space with sufficiently small weight 6.

Then the Floer grading,
14(0) 5 IndezDA(a,0) (mod 8) (2.1)

One can consider the calculation of the index of the anti-self-duality operator as a
boundary value problem with Atiyah-Patodi-Singer global boundary conditions ([3]). We

have

 

__ a. + me) + -h. + p.(0)
2 2

where p1(A) is the Pontryagin form, the term hp is the sum of the dimensions of H‘(Y, V5),

(2.2)

 

mm: ea dim/3) = -2 [mam

i = 0,1, and pg is the p- invariant of the signature operator *dan - dapt over Y restricted
to even forms (cf.[12]). An application of the signature formula to Y x I shows that
pa = [20(0) is independent of the Riemannian metric on Y and is an orientation-preserving

diffeomorphism invariant of Y and 0.

Lemma 2.2.1 For a,- E ’R(Y,) irreducible, we have

1° pao#01(0) = pao(0) + P01(0)

2. ham, = ha, + ha, + 3,
hao#01 = haos
h90#01 = ham

h00#91 = 3~

Proof: ( 1) Consider the cobordism X built by attaching a l-handle to (Yo II Y1) x {1} in
(Yo I] Y1) x I. The boundary of X is Yo#Y1 II —Yo II -Y1. Note that in (X) = «1(Y0#Y1).
So there are natural inclusions ‘R.(Y.-) -> R(Y0#Y1) such that the pair (00,01) can be
extended to a unitary representation of n(Yo#Y1). (In fact, if the a,- are both irreducible,

there is an S 0(3)-family of such extensions.) By Theorem 2.4 in [3], we have

Pao#01(Y0#Y1) " Pao#01 (YO H Y1) = 2819(X) "" Signao#oq (X)!

where H’(X) = 0 and H 2(X ; ada) = 0. So we get the signatures satisfying sign(X) = 0,

signao#a1(x) = 0. Thus pao#ai (Y0#Yl) = Pao(Y0) + P01(Y1)°
(2) Since 010,01 are both irreducible, we have the betti numbers hg, = 0,i = 0,1, and
similarly h” = 0. The Mayer-Vietoris sequence gives:

00*!)

o —. H°(S’,adSU(2)) —. H;o#al(Yo#Y1,adSU(2))

-» H;O(Yo,adSU(2)) ea H;,(Y1,adSU(2)) —+ o

and so hao#a1 = hag + ha, + 3.
Clearly hedge, = 3. So we consider the case of 00 and a1 ,where 01 is irreducible. We

have haw," = 0, 1:50 = 0. Again applying the Mayer-Victoria sequence
0 -» H30(Yo,adSU(2)) EB H21(I’1,adSU(2)) —+ H°(Sz,adSU(2))
_. H3.#.,(Y.#Y1.adsv(2» -» H.‘.(Yo.adsv(2)) ea H;,(n.adsv(2)) —+ o

and using hgo = 0, we have
(3+0)—3+h5.#., - (0+h;,) = 0

1.8. h00#m = her I

Lemma 2.2.2 For irreducible representations 0; E ’R(Y.-), we have the following addition
property for the Floer grading p:

#(ao#01) == Mao) + #(01)

#(00#an) = #(01); #(00#01) = 11(00)-

Proof: For computing a(ag) we can use any connections A.- over Y.- x R which in-
terpolate between 0.- and a,-. We choose A,- to be flat on the regions B3 x R used
to make the connected sum (Y0#Y1) x R. So the A’s match to give a connection
A1#A2 over (Yo#Y1) x R which interpolates from 00#01 to ao#al. By definition,
p(ao#al) = IndezDA(ao#a1,00#01)mod8. Then by equation( 2.2)

 

 

ha a -' Pa #0: he #0 + P9 #0
= _ A _ o# 1 o 1 _ o 1 o 1
“(Got/#01) 2]},me(‘41# 2) 2 2 ,

where Y = Yo#Y1. From our choice of Ag, p1(A1#Ag) = “(A1)+p1(Ag). Since p90,“, = 0
and p9, = 0, our result follows from Lemma 2.2.1. Similarly one checks that 11(00#al) =

”(01) and ”(00#91) = 1400)- I

Similarly one shows:
Proposition 2.2.3 For all 8.- E R(Y;) and a,- 6 R‘(Y,-)

IndezDA(ao#al,flo#fll) = IndezDA(ao, 80) + IndezDA(a1, 51) + 3. (2.3)

Theorem 2.2.4 (Fintushel, Stern [12]) Let 72,, be a connected component of R(Y). Sup-
pose that ‘RO, is a manifold, that aid“ is normally nondegenerate on Ra, and let g : Ra —> R
be a Morse function. Then the critical points of g are basis elements of the instanton chain

complex. Such a critical point b has grading

#(b) = #(Ra) - 119(1)) (2-4)
where 119(0) is the Morse index of b relative to g.

9

We end of this section by giving the following remark which we will use to do calculations

in §3 and 154.

Remark: If 1 = u(ao#m) - u(flo#fli) = (Mao) - 14%)) + (Man) - 14/31)). one sets
that either #(00) — [1(flo) = 0 or = 1. This means that if A;(i = 0, 1) is an anti-self-dual
connection interpolating from a,- to B,- then one of the A,- is a constant flat anti-self-
dual connection on Y,- ( A;(t) = a,- E R(Y.) for all t E R) and the other A, lives in a

l-dimensional moduli space M}/j(0j,fij).

10

Chapter 3

Grafting

The essential step in the calculation of the Floer homology of a connected sum of homology
3-spheres Yo,Y1 is in understanding the structure of the 1-dimensional moduli space of
anti-self-dual connections on (Y0#Y1) x R. This relies on grafting together anti-self-dual
connections on noncompact 4-manifolds. The major problem is the existence of harmonic
2-forms in the gluing region. The difficult point is obtaining estimates on the overlap
relating the “merged” metric with the original metrics g,- on Y,-. For the merged metric 9
we will take a weighted average. The usual Rayleigh quotient for first eigenvalue involves
the d" operator, and in order to get a uniform bound on the first eigenvalue on the
connected sum from one on each side, we have to compare d" and d‘!6 . These operators
involve the derivative term of the weighted average with no control for gluing parameter
a(the neck-length). Thus we adopt Donaldson and Sullivan’s technique for building a right
inverse directly (cf. [11]).

We begin by looking at a special feature of the R-action on the equivalence classes of

connections which will give us a particular way of solving the anti-selfduality equation
Fj+(dj+d:,‘)a+al\a=0

uniquely on the subspace of Q;d(Y x R), which is perpendicular to H}. Then we show
that for all balanced l-dimensional self-dual connections on a single homology 3-sphere
xR there is a uniform lower eigenvalue. Using the parametric method to construct the

right inverse on the connected sum and applying the inverse function theorem, we are able

11

to prove a gluing and splitting theorem for l-dimensional anti-self-dual connections over

(Y0#Y1) X R-

Throughout this section we assume that the anti-self-duality operator is regular. (As
we have mentioned above, this can always be achieved by a compact perturbation of the

anti-self-duality operator. For the sake of simplicity we shall ignore the perturbation.)

3.1 Properties of balanced connections

Let Y be a closed, connected, oriented, smooth homology 3—sphere. For 6 2 0 (to be
determined), let a; : Y x R -> R be a smooth positive function with e5(y, t) = e5Itl for
It] 2 1. Let E' be an S U (2)-vector bundle over Y x R with a translationally invariant
metric and metric-preserving connection. Then following [15], [18], [19], and [29], we define
the weighted Sobolev space LL, on sections 5 of E to be the L]: Sobolev space of e; - 6.
To define Banach manifolds 8(a, b) of paths connecting a and b in By (the L? - S U (2)
connections over Y modulo Lg-gauge equivalence), choose any smooth representatives of
a,b E Ay and a connection C (as below) on Y x R which coincides with a for t S —1 and
with b for t 2 1. Then
As(a.b) = C + Lamas x R))

is an afline space and is independent of the choice of C. The corresponding gauge group

is:

06 = {g E L3,,”(Y x R,.S'U(2)) I there exists T > 0,
E E L3’6(92d(Y x R)) such that g = exp 6 for M _>_ T}.

We need p > 2 to construct the orbit space BYXR = Aim/6g”.
Proposition 3.1.1 1. Let
D., : Lia“? EB 9°)(Y,adSU(2)) -> LEAD] ® 0°)(Y,adSU(2))
be the operator Da(a, fl) = (*daa — dafl, —d:a). There exists a positive A0 such that

for all a E 12"(Y) the eigenvalues of Du satisfy |A(D¢)| 2 A0 .

12

2. If F(A) is in L’ for p Z 2, then there is a constant 0,4 such that
sup [PAIN S CAe'TI'I.
where 7 = 7(Ao) > 0, and C A is continuous in A.

Proof: The first is from [15], and the second is in [10] (see 4.1). I

Choose a positive 6 < min{Ao, 321} and a finite action connection C over Y x R with

limiting values a,b at Y x {21:00}, and use it to define the LI; norm as above. Let us

denote ":1on = llVAulngJ + Hung, (and "anti, = uuuwop.

Definition 3.1.2 : The balancing function b : ByxR -+ R is given by the equation:

5““ 2 °° 2
loo [IF(A)||L2(Y) =/MA)IIF(A)”L2(Y)'

(So the value b(A) is the time which splits the action of A in half.)

Lemma 3.1.3 1. Shifting the connection A in the t-direction, A(t) —* A(t :l: 3), one
has

b(AU + 3)) = 501(1)) - 8» “A(t - 3)) = b(4(1)) + 8

2. Let 80 = b‘1(0) be the space of equivalence classes of connections whose action is

balanced at 0. Then there is a one-to-one map from 80 to B = b‘1(s) for any .9 6 R.

3. If A is not a constant flat connection, the derivative of b is

 

+°° signs - b(a»
D a =/ < r (PF ,a > .
‘b‘ ’ -0. Ila/mam, A ‘

Proof: (1) is proved by a change of variable. (2) follows from (1). For (3):

b(A+sa) 2 +00
/ "PM + 30)||L2(Y) :l:.

