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THESIS

,1 c CO

 

LIBRARY ‘
Michigan State

University J

 

This is to certify that the

dissertation entitled

Torsion, TQFT, and Seiberg-Witten Invariants

of Three—Manifolds
presented by

Thomas E . Mark

has been accepted towards fulfillment
of the requirements for

Ph.D. Mathematics

degree in

MM

Major professor

 

 

 

Date April 20, 2000

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TORSION, TQFT, AND SEIBERG—WITTEN INVARIANTS OF
THREE-MANIFOLDS

By

Thomas E. Mark

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

2000

ABSTRACT
TORSION, TQFT, AND SEIBERG-WITTEN INVARIANTS OF THREE-MANIFOLDS
By

Thomas E. Mark

We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten
invariants of a closed 3-manifold with b1 > 1 to an invariant that “counts”
gradient flow lines—including closed orbits—of a circle-valued Morse function
on the manifold. The proof is based on a method described by Donaldson
for computing the Seiberg—Witten invariants of 3-manifolds by making use

of a “topological quantum field theory.”

ACKNOWLEDGEMENTS

I gratefully acknowledge the unflagging support and assistance of my
advisor Ronald Fintushel, whose help has not only led to the completion
of this work, but also contributed immeasurably to the continuation of my
career. I also thank Peter Ozsvath for his generosity in sharing his great
knowledge of gauge theory. Without his instruction the appendix to this

work would not have been written.

iii

Contents

1 Introduction 1
1.1 Background ............................ 2
1.2 Statement of Results ....................... 8

2 A TQFT for Seiberg—Witten Invariants 13

3 The Pliicker Construction 20

4 Spin6 Structures and Gradient Flows 29

5 Combinatorial Calculations 33

6 Final Results 42

A Appendix: Technical Results 53
A.1 Cylinders and Cylindrical Ends: Linear Theory ........ 55
A2 Gluing Seiberg-Witten Solutions ................. 70

References 80

iv

1 Introduction

Recent years have seen rapid progress in the theory of three- and four-
dimensional smooth manifolds, fueled in good part by the introduction of
Seiberg-Witten gauge theory. One particularly important theorem in the
Seiberg-Witten theory of three-manifolds is that of Meng and Taubes [11],
relating the gauge-theoretic invariants to the algebraic-topological Milnor
torsion. The proof outlined in [11] is based on an explicit reduction of the
case of a general manifold to that of a three—manifold obtained from zero-
sugery on a knot, then use of analytical cut-and—paste arguments to show the
result in this case. Work of Hutchings and Lee [5, 6] has pointed out a way to
prove this theorem in a manner distinct from the original: in particular, their
approach relies first on proving a relationship between the Milnor torsion of
a three-manifold and an invariant that they call I which “counts” gradient
flow lines of a Morse function on the manifold having values in S 1, then re-
lating I to the Seiberg-Witten invariant. The first part of this program was
completed in [5, 6]. The second part, which can be seen as an analogue in
three dimensions of the theorem of Taubes relating the Seiberg-Witten and
Gromov invariants of a symplectic four-manifold, is stated as a conjecture in

[5] and [6]; it is the goal of the present work to give a proof of a large part

of that conjecture—in particular, enough to give the alternate proof of the

Meng-Taubes theorem.

1.1 Background

In order to state the main theorem we will prove, we first need some prelimi—
nary definitions and notation. We begin with the notion of the torsion of an

acyclic chain complex; basic references for this material include [15, 12]; see

also [9].

1.1.1 Torsion

Suppose 0 —+ V’ -—> V —+ V" —-> 0 is an exact sequence of finite—dimensional
vector spaces over a field k. By a volume element 02 for a vector space W
of dimension n we mean a choice of nonzero element 0) E N‘W. It is easy
to show that a choice of volume element on any two of V, V’, V” induces a
volume element on the third. In particular, for volumes 0/ on V’ and w” on

". If 02], 022 are two volume

V”, the induced volume on V will be written w’w
elements for V, then we can write 011 = cwz for some nonzero element 6 E k;

we will write c = cal/022. More generally, let {Ci}?=1 be a complex of vector

spaces with differential 6 : C,- —+ C,_1, and let us assume that C. is acyclic,

i.e., H.(C.) = 0. Suppose that each 0',- comes equipped with a volume
element 02,-, and choose volumes V,- arbitrarily on each image 80,-. Mom the
exact sequence

0 —> 0,, ——> 0,,_1 —+ 80,,_1 —> 0

define T,,_1 = wnun_1/w,,-1. For i = 1,. . . ,n — 2 use the exact sequence
0—)80,+1—>C,—>BC,-—>0

to define T,- = 144,112,- /w,-. We then define the torsion T(C., {w,-}) E k \ {0} of

the (volumed) complex 0.. to be

n—l

710:) = HTi-UHI (1)

i=1

It can be seen that this definition does not depend on the choice of 11,-.