F A + sa 2
_00 (11+...) ll ( )IIL2(Y)
Taking the derivative with respect to s at s = 0 and combining the terms, one has
2 +00 b(A)
||F(A)||L,(YXR)DAb(a) = [M < arm > - j < as“ > .

—oo

13

Now "F(A)"L2(Yxn) = 0 if and only if —%%dt + F. = 0, i.e. if and only if A is a constant
fiat connection, contrary to our hypothesis. Thus (3) follows. I

Definition 3.1.4 Set the balanced moduli space Mega = {A E MYxR C BYxRI b(A) =
0}.

Lemma 3.1.5 For A 6 Mega, y 6 Y, and each p 2 2, there exist constants Mo,Cl,C2
independent of A such that
(i) If dimMyxR 5 1, then Mfg!!! is compact, and

Mltlpr<c / r5lilpr<C£3.
[nae I A' 1 Bg(e)xRe I AI 2

Proof: (i) No sequence of connections in Mgfiln can converge weakly to a limit plus
an instanton bubble, since bubbling needs dimMyxn, _>_ 8. The only other way a se-
quence in M3113 can fail to have a convergent subsequence is for there to exist a subse-
quence {An} limiting weakly to a disjoint union of connections A_°° E MyxR(a, b), A0 6
Myxn(b, c), A+°o E MyxR(c, d) where a, b, c,d denote limiting values and at least one
of of A-°o, A+°° is not constant flat (otherwise {An} actually converges to A0). If, say,
A...” is not constant flat then dimMyxR(c, d) 2 1. Since each A, is balanced, the limit,
A-°°IIAOIIA+°° is also balanced, and it follows that dimMYxR(a, b)+dimMYxR(b, c) 2
1. This is impossible since the dimension of the moduli space M yxR(a, d) which contains
the A, is equal to 1. Thus M19211! is compact.

There exists a constant C independent of A such that CA S C for all A E M32111
from compactness where C A is the constant in Proposition 3.1.l(2). The inequalities follow
from a straightforward calculation by using sup [FAI _<_ Ce‘7l‘l.

(ii) Suppose not. Then there exists a sequence {An} 6 MYxR with [lFAnllLee(YxR) >
n. Thus we have (ymtn) such that [FAul(yn.tn) = n. Let A; = An(t — tn) (rescaling).
So IF A; [(vnfi) = n. Applying Uhlenbeck’s compactness theorem on the compact space
Y x [-1,1] shows that there exists a subsequence {A,-} with a bubble point, and this

14

requires dimMyxR Z 8, contradicting our assumption. I

Remark: For any A E A5(a,b), there is a positive constant M(C.A) such that

M(?:1,A)||“||L, P,(A).. < [lullL’ S M(c, A)"u”L{.,(A)'

If A E MYxR and dimMyxR S 1, then M(C.A) 5 0.. where C. is a constant independent
of A from Lemma 3.1.5.

Proposition 3.1.0 The space 8,,le = {A 6 51!le b(A)— - 0} of balanced connections

is a smooth manifold with codimension I and the moduli space M hill is transversal to

33%,.

Proof: Since an arbitrary A' E M I’xR is not a constant flat connection, it has a translate
A under the R-action which lies in Bkfiln. Note that "FA"2 ¢ 0. Let A = a(t), then if
0 = deA = -(da*%% + 935%) A dt + dad"), we get *F. = 0. Since A is anti-self-dual
%% = *F. = 0 and this A is constant flat connection. But this is not true, so the normal

vector
sign(t) - dj‘FA
MFAII2

to TBYxR at A is nontrivial. By the implicit function theorem for Banach spaces, we

 

have that 83:13 = 4(0) is a smooth codimension 1 Banach submanifold, and moreover
DAb : TAN -+ TOR is an isomorphism where TAN is the subspace of TAByxR spanned
by this nontrivial normal vector. Notice that derivative of b along 8,221,! is zero. We may
consider

TABYXR 9.." 1:489:13 X TAN.

Since Byxn at 83113 x R and D,b(A) = :tId in the time direction, we may identify
TAN E TOR 9.! (TAByxR)¢ the tangent space to Byxn at A in the time direction.

For A E 83:13, the cohomology H}, is a 1-dimensional space. We claim that it contains
{A(t + s) : s E R}. We have

H; = {A(t) + sa(t) : s E R, d'Xa = 0,d;','a = 0}.

15

Define f(s,u) = A(t)+sa(t)-A(t—u). Then f(0,0) = 0, and 350,0) = A’(t) #- 0, since A
is not a constant connection. Hence the implicit function theorem gives a local coordinate
u = u(s) in a neighborhood of (0,0) such that f(s,u(s)) = 0. Le. A(t)+sa(t) = A(t—u(s))
in time-translation form. Let S be the subset of R defined by

S = {s E R : there exists u(s) such that f(s,u(s)) = 0}.

Then S is nonempty (since it contains 0), open (by the implicit function theorem) and
closed (since f(s,u(s)) is continuous in .9). Therefore 5 = R, and so H}, = {A(t + s) :
s 6 R}. Hence H} intersects T481123]! transversely in the point {[A]}. The Kuranishi
technique then implies that locally, solutions of the anti-self-duality equation live in a
l-dimensional moduli space parameterized by H}, i.e. by time-translation. I

3.2 Smallest eigenvalue on Y x R

(i) Some analytical facts

Let d A denote the covariant derivative corresponding to the connection A and d? =
efldjeg be the ad joint of d A with respect to the Lg’s-norm. Floer has proved the following
in [15].

Proposition 3.2.1 (Floer) (i) For positive 6, 9'5 is a Banach Lie group with Lie algebra
(which can be identified with) Lg'5(92d(Y x R)).

(ii) The quotient space 35(a, b) = A§(a,b)/gg is a smooth Banach manifold with tangent

spaces
T[A]36(a.b) = {a 6 L¥,.(9:.(Y X R)) I d?“ = 0}-

(iii) The 2—form F; representing the anti-self-dual part of the curvature of A is smooth

and Gg-equivariant.

(iv) If 6 > 0 is smaller than the smallest nonzero absolute value of an eigenvalue of Du or

D5, then for any anti-self-dual connection A E 85(a, b) the anti-self-duality operator

06 = d? $ if; 2 L’l’,6n¢11d(y X R.) -+ L3,6(n2d @ flgd'+)(Y X R)

16

is Fredholm. Furthermore, D3 = g,- + 03. where
“a "da
Di. =( )
—d: 5

self-adjoint on SHAY) 6 92,,(Y) where 1: is the Hodge operator on the S-manifold Y.

Ifa and b are irreducible nondegenerate flat connections, then one can take 6 = 0.

(v) Let M be the moduli space of all equivalence classes of nonflat anti-self-dual connec-
tions A on Y xR whose action ”@511”: is finite. There is a first category set of metrics
on Y such that the anti-self-duality operator Di is surjective for all A E M n 85.

Remark: Proposition 3.2.1(v) implies that (Dis). has trivial kernel. From the ellipticity
of the anti-self-duality operator we have

cluv e ulna, s II(Df.)‘(v e u)IILg,

for v 6 u 6 (92d 6 de,+)(Y x R). Thus, for any such A 6 MI’xR n 85, by taking
p = 2, v = 0, there is a positive real number C(A) such that

( ) YxRe I I - YxRe I A I ( )
for all u 6 Q3". (Y x R).

The following definitions are combined from [9], [10], and [15].

Definition 3.2.2 : An ideal anti-self-dual connection (tmjectory) over Y x R, of Chern

number k, is a pair
(A; (2,, ...,z,)) e M;;'R(a,b) x s'ar x R)

where A is a point in M[‘,':R(a,b) n 85 and (1:1, ...,z;) is a multiset of degreel (unordered
l-tuple) of points of Y x R.

Let {An}, 11 e N, be a sequence of connections of charge I: on the S U (2) bundle P over
Y x R. We say that the gauge equivalence classes {An} converge weakly to a limiting ideal

anti-self-dual connection (A; (31, ...,:r()) if

17

(i) The action densities converges as measures, i.e. for any continuous function on Y x R,

l
[m f|F(An)l’du -+ [m f|F(A)|’du + 81221061)-

i=1

(ii) there are bundle maps

pfl : PlYXR-\{3lrn-1xl} —* PIYXR.\{31,...,3;}

such that p;(A,,) converges to A in C°° on compact subsets of the punctured man-

ifold.

Definition 3.2.3 : Let a and b be flat 5 U (2) connections over Y. A chain of connections
(81, , 8,.) from a to b is a finite set of connections over Y x R which limit to flat
connections c.-_1,c,- as t —. 4:00 such that a = co, c,, = b, and B,- connects c,-_1,c, for

OSiSn.

We say that the sequence {A5,} 6 M§xn(a,b) is (weakly) convergent to the chain of
connections (B;,...,B,,) if there is a sequence of n-tuples of real numbers {taJ S . . . 5
ta,,,}a, such that to“,- — tam--1 —1 00 as a —> co, and if, for each i, the translates t;,,-Aa =

Aa(o — to...) converge weakly to B,-.

We need to combine the notion of chain connection with the notion of an ideal con-

nection.

Definition 3.2.4 .' An ideal chain connection joining flat connections a and b over Y is
a set

(Ajizjli-"rzjlthjSJ
where (Aj)15js_] is a chain connection and for each j, (Aj;$j1,...,$j(j) is an ideal con-

nection.

In this set-up, there is a version of the Uhlenbeck compactness theorem. We state it in a

form proved by Floer in [15].

18

Theorem 3.2.5 (Uhlenbeck compactness on Y x R) Let A, 6 M§XR n 85(aa,ba)
be a sequence of anti-self-dual connections with uniformly bounded action. Then there

exists a subsequence converging to an ideal chain connection
(Aji 251. ..., $11915ng-

Moreover, one has
J
20:,- + (j) = lc, cg(A,-) = 1:,- (not necessarily an integer).
i=1

(For more discussion and details, we refer the reader to [10] and [15].)
(ii) Smallest eigenvalue estimates

(a) We want to prove the existence of a uniform lower bound for the eigenvalues of Ai’l'
for all balanced l-dimensional anti-self-dual connections A over Y x R which are asymp-
totically flat at the ends.

Theorem 3.2.6 Suppose dim/\rlyxn = 1 . Then there exists a positive constant C such
that for all A E M3233, and for all p 2 2,u e L3.5(93d'+(Y x R)) we have

(3.] eré-ltl M" </ ere-M ldxulp-
YxR ‘ YxR

Proof: For p = 2, the result follows from inequality 3.1 and Lemma 3.1.5. For p > 2,
we use the inequality in the remark after Proposition 3.2.1 for v = 0. The constant CA is

continuous in A. Hence the result follows by using Lemma 3.1.5. I

Remark: The above estimate also holds when A is the trivial connection. On a fixed
homology 3-sphere (i.e. with a fixed Riemannian metric) the standard Laplacian on the
self-dual 2-forms has a strictly positive first eigenvalue by the Hodge Theorem (see Chapter
5). We can use this to get the bounded right inverse for d+. This will allow us to glue one

side l-dimensional trajectory flow together with trivial connection on the other side.

19

(b) The flattening construction: We first describe a special gauge suited to our con-
structions. Fix A 6 M833, and choose a trivialization of the fiber at a base point y E Y.
Parallel transport first along the R-direction, and then outward in normal coordinates
in Y at each fixed time slice. This defines a gauge for A 6 Mega which we call the
cylindrical gauge. In this gauge A; = 0 on {y} x R, and A, = 0 where r is the radius on
Y centered at y.

Lemma 3.2.7 In the cylindrical gauge in 83(5) x R, we have [A(x,t)[ 5 rllFAlloo.

Proof: Let (271,32, x3, t) be coordinates in 83(5) x R. For 1 5 i 5 3, we have |A,(x,t)| S
gmax|(,,5)|<,|F(x,t)| ( c.f. [31]). Since A, = 22:1 xkAk = 0 we get 22:, x5954} = 0, thus
22:, 21.1% = r£;A¢ 432:, “2;; = rgAg. Also [0' 387A, = A¢(x,t)—A¢(y, t) = A¢(x,t).
Thus 3

mm): s l f .2 freer: s rmu.(.,.).<.lr(z,t)l.

We next need to describe how to flatten a connection A E M33211! along By(ro) x R.
Let x = x(ro,e) be a smooth cutoff function satisfying

Co

x E 0 on By(ro), x .=.1 on Y\By(ro + e) and |dx| _<_ —€—

for some constant Co.

Definition 3.2.8 For A 6 Mg’n define A E BYxR to be the connection on E which is
equal to A outside 8,,(ro + e) and on 8,,(1‘0 + e) is A = x - A as connection matrix in the

local trivialization of E given by the cylindrical gauge.

Lemma 3.2.9 There exist so and C (independent of A) such that for 0 < e < 60 and any
A E M3233 with dimMyxR $1 and any p,q 2 2

.. it! 2
HA - Alumna, 5 Ce . . "alumna, s Ca»

20

Proof: Take x = x(e,e) and A = x - A, we have F; = (dx A A)+ ~1—(x2 - x)(A A A),
since A is anti-self-dual. This has support on 83(25) x R, and using Lemma 3.2.7 and
Proposition 3.1.1, we have the pointwise bound

”1| 5 CoE“|AI + IAI’ s 005-125lFAI1' 462ml“ 5 03an s Case-.5.
where from Lemma 3.1.5 060 is independent of A. Hence

_ 6|1| + 1 i
IIFXIILgMYxR) ‘ (/33(2€)XR [6 FA [9)» S C2(6)Er.

The bound on A— A is similar, [A— AI 5 [Al 5 2e|FA| 5 Gee-‘7'". Thus the result follows.
I

(c) The Neighborhood of M31111 : Assume throughout this subsection that dimension
of moduli space dimMYxR 5 1. Fix p,q > 2. We are going to show that the uniform

lower eigenvalue estimate also holds for nearby anti—self-dual connections.

Definition 3.2.10 : Set
U5l = {B E ByXRIthere exists a A E Mg’Rsuch thatllA — Blng, < 61, "FEHLK, < 61}

Note that Lemma 3.2.9 implies that if A E M83111, then for sufficiently small 6 the

flattened connection A lies in U51 .

Lemma 3.2.11 There exists 60 such that for 0 < 61 < 60 there is a C5 independent of 61
such that

llulILf'JYxR) S CsllWhW'tlnggnn) for a" B 6 ”6:

Proof: "(th‘ulngwam 2 "(dll'mllcg'm'xm - “(A - B) *6 “llLo’J(YxR) Where A is

an element in M833. which is 61-close to B.

"(A - B) n uIILg,(y.m s "A - Bllu llullu

0.6/2 0.6/2

S C 61"“"Lf’6

21

by Halder’s inequality and the weighted Sobolev embedding theorem [19].
Since A is anti-self-dual, the Weitzenbéck formula gives dj(dj)" = VXVA + R, which
implies that

nuns, s annulus s c.0(p)n(d;)~uu.;, + cuullrg, s anxrwurg, (3.2)

The first inequality is from the remark after Lemma 3.1.5, and the last from Theorem 3.2.6.
Choosing 60 such that CC60 < l, we have

. 1 .
"(43) ‘ulngJ 2 5mm WIILOP, (33)
Thus from ( 3.2) and ( 3.3), we have

"‘4th S Cllwn'mllzg’, S QCIIWEYWIILgJ

From Lemma 3.2.11 and the weighted Sobolev embedding theorem LI; e—o Lbs, for
i + 3- 2 %, the bounded right inverse operator Q 3( = (d§)‘5(d§(d‘§)“)'1) satisfies

"08"“ng s cuoaunrg, s CIIuIILg, for an H 6 U6.-

(d) Changing metrics : We want to show that there is also bounded right inverse for
flattened connections with metric Co close to the original metric. Pick a point yo 6 Yo.
For simplicity we assume that the metric on Yo is flat in the 3-ball 83(ro + e) centered at
ya with radius r0 + e. For r1 < r0, let N,I,,.o,,.,(go) be the set of Riemannian metrics g on
Yo \ 83(r1) which satisfy

0) 9 = 90 0“ Yo \ 33(70):

(ii) "9 - gollce < 5' 011 3300) \ Ba("1)-

The annulus 83(ro) \ 83(1'1) will be used as the gluing region in forming connected
sums.

(1) Let it: be the projection onto self-dual 2-forms with respect to the metric 9. Note

that xi is a continuous map with respect to the metrics, i.e. Ilrri - If?” S C [lg — 90”00-

22

(2) For the metric go on Yo, there is right inverse Go for the operator d}:. Let
S = d3” 00. Then
+
dgroom) = “o, IISWIIQJM) S CplluOIILgJuoy

where [I - "L550“ indicates the Sobolev space with metric go for forms with support in
(Y0 \ 3301)) X R.
(3) For g E N,o,,°,,.,(go), the Lg'a—norms are equivalent, i.e.

Cgllluolly (,0) _ <IIUo||Lr (g)- <Culluo||LP (90)
0.10.5 5

where C‘s —v 1 as e’ -> 0.

Lemma 3.2.12 For self-dual 2-forms uo with support in the (Yo \ 830-1)) x R and g E
N,:,,o.,,(go) with sufl‘iciently small 5’, d}: has right inverse Q: with
”qullLf’Ag) S Cllulngmr
Also. no: “lire (,5 < cuuur 3,5, for: - + 1> >3;
Proof: We will construct the right inverse by arranging that d+'Qo - Id is a contraction
mapping on L’ 0,,(g)(ft+ (Yo x R)). We have d+'Qouo= d2 ”Qouo + (dI' — din '° )Qouo and
from (2)
+ _ + +
(«1,300 - mu. — ((1,; - «1,: W000-

By the definitions of g and the flattening construction for A0 with Xl[0.ro] E 0, one has
d+'— =d+9 — area = (51 — 1r”)(d"'eo + d'm).
From (1), (2), and (3) above we have

||(d+'Qo - ”Molly" ,(g)_ S C 0215'“ + Cp)""0“L' ,(g)

For 5’ small enough that 003.511 + 0,) < %, the operator sz0 is invertible, and the
right inverse for a}: is Q9 = 041on)“. I

For B 6 U5, and g E N51,,o,,.,(go), we also get a bounded right inverse for the operator
d? by combining the proof of Lemma 3.2.11 and Lemma 3.2.12.

23

3.3 Structure of the trajectory flow on the connected sum

(i) Forming the connected sum

(a) Let Y,- be an oriented homology 3-sphere with Riemannian metrics g;,i = 0, 1. Choose
basepoints y.- 6 Y.- and suppose for simplicity that the metrics g,- on Y,- are flat in neigh-
borhoods of the y,-. Using these flat metrics we identify neighborhoods of the points y,- in
Y,- with neighborhoods of zero in the tangent spaces Tng. Precisely, for any real numbers
e,T > 0, we set N,,(e,T) = {(r,0) : T'le 5 r 5 Te} C Tng \ {0}, where e eventually
will be made small and T(> 1) is another parameter ( to be fixed later in the proof) with
T5 less than half the radius of injectivity of y.-. Then define

feg- : Nw(e,T) ——+ N”, (e,T)

by f,,1~(r,0) = (-r + Te + T"e,0). Let U.- C Y,- be the annulus centered at y,- with inner
radius r1 = T"e and outer radius r0 = Te. The “linear inversion” map f5; taking the
inner radius of U0 to the outer radius of U1 induces an orientation-reversing difl'eomorphism
from U0 to U1. Let Y,-' C Y,- be the open set obtained by removing the T45 ball about y,-.

Then, in the usual sense, we define the connected sum Y = Y(e,T) to be
Y = Y0#Y1 = Y1; Urey Y1,
where the annuli U.- are identified by fax

(b) Let (Y,-,g,-),i = 0,1, be oriented Riemannian 3.manifolds as in (a). To construct a

Riemannian metric on Yo#Y1, we fix a cutoff function, (I) E C°°([0, +oo)), which satisfies

T + T‘1 1
¢|[o,T—1¢] E 0, ¢(-—2—€) = 5, and ¢I[T,,+°°) E 1.