Note that in the case that our complex consists of just two vector spaces,
C=O—>Cli>Co—>O,

we have that T(C) = det(6). We extend the definition of 7(0) to non—acyclic
complexes by setting 1(0) = 0 in this case.

As a slight generalization, we can allow the chain groups C,- to be finitely
generated free modules over an integral domain K with fixed ordered bases

rather than vector spaces with fixed volume elements, as follows. Write

Q(K) for the field of fractions of K, then form the complex of vector spaces
Q(K) (SK 0,. The bases for the 0,- naturally give rise to bases, and hence
volumes, for Q(K) gm 0,. We understand the torsion of the complex of
K -modules 0,- to be the torsion of this latter complex, and it is therefore a
nonzero element of the field Q(K).

Suppose now that X is a connected, compact, oriented smooth manifold
with a given CW decomposition. Following [15], suppose (,0 : Z[H1 (X; Z)] —>
K is a ring homomorphism into an integral domain K. The universial abelian
cover )2 has a natural CW decomposition lifting the given one on X, and
the action of the deck transformation group H1(X; Z) naturally gives the
cell chain complex 0.0?) the structure of a Z[H1 (X; Z)]-module. As such,
0,-(X) is free of rank equal to the number of i-cells of X. We can then
form the twisted complex 02’ (X) = K ®¢ 0.0?) of K -modu1es. We choose
a sequence 6 of cells of I? such that over each cell of X there is exactly
one element of 6; this gives a basis of 0f(X) over K and allows us to form
the torsion 7,,(X , e) E Q(K) relative to this basis. Note that the torsion
71,,(X, e’) arising from a different choice e’ of base sequence stands in the

relationship T¢(X,e) = :tgo(h)'r¢(X,e') for some it E H1(X;Z) (here, as is

standard practice, we write the group operation in H1 (X; Z) multiplicatively

when dealing with elements of Z[H1 (X; Z)]). The set of all such torsions
arising from all choices of e is “the” torsion of X associated to (p and is
denoted r¢(X).

We are now in a position to define the torsions we will need.

Definition 1.1 1. For X a smooth manifold as above with b1(X) 2 1, let (I) :
X —+ S1 be a map representing an element [(15] of infinite order in H1 (X; Z).
Let 0 be the infinite cyclic group generated by the formal variable t, and
let (,01 : Z[H1(X;Z)] —> Z[t,t‘l] be the map induced by the homomorphism
H1(X;Z) —> 0, a +—> tl‘m“). Then the Reidemeister torsion r(X,¢) of X
associated to 05 is defined to be the torsion r,,,,(X).

2. Write H for the quotient of H1 (X; Z) by its torsion subgroup, and let
(pg : Z[H1(X; Z)] —> Z[H] be the map induced by the projection H1(X; Z) —)

H. The Milnor torsion r(X) is defined to be r¢,(X).

Remark 1.2 1. Some authors use the term Reidemeister torsion to refer
to the general torsion r¢(X); and other terms, e.g., Reidemeister-Franz-
DeRham torsion, are also in use.

2. The torsions in Definition 1.1 are defined for manifolds X of arbitrary
dimension, with or without boundary. We will be concerned only with the case

that X is a closed manifold of dimension 3 with b1(X) > 1. In this special

case, work of Turaev [15] shows that r(X) and T(X,¢), naturally subsets of
Q(H) and Q(t), are actually subsets of Z[H] and Z[t,t‘1]. We will usually
think of r(X,¢) as an element of Z[t] defined up to multiplication by it“
and similarly T(X) as an element of Z[H] defined up to translation by ih

forh e H.

1.1.2 Sl-valued Morse Theory

We briefly describe the result of Hutchings and Lee that motivate the theorem
we will prove. From now on we will fix a smooth, connected, closed, oriented

three-manifold X having b1(X) > 1 and a function 915 : X —> S 1 that satisfies
0 ¢ is Morse, i.e., (b is smooth and has only nondegenerate critical points;
0 ¢ represents an element [05] E H1(X; Z) of infinite order;
0 (b has no critical points of index 0 or 3.