Definition 3.3.1 : The Riemannian metric g on the connected sum Yo#Y1 is defined as
follows:
1 011 Y.- \ 8,,(Te), set 9 = g; for i = 0,1.
On the overlap annulus Nm(e,T) 2! N,,(e,T), g = ¢go+(l—¢)f:’1-g1 = ¢go+f:,1~(¢gl)
(because of the linearity of [5,1- ).

Lemma 3.3.2 Let 6’ be the constant of Lemma 3.2.12. There exists To > 1 such that for
all 1 < T 5 To 38 with T5 < §injectivity radius, we have N,.,T,,T-1,(g.-) at 0. Furthermore

we have Ns’,Ts,T-’1c(go) n Ne',T¢,T"1e(gl) # 9

24

Proof: We just use the metric from Definition 3.3.1 and calculate the C0 norm of g — go

on the annular region.

T T“1 e
9,, = (go)... 900 = {<1 + (1 — ¢)(-1 + LE—lmgo)...
Then VT S To
"9 - sollco .<. 11081431“| - 1. IT" - 1|}
Choose To close to l enough to make [lg - gollco 5 5’. Then the result follows. I

Remark: Lemma 3.3.2 tells us that we may glue the two manifolds by an orientation-
reversing isometry on the tiny overlap region. Therefore for forms u supported on Y,', we
have
glluIILgflm, s IIuIIL;,(.,(y,m s 2IIuIIL,,,,.,,,¢,.

(c) We next use the S U (2)-bundles P,- over Y,- to define a bundle P over Y. Using the
projection map 1r; : Y,- x R -+ Y.-, we pull back the bundles P,- to get bundles x1‘(P,-) over
Y.- x R. Let A0 be a flat connection on Yo XR, constant in the sense that Ao(t) = a E ’R(Yo)
for all t E R, and let A; be an anti-self-dual trajectory from 8 to 7 (i.e. a anti-self-dual
connection lying in a one dimensional moduli space) on Y1 x R. Set A.- = xA,-, i = 0,1

(using the flattening procedure on each side as in section 3.2 (ii) (b) with x = x(Te,e)).

Choose an S U (2)-isomorphism of the fibers:

P i (130)!» —_’ (Pllm

Using the flat structures A,- (both are flat on the overlap), we can spread out this iso-
morphism by parallel transport to give a bundle isomorphism g, between the P,- over the
identified part (an annulus or conformally spherical tube) covering ft]. We call such a
bundle isomorphism g, a gluing map. Use this gluing map to construct a bundle Po U, P,
over Y = Yo#,,TY1 and also the pull-back bundle rrf(Po U, P1) = E(p) over Y x R. The
gluing map g, respects the connections A,- so we get an induced connection, A, = Ao#,A1
on E(p). Thus A, = Ao#,A1 is A,- on (Y,- \ 83(T‘le)) x R. Note that A, is trivial over
the region identified by the gluing map.

25

The connections A,, for difl'erent p, are not in general gauge equivalent (even though
the bundles E(p) are obviously isomorphic). Let PM he the isotropy group of A.- over
Y; x R and let 1" = I“, x 1‘4, . The equivalence classes of connections constructed in this

way are in one-to-one correspondence with

HomsU(2)((Po)m(P1)m) == SU(2)/r.

the space of “gluing parameters”. When the A,- are irreducible, I‘ = {i1} so the space of
gluing parameters is 50(3).

The following proposition can be found in the text of Donaldson and Kronheimer ([9]
page 286, for a proof see [6] Lemma 4.31, page 314).

Proposition 3.3.3 The connections A,,, A,2 are gauge equivalent if and only if the
parameters m, p; are in the same orbits of the action of I‘ on S U (2)

The following proposition follows from the above Lemma 3.3.2 and Lemma 3.2.12.
Recall the constants so of Lemma 3.2.9 and To of Lemma 3.3.2.

Proposition 3.3.4 For 0 < s < so and 1 < T < To, there is a constant C independent
of s such that the operator d}: has a bounded right inverse G with

IlGullLf,5(9)(Yo#¢.TY1) S ClluHLo',5(9)(Yo#¢.TY1)

and

l
”GullLI'Jg) S Cllullme 4 +

0

.>.

alt-b
”all-1

Proof: The right inverse for d}? is 0?. Then using the definition of A,, we define Gu =
guo + qul which is the right inverse for the operator dz. Here uo = nu, 111 = (1 - 17)u
and q is a smooth cutoff function on the annulus Uo n U; which obeys filmy-1,] = 0 and

’ll{Tc5r} = 1' -

(ii) Gluing and splitting

26

Our goal is to deform the “almost anti-self-dual” connection A, to a nearby anti-self-

dual connection A, + a,. This entails solving the non-linear anti-self-duality equation
F+(A,) + djpa + (a A a)+ = 0.

The upshot of Proposition 3.3.4 is that we are able to solve the linearized anti-self-duality
equation din = b over Y = Yo#,,TY1, as long as A is irreducible (HR = 0) and regular
(Hg = 0), and furthermore there are estimates on the solution of the corresponding
linearized equation which are independent of s. We shall use the inverse function theorem

to deform the almost anti-self-dual connection A,.

Lemma 3.3.5 (c.f. [15]) Let f : E —> F be a 01 map between Banach spaces. Assume
that in the first order Taylor expansion f(£) = f(0) + D f(0)£ + N (f), D f(O) has a finite
dimensional kernel and a right inverse G such that for f,( E E

IIGN(£) - GN(C)|IE S C(IIL‘IIE + IICIIE)IIE - (Hz
for some constantC. Let 61 = (80)“. Then if ||Gf(0)||E 5 :531, there exists a 01-function
¢ : K5, —. ImG

with f(£ + 45(6)) = 0 for all 6 E K5, and furthermore we have estimate

"song 5 guanoms + gurus

where K5, = Keer(0)n {6 E E: IIEIIE < 61}.

Applying Lemma 3.3.5 to f(a) = F+(A,) + djpa + (a A a), with f(0) = F+(A,), N(a) =
(a A a)..., D f(0) = dip (with the bounded right inverse G from the Proposition 3.3.4),
E = LL, n Lg’5(TAP8) and F = L3’6(01(Y x R, ad)) , we have the following

Theorem 3.3.0 Let Y,~(i = 0,1) be homology 3-sphere and A.- E Mauln. Assume
dimMyoxn = 0, dim.My,xR = 1. Let so be the constant of Lemma 3.2.9. Then if
0 < s < so and 1 < T < To,(To is choosed from Lemma 3.3.2) we can deform A, to a

smooth anti-self-dual connection over (Yo#,,1-Y1) x R.

27

Proof: Using Proposition 3.3.4 and Lemma 3.2.9, we have
- ~ 2
Increasing, .<. cum/1mm, .<. calm/10m, + urn/anus, s c.»

and N(a) -— N(b) = ((a — b) A a)... + (b A (a — b))+. We use weighted Hélder inequality and
Lemma 7.2 in [19]

"((0 - b) A ¢)+l|Lg', S “a " but.“ Halli.“ S 06"“ - blng’Jlalng,

where 05 = c(Vol(Yo) + Vol(Y1) + 1)/6. So
uGN(a) - among, s 006”“ - bllrg,(llallz.g, + "bung.

Thus by Lemma 3.3.5 with 61 = (8005)“, there exist at : If}, -> ImG with f(£ + 43(0) =
0, here ¢(A,) = a,. So A, + a, is anti-self-dual connection over (Yo#¢,TY1) x R with
[[919]ng, small, and is smooth by standard elliptic regularity (cf.[2l]). I

Remarks: (i) The restriction on dimensions of moduli spaces is from Lemma 3.2.9 and
Proposition 3.3.4 to be able to get the bounded right inverse. Also from the proof above we
can glue two 1-dimensional anti-self-dual connections into a 2-dimensional anti-self-dual
connection.

(ii) Using the remark after Theorem 3.2.6 and the construction in Proposition 3.3.4,

we also can deform the Ao#A1 into anti-self-dual connection when one of A,- is trivial.

To incorporate the gluing parameter S 0(3), we apply the parameterized version of
Lemma 3.3.5 which states that the solution depends smoothly on the parameters and is
well-behaved under gauge transformations (see [5] Chapter X). That gives the description

of a model for an open subset in the moduli space M l’o#Y1 xR'

Theorem 3.3.7 Given a constant flat anti-self-dual connection Ao and a I-dimensional
anti-self-dual connection A1 with each DA, surjective in the weighted Sobolev space, then
for small enough s and all gluing parameters p, there is a smooth anti-self-dual connection
(Ao#,A1) +a,(t). If m, pg are in the same orbit under the 1" action on the space of gluing

parameters S U (2) , the corresponding anti-self-dual connections are gauge equivalent.

28

The restrictions on s and T imposed in Theorem 3.3.6 mean that the “neck” region of
the connected sum must be narrow with very small radius. Conversely, when our metric
satisfies these conditions, we can characterize the anti-self-dual solutions found by our

gluing construction. Define
GIc :M'findL x 50(3) x Mill“, -> 30’“me

by Gl.(Ao, p, A;) = Ao#,A1 as in §3.3 (i) (c), where io _>__ 0,i1 2 0,io + i1 = 1. Now for
62 > 0 in the proof of Theorem 3.3.6, let U5,(s) C BYxR be the open set

U52“) = {Al BEiIIIlIIGl "A - BllL',(g)((Yo#.,q-Y;)XR) < 52. "PI "Lo",(g)((Yo#.,TY1)xR) < 52}-

0.

The solutions to the anti-self—duality equation obtained from Theorem 3.3.6 lie in U5,(s),
and any element in U52 can be deformed to a unique anti-self-dual connection by Lemma

3.3.5 (The uniqueness follows from the contraction mapping principle on T832911).

Theorem 3.3.8 For s,T as in Theorem 3.3.6, any point in U5,(s) n M],O#y,xR(9¢) can
be represented by a connection A of the form Ao#,A1 + ¢(Ao#,A1) where A,- is (0 or
I)-dimensional anti-self-dual connection on MKxR and 43 is the 01-difl'eomorphism in

the proof of Theorem 8.3.6 with ||¢(Ao#,A1)||L3‘ < 62.

Proof: Suppose the contrary. Then there exists a sequence s, -> 0 with s“ < so,
{[An]} 6 U545") n M],O#len(g¢,,) where Ufa(s,.) is complement of U5,, i.e. A,, are not
in such a form.

By Uhlenbeck’s compactness theorem applied to the balanced anti-self-dual connec-
tions, we have a subsequence converging to A0 V A1 , where A; is a anti-self-dual connection
on (Y.- \ {y;}) x R ( since l-dimensional moduli space is compact up to time~translation
by Lemma 3.1.5). The connection A,- has a singularity along a line {3);} x R. Since this
is codimension 3 , it can be removed by Sibner’s theorem [26]. Let the extended anti-self-
dual connections still be denoted by A,. By the flattening construction, for small enough

s" we have

- , - 5
Ai(5n) = X(T5n15n)Ai With ”Ai(5n) - Ailng',(gn)(Y.-XR) < —82-’

29

and let A,(s,.) = Ao#,,,,,A1 as in §3.3 (i) (c). Then

IIAn - Arlen)IILg,,(g.)((Yo#..,rY1)xR)

IA

||A,, - A.0051:)llr.g,,(g,.)((11,\s;,(r-1..r..nun)'1' ||A,, - 151(En)||r.g.,(g,.)((v,\33(T-1¢,.))xn)
1
S 2 "An - Ad||L:,.(g.)(m\s.(r-rc.))xn) + "41(50- 4"||L3,.(a.)((n\B.(T-'e.»xx)
i=0
For 11 large enough we have "A,, — A.-|| L3,(,n)(m\g,(1-1,n))xn) < 15} from convergence.

Thus "A" - AP(E")IIL'.,(9n)((Yo#..,1°Y1)XR) < éf. Since A" E MlYo#.,,,1-Y1)XR(9")’ Of

0

course F+(A,.) = 0, so we have A,. E U ?(s,,) which contradicts A,. E U5“2 (sn).
Thus for sufficiently small s, a l-dimensional moduli space can be represented by the

one deformed from the gluing process. I

Corollary 3.3.9 Under the assumption of Theorem 8.8.8, there is a unique small solution
to the anti-self-duality equation. 50 U52 (s)f‘IM],0#Yi m is equal to the image of the gluing

map.
This is the main analytic result of this thesis. We summarize this section in the following

Theorem 3.3.10 Suppose A,- is an anti-self-dual connection on Y, x R, and we consider
the connected sums Z = (Yo#,,1-Y1) x R for fixed 0 < s < so,1 < T < To. Then for

sufliciently small s one has maps

Ll MiexR(°i1/3i) x80(3) Micrxnhvds Glezwiii‘rmfiiifiwl
i+i’=1

and

M‘z(ai#7mflr#7r) Spit-tins L] M‘,x3(ai.flr) X50(3) Miamflwnvh
i+i'=1

which are inverse maps to each other.

Note that the splitting operation takes an anti-self-dual connection A 6 M l’o#Y1XR
with asymptotic values a,#7,-r to 8,-#7,-r one obtained by deforming Ao#,A1 where

(Air A;') E Mifixniairflt) X MIC: XR.(7i'1 71")

and they are glued together by constant (in R) gluing parameter p E 50(3).

30

Chapter 4

The Floer homology of Y0#Y1

4.1 General properties of the Floer homology of Yo#Y1

From Chapter 2 we know that the nontrivial representations of 1(Yo#Y1) in 5 U (2) fall
into two classes: the nondegenerate representations ao#01,0o#al and the degenerate
representations ao#,a1, p 6 50(3). From Lemma 2.2.2 we have the gradings #(90#01) E
p(a;), p(ao#01) =_= p(ao) and p(ao#a1) E p(ao) + p(a1) (mod 8). To compute the Floer
homology we first need to perturb the 50(3) components C(ao#al) = {ao#,a1 | p E
5 0(3)} to make the critical points nondegenerate. This is accomplished as follows. (See

[12]).

Each C(ao#a1) is a smooth closed submanifold of 8yo#y,; so we have the following

exact sequence which splits in L2:

0 —> TC(00#01) —P TBY0#Y1IC(00#01) -* NC(00#01) —* 0.

The Chern-Simons Hessian is degenerate along TC(ao#al) but is nondegenerate in the
normal direction. Identify the normal bundle N (ao#a1) of C(ao#al) with the total space
of NC(ao#al), and let

N,(ao#al) = {u E NC(ao#al) | [lullLa < e}.

Let s; < s: and define a cutoff function x on Byo#y, such that x = 1 on N,,(ao#al)
and x = 0 on the complement of N,,(ao#a1). Let 1r : N(ao#m) —-> C(ao#al) be the

31

projection. Let g : 50(3) —> R be the standard Morse function with one critical point in

each dimension. Set s3 = (so -— s1)2 and define
E(“) = €3X(0) °9(*(0)) = 3mm "* R

and let e be the L2 gradient of E. On N.,(ao#al) we have e = Vg, on the complement
of N.,(ao#a1),e = 0, and on N,,(ao#a1) \ N,,(ao#al):

 

||e(a)llco 2 83(ldX|||9(a)”co — lxlllV9(a)lloo) .>. €a( ll9(a)llco - ||V9(a)llco)-

£2 - 61
This points out that provided we choose s2 - s; sufficiently small, there will be no critical
point of e inside N,,(ao#al) \ N,,(ao#a1). Set f(a) = *1“, + e(a) over Byo#y,. In [12]
Lemma 4.1 it is shown that the zeros of f are the zeros of 1F, outside of N,,(ao#a1) and

inside N.,(ao#a1) they are the critical points of g, and all zeros of f are nondegenerate.

After performing this perturbation for each C(ao#a1) in ’R(Yo#Y1), the Floer chain
groups 0,-(Yo#Y1) are the free abelian groups generated by elements

' 003901 in grading 11(00)-

- 00#01 in grading 11(01)-

' (ao#01)j in grading #(00) + #(01) — J" j = 0, 1,23-
(See §2 and [12].)

The boundary operator of 0.(Yo#Y1) is computed by counting the trajectory flow line
spaces A250,,“ (a, b) for a, b generators in the list above.

We start with a graded differential group 0. = 6,0,, 9 E Z3, with 00., C 0 -1, where
0,, = {a E R‘(Yo#Y1)| a(a) = q}. There is an associated filtration compatible with the
grading, FpC, C F,+10, (increasing filtration). Define the F,0... as follows:

F,0 = 61,5,{2 < a > | a = (ao#a1),-, p(ao#a1) = k, a.- E R(Y.°),i = 0,1}.

Note that the perturbed Chem-Simons functional is non-decreasing along the gradient
flows (anti-self-dual connections). It follows that Floer’s boundary map 0 : 17,0, '—->
F,0,-1 preserves the filtration. Because R‘(Yo#Y1) is compact each 8,1,“, is finitely gen-

erated. Note that ao#01 generates a free Z-summand of E)”, for p = p(ao),q = 0,

32

 

similarly 0o#01 gives a free Z-summand of E}, for p = ”(01),q = 0. An 50(3) compo-
nent coming from ao#01 gives free Z-summands of E)“, for p = p(ao) + [4(01) and q = 0

and -3, and a Zg-summand Eh(ao)+u(a1).—2'

In particular the filtration is bounded. Since U F,0 = C. we get a spectral sequence.
Theorem 4.1.1 There is a (fourth quadrant) spectral sequence with
Erin a Hr+q(Fer/Fr-lC-)

which converges to H F..(0., 0).

Clearly d, = 0 for r 2 5, i.e. the spectral sequence (EI,,,,d,.) collapses at 5th term.
Therefore E2... gives the graded Floer homology H F.(0.(Yo#Y1), (9). The next section
will be devoted to a complete description of d, which will be enough to calculate some
examples. The description of d2, d4 will be the subject of a future paper. We end up this

section by showing d3 = 0.

Proposition 4.1.2 In the spectral sequence (EI,,,,d,) built from the filtration of Floer’s
chain complex for connected sums of homology 8-spheres, the third differential map d3 :

3 3 -
EM —> Ep-3.q+2 is zero.

Proof: Note that E}, = 0 for q > 0 and q 5 —4. Also, we have seen above that
E35, = 0. So the only possible nonzero case is do : Eff, —+ E: 3.0. Suppose d3(a#fl)2 =

n(a'#fi')o 7‘- 0. Then for the boundary Operator 0 of 0.(Yo#Y1), the coefficient of

(a'#fl')o in 0(a#fl)o is n 76 0. Then 62(a#8)1 = 26(a#8)2 79 0, which is impossi-
ble. -

4.2 Description of d1

We are going to define two special maps which are not used in the definition of Floer

homology.

33

Definition 4.2.1 d : 01(Y) -> Z < 0 > is defined by
do = #M1(a,0)0
and 6 : Z < 0 >—> 04(Y) is defined by

5601(Y)

Lemma 4.2.2 d0 = 0, 06 = 0.

Proof: This is proved in [10]. The proof is similar to the proof that 8’ = 0 in the Floer
complex. I

Let 0.- : 0..(Y,-) -> 0.-1(Y.-),i = 0, 1 be the boundary map of the Floer chain complexes
of Y.-,i = 0, l. The main result of this section is a description of d1 for the spectral sequence

of filtration built from the Floer chain complex 0.(Yo#Y1). Our result is:

Theorem 4.2.3 The difi'erential d1 of the spectral sequence (Elwdl) for the Floer chain
complex 0.(Yo#Y1) is given in terms of the listed basis in §4.1 by

41 =3o®ldlildo®51 fast-(181111 ildo®d+5®ld1 ildo®6,
where 03 is boundary map of the standard cellular chain complex of 5 0(3).

The notation in Theorem 4.2.3 and the determination of signs can best be explained

by two simple examples:
dn(a#fi).° = (3001153); + ( -1)“(°')(a#31[3)i + (-1)““°’”)‘)33(a#fl)s

d1(00#91) = 300#91 + (-1)“(°')(00#591)o-

We have extended our notation here in the obvious way: (2 mj7,)#fl = 2 m,(7j#8).