Our topological assumptions on X guarantee the existence of such functions
qb. Given 9b, we fix a smooth level set 20 = ¢‘1(pt) C X once and for all.
Upward gradient flow of 05 defines a return map f : 20 —> 20 away from the

descending manifold of the critical points. The zeta function of f is defined

by the power series

. t"
<0) = exp (2 Mm?)

where F1x( f") denotes the number of fixed points (counted with sign in the
usual way) of the k-th iterate of f. One should think of C (f) as keeping
track of the number of closed orbits of (b (which correspond to fixed points
of iterates of f), as well as the “degree” of those orbits (by which we mean,
for a closed orbit 7 C X, the degree of the map it : 7 ”-35 S1 —+ 5‘).

To “coun ” gradient flows of (1) connecting critical points, we introduce
a Morse complex. Write Z[[t]] for the ring of Laurent series in the variable
t, and let M ‘ denote the free Z[[t]]-module generated by the index-i critical

points of d). The differential d : M ‘ ——) M “”1 in this complex is defined to be

dz” = Zaw(t)yu

where 2:” is an index-i critical point, {yy} is the set of index-(i + 1) critical
points, and a“,,(t) is a series in t whose coefficient of t" is defined to be
the number of gradient flow lines of ()5 connecting 3,, with y,, that cross 230 n
times. Here we count the gradient flows with sign determined by orientations
on the ascending and descending manifolds of the critical points; see [6] for

more details.

In our case, where X is three-dimensional and 45 has no index 0 or index 3
critical points, there must be the same number of index 1 and index 2 critical
points—i.e., that d : M1 —> M 2 forms a square matrix after choosing ordered
bases for the M ‘. Fixing such a basis allows us to form det(d) E Z[[t]]. This

data is equivalent to the torsion r(M") defined in the previous section.

Theorem 1.3 (Hutchings-Lee) In this situation, we have the relation

C(f) det(d) = 7(X, <15) (2)

up to multiplication by :tt".

1.2 Statement of Results

As mentioned above, it has been shown [11] that the Seiberg-Witten invariant
of X (after summing over the action of the torsion subgroup of H1 (X; Z))
can be identified with the Milnor torsion r(X). In particular, if we use our
Sl-valued function 03 to “average” with respect to 20, this result implies
Z swm) Mam/2 = WOW 7(X, a»). (3)
aEspin°(X)
This statement, compared with equation (2), shows that the “counting” in-

variant C (f) det(d) is related to the Seiberg-Witten invariant. Conversely, a

proof that the left hand sides of (2) and (3) are related would provide an
alternate, independent proof of the Meng-Taubes theorem. Strictly speak-
ing, this would prove an “averaged” version of the theorem, but that (see
[6]) is sufficient to recover the full relationship between the Seiberg—Witten
invariant and the Milnor torsion stated in [11].

The main result of this work is a proof that the left hand side of (2) is
indeed equal to t““"(z°)/2 times the left hand side of (3), as conjectured in

[5, 0].

Remark 1.4 Hutchings and Lee’s conjecture is more general, in that they
hypothesize that their counting invariant I of spinc structures should agree
with the Seiberg- Witten invariant. The present work proves this statement

“modulo torsion. ”

Our proof of the Hutchings—Lee conjecture is based on ideas of Donald-
son for computing the Seiberg-Witten invariants of 3-manifolds. We outline
Donaldson’s construction here; see Section 2 below for more details. Given
d) : X -> S1 a generic Morse function as above and 20 the inverse image of
a regular value, let W = X \ nbd(20) be the complement of a small neigh-
borhood of 20. Then W is a cobordism between two copies of 20 (since

we assumed d) has no extrema—note we may also assume 20 is connected);

consider the Seiberg-Witten equations on W. Note that two spin“ structures
on X that differ by an element a E H 2(X ;Z) with a([}30]) = 0 restrict to
the same spin" structure on W, in particular, spinc structures 3 on W are
determined by their degree (c1(s), 20).

Now, a solution of the Seiberg-Witten equations on W restricts to a
solution of the vortex equations on 20 at each end of W (more accurately,
we should complete W by adding infinite tubes 20 x (—00, 0], 20 x [0, co) to
each end, and consider a finite-energy solution on this completed space)—see
[3], [13] for example. These equations have been extensively studied, and it
is known that the moduli space of solutions to the vortex equations on 20
can be identified with a symmetric power of 20 itself: see [2], [8]. Explicitly,
if we use a spine structure 3 on W with (c1(s), 20) = 2m for some m E Z,
the vortex moduli space is identified with Symg “1“"20. Donaldson uses the
restriction maps on the Seiberg—Witten moduli space of W to obtain a self-
map It", of the cohomology of Symg—ngo, the alternating trace Tm", of
which is identified as the sum of Seiberg—Witten invariants of spine structures
on X that restrict to the given spine structure on W. For a precise statement,
see Theorem 2.2.