Theorem 4.2.3 gives a full description of d1. It is built from the contributions from
each homology 3-sphere and the gluing parameter space 50(3). The trivial connections
of both homology 3-spheres also contribute via the special maps d,,6,-,i = 0, 1. Now we
proceed to the proof of Theorem 4.2.3.

34

Let a#,,.,6 correspond to (aft/3),, where p,- is in the gluing parameter space 5 0(3). For
the boundaries in the Floer chain complex we only consider l-dimensional trajectory flows
with finite action. Fix metrics go,g1 and parameters s,T which satisfy the hypotheses
of Theorem 3.3.6 and Theorem 3.3.8. This fixes a metric on Yo#Y1 by §3.3 (i) (b).
If A; E kan(fl,7) and Ao E Mgoxn(a,a) is constant, Ao(t) = a, then for fixed
p 6 50(3), Theorem 3.3.6 gives a unique element in M(yo#y,)xg(a#,fl,a#,7) and the

Splitting Theorem 3.3.8 gives the converse. Therefore we have

#MIY0#Y1)XR(0#9590#P7) = #MI’1XR(397)’

Let 16182;th denote the quotient of the moduli space of perturbed anti-self-dual con-

nections by time-translation. We want to show that

#MII?#Y1)XR(O#Pfl’a#P7) = #MIY0#Y1)XR(0#Pfl’a#p7)°

The following proposition says that the compact perturbation we used in the construction
neither changes the algebraic number of trajectory flows between critical points a#,8 and

a#,7 nor does it create new l-dimensional trajectory flows.

Proposition 4.2.4 For so sufliciently small, we have
(1) If kanmn) at 0, then for p E 50(3) 0 critical point for the Morse function
g : 50(3) -> R

#MI£3#Y1)XR(C'#PB’ 0#P7) = #MIY0#Y1)XR(O#P[3’G#P7)°

(2)1fMir

1

“(5,7) = 0, then M[,';;,,1,x,,(a#,,s,a#,,7) = 0,

Proof: (1) Using the identification between trajectory flows and anti-self-dual connec-
tions, we consider the bundle AB%1#Y1)XR Xg 01((Yo#Y1) x R, ad) -> 8o(a#fl,a#7).

The self-dual curvature F} induces a section of this bundle whose zeros are

M(Y0#Y,)xa(a#fl.0#7) = M(Y0#Y,)xn(0#fl.a#7)/R-

For l-dimensional anti-self-dual connections, this is a finite set of points. The orienta-
tion on these points is precisely the orientation on the trajectory flow used in defining

the boundary map of Floer homology. Thus the algebraic number of zeros of F)” is

35

 

#MIY0#Y1 )xn(a#fl,a#7). Consider a trajectory with asymptotic values a#,8,a#,7.
Consider s E I = [0, 1]

rm : I x 8o(a#,fl,a#,7) -+ I x 13901,“,,,,(a#,s,a#,o x, saunter” x 11, ad)
(8,4) H F} + ‘74 +8‘I’A

where 6,4 = %(r,e(a(t)) + e(a(t)) A dt), and where 0,4 = w(a(t)) is a perturbation con-
structed so that ‘11,,4 has transverse zeros. For s = 1, the zeros of F: + '14 + {a are the

solutions of the perturbed anti-self-duality equation

6‘5?) - on“, - e(a(t)) - e(a(t)) = o.

 

Let M” denote the zeros of Tug. Then 0,1 E I are regular values of the projection
in : M‘" -> I since the 1-dimensional moduli space is transverse to 8o by Proposition
3.1.6. So the parametrized moduli space M 1" is a one dimensional submanifold of I x
8o(a#,,8,a#,7) with oriented boundary components -M1'0 and M”. Each M1" is
compact by [15, Theorem 3]. This means that M 1'0 and M "1 are oriented cobordant; so
(1) follows.

(2) Suppose M [,1 me’ 7) = 0 but there exists a 1-dimensional trajectory flow between
a#,,,8, a#,,7 after performing our perturbations. We have a solution A,3 of the perturbed
anti-self-duality equation, and Ae, lives in a 1-dimensional moduli space with asymptotic

values a#,ofl,a#,,7. So F13 = —%(e(a,,(t)) + e(a,,(t)) A dt). We have
naming, s uGe(a..(t))urg, s c - as.

The last inequality holds because we have a uniform bound for G by Proposition 3.3.4
and because in constructing our perturbation, we have used a smooth Morse function.
By choosing so small enough so that 0 - 63 5 153‘, then we can (by Lemma 3.3.5 and
Theorem 3.3.6) deform A,,3 to an anti-self-dual connection A with asymptotic values
a#,ofl,a#,,7. For any metric on Yo#Y1 for which the splitting Theorem 3.3.10 holds,
A is obtained from Ao#,A1 where Ao E MgoxR(a,a) and A1 E thn(fl,7). This
contradicts the hypothesis. I

36

Proposition 4.2.5 For generators a,b of 0..(Yo#Y1), elements of Mll53#1’1)(a’b) (1-
dimensional trajectory flow lines) occur only as follows:

I. If both 015,8), are irreducible, there is a 1-dimensional trajectory from (ao#al),- to
(509551))“ 1'f and only if i = j and either

(i) co = 8o and there is a 1-dimensional trajectory from al to 81, or
(ii) 01 = 81 and there is a I-dimensional trajectory from no to ,Bo.

2. There is a 1-dimensional trajectory from (ao#a1),- to 8o#01 if and only if co = 80
and there is a I-dimensional trajectory from 111 to 01 and i = 0. A similar statement

holds for 0o#fll.

8. There is a 1-dimensional trajectory from ao#01 to (Bo#81 ), if and only if no = 80
and there is a I-dimensional trajectory from 01 to [31 and j = 0. A similar statement

holds for 0o#01.

4. There is a I-dimensional trajectory from ao#01 to 8o#01 if and only if there is a
I-dimensional trajectory from no to ,Bo. A similar statement holds for 0o#01 and

9011951-

5. There is no I-dimensional trajectory from ao#01 to 0o#81 or from 0o#a1 to flo#al .

Proof: (1) It follows from the proof of Proposition 4.2.4 and the remark after it, that for
p 6 50(3)

#Mm’#y,)xn((ao#a1 )1. (30419305) = #Mly,#y,)xniao#p01,3o#pfl1)-

For any metric on Yo#Y1 for which Theorem 3.3.10 can be applied, we have that any
A E M[Y°#Y!)xn(ao#,a1,flo#,fll) is obtained from Ao#,A1 where A1,,k = 0,1 which
are (0 or l)-dimensional anti-self-dual connections in Myka. Hence a], = [it or there is

a 1-dimensional trajectory flow from a), to 5!: which is realized by A,,. So

0 S #(ak) - ”(51) S 1

and

fl((ao#01)i) - #((flo#51)j) = 1, #(00#01) - [4503551) = 1-

37

Therefore we have that i = j and one side is constant and the other comes from a 1-
dimensional trajectory flow. (2), (3) and (4) follow by a similar argument.

(5) By Theorem 3.3.8, there is no l-dimensional trajectory flow from ao#01 to 005581.
I

A straightforward calculation shows:

Lemma 4.2.0 0533 = 030.31. = 0,1.

Proof of Theorem 4.2.3: In Proposition 4.2.5, we have listed all possibilities for 1-
dimensional trajectory flows. So the boundary map for C..(Yo#Y1) includes all possibilities.
We need to check that d? = 0.

(1) For basis element (a#fl).- with 0,8 both irreducible, we have

di(a#fl): = (aoa#fi): +(—l)“(°)(a#01fl)i+(—1)”(°’#”)‘83(a#fl)i

(a) If Mhoxnlaflo) = 0 and thRM, 01) = 0, by the definition of d, we have

df(arrm. = firearm-1 + a[(-1)P‘°’(a#am.~1 + a{(-1)P<°*P)-'a.(a#/3).-I =

(03mm- + (-1)"“’°°’((aoa#a.m.- + (-1)P‘3°°*”"63((6oa#fl).~)

+(-1)P‘°’{(aoa#am.- + (—1)P‘°’(a#a.’m.- +(-1)P“°*31"M6.((a#am.-)}

+(-1)P‘°*""{6a(aoa#m.~ + (—1)"‘°)aa(a#a.m.- + (-1)"W’aetfiwaflw#5).»
We use Lemma 4.2.6 in the last equality. Clearly it shows d¥(o#8),- = o.

(b) If Mi’oxR(0100) 9t 0 and kan(fl,01) = 0. Note that 33(8oa#fl) = 0. By
Proposition 4.2.5 (2), i = 0 in this case. Then we have

«mama = (daoarrm + (aortas)
+(-1)u(a)(aoa#ag) + (-1)2“(°'l(0#3123)o + (_1)#(0)+u((°#315)o)33(a#313),,
+(—1)“(°’#p)°03((6oa#fl) + (_1)u(a)(a#a, mo) + (_1)u(a#fi)o+u(3a(a#fi)o)ag(a#g)o

38

Then it shows that d'f(a#8)o = 0. For 6oa 9t 0o,618 = 01 and floor = 0o,618 = 01, the
d? = 0 follows from a similar argument.

(2) For another kind of basis like ao#01 or 0o#al, We verify one of them, ao#01. The
other one is similar. By Proposition 4.2.5 we have

d1(ao#91) = aoaoaro, + (-1)"(°'°)(a#591)o-
Based on the definition of Floer homology, the trivial connection 0o#01 of Yo#Y1 is not
taken into account. So Boao 76 0o.
di(00#91) = (230510, + (-1)P<”°°°)(a.ao#ao.)o

+(-1)“(°)(300#591)o + (011901591 )0 + (-1)“(°)+“((°#“‘)°)33(ao#591)o-
Using the definition of 63 , Lemma 4.2.2 and Lemma 4.2.6, we have the differential d1
satisfying d? = 0. I

4.3 Examples

Even with our formulas of §4.1, examples remain difficult to compute since it is an ex-
tremely nontrivial problem for a fixed homology sphere Y to compute all possible 1-
dimensional moduli spaces M§XR(a,8),M],xn(a,0),M],xR(0,a).

Example 1: Calculation of H F.(E(2,3,5)#2(2,3,5))

Let us consider the connected sum of the Poincaré 3-sphere with itself. The Floer
homology of the Poincaré 3-sphere is generated by a E 01,,6 E 05 (dimMy(a,0) =
1,dimMy(fl,0) = 5) with all the boundary difl'erentials trivial, so for 2(2,3, 5):

01: HF1 E Z, 05 = HF5 E Z, 01': HI} = 0, j 516 1,5 (mod8).
This follows from work of Fintushel and Stern [12].

Proposition 4.3.1 Let Y be the Poincaré 8-sphere B(2,3,5) and let a be the irreducible
representation with p(a) = 1. Then the 1-dimensional moduli space of anti-self-dual con-
nections with asymptotic values 0,0 (denoted by M].(a,0)) is nonempty and #M],(a,0)

= :l:1, where “11%” denotes a count with sign.

39

Proof: Let X be the 4-manifold obtained by plumbing the (negative definite) Eo-diagram.
Then 8X = Y. The intersection form of X is the form E3. Let P be the principal 50(3)-
bundle over X with p; = 2 and 1.02 represented by the Poincare dual of one of the 2-spheres
corresponding to the nodes of the Eo-diagram. The moduli space M x(0) of anti-self-dual
connections with asymptotic value trivial (0) on the boundary Y has dimension one.
There are two kinds of ends of M 15(9). The first corresponds to reducible anti-self-dual

connections on P. These are in one to one correspondence with
{:te E H2(X, Z)| e2 = —2, e 5 w3(P) (mod 2)}.

There is a unique such reducible anti-self-dual connection (cf.[l3]). The other ends come
from splittings M9,(a)#M “(1,0), where M9, (a) is a compact zero dimensional moduli
space with asymptotic value a. Let no = #MgAa). Thus M x(0) is an oriented (see
[7]) l-dimensional manifold whose noncompact components fall into two classes. First,
those components such that neither end comes from a reducible connection. Each end of
such a moduli space corresponds to an M Ma, 0) which contributes inc to the counting
#Mb(a,0). The other end of this same component of M x(0) contributes 15:11,, to the
count; so they cancel out. There is one other noncompact component of M 15(0) with ends
corresponding to the unique reducible anti-self-dual connection on P and the remaining

component of M],(a,0). Thus #M],(a,0) = :lzl. I

We next want to calculate H F.(E(2, 3,5)#E(2,3,5)) by using the spectral sequence.
From §4.1 we have E}, as in the figure. As we described in §4.2, 0.- = 0,6,- : 0,i = 0,1
and the only nontrivial contribution is from dy,a = 10 and 63 from the gluing parameter

50(3). I.e. d1 = 1 ® dy, :l: dyo ® 1 + 63. Thus we have

13?, = Z < a#0 d: 0#a >, E3, = Z < (MM); >,
193,-: = Z 6 Z, for p = 2,6, E;_, = Z2 6 Z2 for p = 2,6,

2 - 0
EM -— 0 otherwnse.

Obviously, (I: = 0 and d3 = 0 as well from Proposition 4.1.2. Thus E3, = E2, = E1,”

The d. : E353 -> E3, is the only possible nontrivial differential. But E3, is generated

40

l
WHICH l 3 45
“‘1 M 3:11:34

11 so! 12 56"
D t

0 U M ‘4- Z+Z 0 Z
d l
.1 1 -l

 

 

-2 2s 1+1. -2

 

 

 

-3 M M -3 2+2 2+2
«arms;

by (8358): and dimM(a#,6,fl#fl) = —4, so d4 = 0. Hence the spectral sequence is

convergent at E3...

Theorem 4.3.2 For the connected sum of Poincare' 8-sphere with itself, the Floer homol-

091458
HFo=Zz®Z2, HF1=Z, HF2=Z, HF3=Z6Z

HF4=Z2622, HF5=0, HFo=0, HF7=Z6Z

Note that this theorem shows that Floer homology is not in general 4-periodic.

Example 2: Calculation of H F.(2(2,3,7)#2(2,3,7))
From [12], we have

Hr,(2(2,3,7)) = 01(s(2,3,7)) = z < a >, HF5(2(2,3,7)) = 05(2(2,3,7)) = z < b >

dlmMg(2'3,7)(a, 0) '=' 3, dlmMg(2,3'7)(b, 0) E 7, dlmMg(2.3’7)(b, a) E 4, mod8,

andweget
HFO=ZQ®Zg ”Fl-=0 HF2=Z HF3=Z®Z$Z
HF4=22®Z2 HF5=U HF6=Z HF7=Z®ZEBZ

Example 3: Calculation for H F.( E(2,3,5)#E(2,3,7)).

41

An easy calculation shows
HFO=Z HF1=Z®Z$Z HF3=Zg®Z2 HF3=U

HF4=Z HF5=Z$Z®Z HEB-222922 HF7=0.

42

Chapter 5

On spectral properties

5.1 The Laplacian on a connected sum

As we have explained in section 2, in order to calculate the Floer homology for a connected
sum, one has to cut and paste anti-self-dual connections by cut-off functions which make
the connection trivial on the tube 52 x I x R. Then the uniform lower boundedness of
the Laplacian on anti-self-dual 2-forms on the tube plays an important role for solving
the resulting anti-self-duality equation for the glued connection. We will show that the
uniform lower bound on the tube goes to 0 as we stretch the length of the interval I to 00.

We consider a homology 3-sphere Y with an Open 3-ball, B,(r), removed, where r is
the radius of the ball centered at y, and boundary of this manifold is a 2-sphere 5’. Let
(2"(Y \ B,(r), 5’) denote the space of smooth p-forms on Y \ 8,,(r) which vanish on 52,
and let A" be the Laplacian acting on p-forms.

Lemma 5.1.1 1. The Operator AP : L3(D’(Y\B,,(r), 52)) —. L3(QP(Y \ 8,,(r), 52)) is
injective.
2. The volume form 1.60 of S2 is the unique generator for H 33(53 x I), up to a constant.

We omit the proof, which follows from Hodge theory.

Suppose Yo 311d Y1 are oriented 11011101088 3'BPheres with Riemannian metrics Which
are flat in fixed small balls. We use an identification of these balls to define a connected

43

 

sum Yo#Y1. Thus if we have locally oriented Euclidean coordinates f centered at yo E Yo
and n at m E Y1, the identification map on an annular region is given by the conformal
equivalence q = f.(£) = eff/KI"). Here 5 -r I is any reflection and s is a parameter which
will eventually be made sufficiently small. We introduce another parameter N such that
N - s is less than the radius of injectivity of both Yo and Y1.

Let U: C Y; be the annulus centered at y,- with inner radius N “s and outer radius N s.
The conformal equivalence map induces a diffeomorphism from Uo to U1. We let Y,’ C Y,-
be the open set obtained by removing the N "s ball about y,-. Then, in the usual sense,
we define the connected sum Y = Y(s) to be

Y = mm = Y,’ 1),, Y,’

where the annuli U,- are identified by f,.
We shall also need to use another model for the connected sum. This depends on the

conformal equivalence:

d:52xR—+R3\{0}

given in ‘polar coordinates’ by d(s,t) = s - e‘. Under this map the annulus U with radii
Ns,N"1s goes over to the tube d"‘(U) = 52 x (logs - log N,logs + log N). Thus we can
think of the connected sum as being formed by deleting the points y.- from Y5, regarding
punctured neighborhoods as half cylinders and identifying the cylinders by a reflection.
(See [9])

From this point of view, we are working with the manifold Y,- \ 8,,(N'1s) and in the
annular region we relabel the length by shifting logs units and setting T = log N. Thus
we have identified the annular region with the tube 52 x [-T, T]. Define a smooth cut-off
function 8 with support in [—T,T] and fl = l on [—T + s,T — s].

Proposition 5.1.2 Let (Y\B,(r), 6) be a homology 8-sphere with a 8-ball removed and fix
a cut-ofl' function 8 on the neck as above. Then there exists a unique 1) E 02(Y \ 8,,(r), 6)
such that

A1] = “—51er

17:0 on a

44

Remarks:

1. A is the Laplacian on 02(Y \ 8,,(r), 6). To prove this proposition we shall need min-
imize the Lagrangian L(n) = fYo [d17|3+|d'17|2 +2fluwo A #17, whose formal variational

equation is the equation we expect to solve.

2. Injectivity of the Laplacian gives the nonzero first eigenvalue

A, = inf lld'flllia + Ildfllliz > 0.
Halli:

 

We also have the Poincaré inequality
/17A-rr] 5 0-/ qutdr7+d‘r)A*d‘n.
Yo Yo

Proof: The proof will follow the standard variational minimizing method which will give us
a weak solution and then elliptic regularity will imply that the solution is smooth. This will
sive no 6 C°°(9’(Y\By(r).3)) and Mo = -fl"wo- So A(flwo-no) = 0, (flwo—nofla = wo-
So 6wo — no is a harmonic 2-form with the correct boundary condition for patching the
harmonic 2-forms together. We will follow the proof in [21] page 294 and [25].

We first show that for r) E L§(Q’(Y \ 8,,(1'), 6)), the Lagrangian

L0?) = / ldnl’ + ld'vl2 + 2fl"wo A *n
Y\Bs(')

has a lower bound. Because the inequality 2ab 5 t"‘a2 + tb2 holds for all positive t we

have
l2(-fl"wo.n)l S 1“” - {3"wolliz + tllnlliz-
Thus
Lo) 2 "an11. + ud'nui. — t"u - mu}. - tllnllia
for all r) E L§(fl°(Y \ B,(r), 6)).
Next use the Poincare inequality to continue the inequality

2 (1 - tC)(lld'llli.2 + lld’fllliz) - t"||fi"wo||},

1 .. .
= 5(lldflllip + Ild'flllizl - 20% walliz (letting ‘ = 2%)
-2Cllfl"wo||f.2

IV

45

.-. ‘t.’_.l_‘i a: 1"

 

Since 6,wo are fixed, and we have Rang L C [-20||6"wo||i,,oo), we have the desired
lower bound. Hence Lo = inf {L(m,) : m, E L§(fl’(Y \ 8,,(r),6))} exists.

Now choose a sequence {’71. E L§(03(Y \ B,(r),6))} such that L(m,) -* Lo as k —+ 00.
The above inequalities show that

lldmlliz + Ild‘mllf; S ”0 + 4Cl|fi"wollia + 1’ and
llmllip S C(Ildmlliz + lld‘mlliz)
.<. C(2Lo+4Cl|/3"wollf,2 +1)

for k > N. An a priori estimate (see Morrey chapter 6 in [21], Appendix in [9] and
Nirenberg [23] page 153) tells us that

Ilmllrg S C'(|lfi"wolli.2 + Ilflkllm)

So using the above inequalities we see that the sequence {m} is uniformly bounded in L3

norm, since the unit ball in Lg is weakly compact. Thus there exists a subsequence

n, i3 a. 6 Laws \ am»

Since Yo is a 3-dimensional compact manifold, Lg G» 0° is a compact embedding; so we
may assume that {pp} —+ no in 0°. Thus no = 0 on 6Yo, and hence no E L§(fl2(Y\
B,(r),6)). Weak convergence gives that L(r)o) 5 lim L(r7,,r) = Lo which means that no
minimizes L over L§(92(Y \ B,(r),6)).

Hence for any 6 E C°°(Y \ B,(r),6) and t 2 0, L(r]o) 5 L(no + t5)

I[(170 +16) = L(no) + 2t [NB ( )(d'lo A *dt + 4‘00 A rd't + New A *6) + 0(12).
y 1'
where 0(t’) = t2(”d£"iz + Ild‘flm) Since this holds for all t 2 0,

0 S (dm.d£)+(d‘no.d‘t)+(fl"wo.£)
(d‘dno.£) + (was. + (dd‘nM) -'(d'm.£)a + (fl"wo.£)
= (Ano+fl"wo.€)

Replacing f by —6 shows that Ano + fluwo = 0 in the weak sense, and elliptic regularity
then implies that no E C°°(fl2(Y\B,,(r),6)) and Arjo = -6"wo. This completes the proof.

46

Note: Note that we now have A(flwo -— m) = 0, (flwo - m)|3yo\3m(N—r¢) = Lac, 0 =
S2 x {T — 5}.

Using this proposition, we have solutions for A1), = -—fl"wo, mla, == 0 for i = 0,1 where
6,- = 32 X {T- 6}. Define

awe - no on (Yo \ Bum-15m)
u: can onS’x[—T+£,T—e]
flwo - m on (Y1 \B,,1 (N’le),61).
We have u E C0(02(Y0#Y1)), and Au = 0 in the weak sense. Choose the eigenfunction, say
u, corresponding to the first eigenvalue for the one dimensional Laplacian with Dirichlet
boundary condition on [-’-:-, 1:1]. We use the 2-form p - n which has compact support
on Y x R. In this case, the L3,, norm is equivalent to L2 norm, so the operator Af/xR
is -3872; + Ay. This means that the first eigenvalue of A§XR is _<_ (fir)? Let w :—
uu +*pu A dt E 91(Y0#Y1 x R), where * is the 3-dimensional Hodge star operator. Since
pu is perpendicular to *pu A dt and A+ = i-A on (21, we have

1 u ' E
(nu,A+(uu))L2 = -(Im. —u u)L2 = 0 ~ (-—)’(uu.uu)u
4 N

where ’ denotes the derivative with respect to t — the R. factor. So the same holds for
the self-dual 2—form w. This implies that the first nonzero eigenvalue for 13* is less than
or equal to C - (73)”; so in particular it approaches 0 as the length of the tube goes to
00. Another way to see this fact is to work directly with the weighted Sobolev space L3,.

Then one has the one dimensional Laplacian

d'2 . d

with Dirichlet boundary conditions on [—§- fl], and one sees that for this boundary value

c ’ 3
problem there is no zero eigenvalue. Then the eigenfunction of the first nonzero eigenvalue
A is
f = “9-?!” + beg-3'5ltl
where k 2: V65 -- 4A. We have I E C‘, and f(t) = f(-t); so 5RD) =2 0. Together with the
boundary condition, this gives us

fig.a+§_—;—£.b=0, ae$°¥+bci§£'

“I:

=0.

47

Combine these equations to get

g),
Stretching the tube length to 00 corresponds to letting a —* 0 or N -> 00, which means
that I: —> 6, and thus A -v 0. (In T. Mrowka’s thesis [22] he proves the existence of a lower
bound for the first eigenvalue of this operator when the tube length is fixed.) The fact that
the first eigenvalues of A+ go to 0 as the tube gets stretched to 00 gives an “obstruction”
to solving the anti-self-duality equation when one connects two anti-self-dual connections
on Yo x R and Y; x R. One may try to build an obstruction bundle as Taubes did in the
case of a compact 4-manifold with an indefinite intersection form [28], but the following

propositions tell us that we cannot do the same thing for a noncompact 4-manifold.
Proposition 5.1.3 If H‘(Y) = 0, then Lg’sH‘+1(Y x R) = 0.

Proof: This follows from Lefschetz duality and a theorem ([3],Proposition (4.9)) of Atiyah-
Patodi-Singer.

For example, if Y is a homology 3—sphere, then L351! 2(Y x R) = 0, and L3’6H3(Y x
R) = 0; so we do not have an L3,; harmonic 2-form. However we do get the small

eigenvalue for the Laplacian on self-dual 2-forms from Proposition 5.1.2.

In the statement of the next proposition we let 0 denote “spectrum”.

Proposition 5.1.4 Let A0 = min{a(A§,),o(A§,-1)}, then we have a(Agma) = [A0, +oo).
For a compact 3-manifold Y the spectrum consists of eigenvalues, and for the noncompact

manifold Y x R the specth is essential.

Remark: This proposition points out that it is impossible to form an obstruction bundle

in the manner of Taubes with a finite-dimensional fiber.

Proof: In this proof we will drop the weighted Sobolev norm notation and denote it simply
by [I - ||. Recall that A E a(A) for a self—adjoint operator A on a Hilbert space if and only

if for arbitrary positive 5, there exists a nonzero u¢ such that
"(A - mu." .<. 6 - lluell

48

(c.f. [33] Theorem 5.24).

Given A E [Ao,+oo), write A = A0 + p, u 2 0. It is well-known that in Lag-norm the
operator A, = —%(e‘5l‘lg;e5"|) has essential spectrum [0,oo) (see [33] section 10.2). 80
there exists I. such that

"(At - #1))?" S E ' ”fall
If w; + wgdt E Q‘(Y x R), then one has

AYXR.(f¢(w1 + ”2‘11” = Atfcofll + wzdt) + [C(Ag’wl + Alf-lmdty

Suppose that A0 = min a(A‘f"). Let w] = 0 and w; an eigenform corresponding to A0.
Then Ab'lwg = Aowg. Therefore we have
llAlrxnUcwzdt) - (A0 + Mfcwzdtll

"(Am -— sf.)w2dt + f.(A§7‘w2 - Amman
Ellfmdtll

IA

because of our choice of f,. So A0 + a(At) C a(Abe), i.e. [A0, +00) C ”(Ai’xR)°
When A0 = min a(A§,), we get the same result by choosing an an eigenform corresponding
to A0, i.e. Abwl = Aowl, w; = 0.

Next we need to show that ”(Ai’xll) C [Ao,+oo). Given any A E a(Aan) there is
a u, with ||(A§,XR - AI)u¢|| 5 E - ||u,||. Write u, = «21(6) + wg(e)dt, where (01(8) and
w2(e)dt are in L3’59‘(Y x R) and are mutually orthogonal. Then also Ahxnwfis) and
Abxnwg(£)dt are orthogonal. Thus we have the

||A§xnw1(s) — Awfls) + A§xnm(£)dt - Aw2(s)dt||’

llAirxnw1(6)- Aw1(€)||’ + IIAirxnw2(8)dt - Aw2(€)dt||’
S £2lluell2

g £2(||w1(e)||2 + ||w2(e)dtll’)

We have
either llAi’wal(5) - Aw1(£)||2 S 52l|“’1(’5)"2

or [[Abxnwfiswt — Ac..22(€)dt||2 S ef"||an(e)dt||2

49

In the first case where ||A§x3w1(e)— Aw1(£)||2 S £2||w1(e)||2.

Let {(1),} be an orthonormal basis of LgvsflflY) consisting of eigenforms of N]; with
eigenvalues {Ag}, and write 1.111(5) = 2:,- f,-(t,£)¢,-.

A'irxii.‘~'1(€)= Z: Aims-(1.6M,- = 2(A,f,~(t,e) ° «15,- + A,f,(t,e)¢,-)
J 1

Airman“) - “1(6) = ZMJjUfl) ' ¢j + Ajfj(ti€)¢j - Afj(t,€)¢j)-
Since the basis {45,-} is orthonormal, we have
;||A1f1(i,€) + (A,- - A)f,-(t,e)||’ .<. 5’ X1; ||fj(t,€)||2-
Thus there exists a k such that
||A1Ik(t,€)+ (AI: - A)fk(t,€)ll2 S £2||.i'1c(tafi)||2-

where "[1,“,5)" 95 0.
If A < A0, recall A E a(ALXR), so we have A], - A 2 A0 - A > 0, therefore A — A}, < 0;
so A — A), is not in the spectrum of Ag, i.e. A; — (A - A1,)! is invertible. Thus there is a

positive constant c (independent of a) satisfying

c ' Ilfk(t.6)|| 5 "(A1 - (A - Ak)1)fk(t,€)ll S 6 ' llfk(t,€)||-

Choosing a small enough 5 we get a contradiction. So A _>_ A0, i.e. ”(Ai’xR) C [A0, +00).

Similarly one can reach the same conclusion in the second case. So we have shown that
a(ALxR) C [A0, +00) in both cases. This completes the proof that a(ALXR) = [A0, +00).
I

In particular, for a fixed homology 3-sphere Y (i.e with a fixed Riemannian metric),
the spectrum is strictly positive, i.e.

min{a(A¥,xR), 0(A§xn)} = A0 > 0.

50

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