Our main result is the following.

10

Theorem 1.5 Let X be a Riemannian 3-manifold with b1(X) > 1, and fix

an integer m E Z as in the previous paragraph. Then we have

Trnm : [7(M‘) C(fllg-Hm (4)

where Tr denotes the alternating trace and [],, denotes the coefficient of t”

of the polynomial enclosed in brackets.

The “translation” (m H g —— 1 + m) corresponds to the factor flaw/2 in
equation (3), and arises because a Seiberg—Witten solution on W in a spine
structure whose determinant line has degree 2m restricts to a solution of the
vortex equations on 20 in a bundle of degree 9 — 1 + m.

The main idea of the proof is to identify both quantities in equation (4)
as a Lefschetz-type intersection between the graph of a gluing map and a
diagonal-like cycle in a product Sym9‘1+m+N20 x Sym9'1+m+N20 of sym-
metric products, where N is the number of index 1 critical points of gt. On
the left-hand side this is fairly straightforward algebraic topology, thanks to
further results of Donaldson; on the right-hand side it involves some combi-
natorial calculations and genericity arguments.

A key point, however, is that we can calculate Trnm explicitly, given

information about the intersections between ascending and descending man-

11

ifolds of the critical points of d). The result is stated in Theorem 3.1, and
gives an entirely topological description of km. The proof of this theorem,
which we do not include here but refer to [3], is remarkable in the fact that
one need not solve the Seiberg-Witten equations in order to obtain the result.
Indeed, Donaldson shows in [3] that km is determined by formal properties

of the theory, using general algebraic arguments.

12

2 A TQFT for Seiberg-Witten Invariants

In this section we describe Donaldson’s “topological quantum field theory”
for computing the Seiberg-Witten invariants. We use the notation from the
introduction: gt : X ——> .S'1 is a Morse function without extrema, 20 a smooth
level set, and W = X \ 2 is a cobordism from 2 to itself that comes with
an identification 0+W —> 0_W recovering X. We will find it convenient to
complete W by adding infinite cylindrical ends )3 x (—00, 0] and E x [0, co) to
the boundary, forming a noncompact space that we denote W. We take the
ends to be Riemannian products of the metric on 20 with the usual metric
on IR.

Recall that a spin6 structure on a Riemannian 3—manifold Y is a lift of
the SO(3)—frame bundle to a spin°(3) = SU(2) x U(1)/ :l: 1 = U(2) principal
bundle. Such lifts are in 1-1 correspondence with elements of of H 2(Y; Z).
Each spinc structure a has a determinant U (1) bundle det(a) defined using
the obvious representation spin°(3) —> U(1), (z, e”) H e”. This gives rise
to a map from the set spin"(Y) of spinc structures into H 2(Y;Z) by a +——>
c1(det(a)) that is in general not 1-1; the indeterminacy is described by the
2-torsion in H 2(Y; Z). Note that in our case H 2(W; Z) = Z.

Now, a choice of spin‘ structure (1 gives rise to a hermitian 2-plane bundle

13

S, the spinor bundle, using the usual representation of U (2) = spin°(3).
There is a map 7 : TY —-> End(S) satisfying the Clifford relation 7(2))2 =
—|v|211; this is known as Clifford multiplication and we will usually write
7(v)(s) as v.3.

Given a spinc structure a with determinant line bundle L of degree 2m,
we consider the space of pairs (A, \I'), where A is a connection on L and ‘II is
a section of the spinor bundle S associated to a. The choice of A determines,
together with the Levi—Civita connection on W, a connection on S that is
compatible with Clifford multiplication, and an associated Dirac operator

DA. The Seiberg-Witten equations for (A, ‘11) are

D pl! 2 0

*FA = ir(\Il,\Il);
we will consider only finite-energy solutions on W. Here r(-, ) : S ® 5 —+
01(Y) is the adjoint of Clifford multiplication, defined by (a,T(¢,’l/))) A1 =
-.:;(¢,ia.z,b)5. The equations are invariant under the natural action of the
gauge group g = Map(W, S l); we can then form the moduli space MW of
solutions modulo gauge. After appropriate perturbation of the equations

(a technical point that poses no difficulties for the argument to come, and

which, therefore, we will ignore) and use of appropriate Sobolev norms to

14

fit the theory into the usual Fredholm “package,” the space MW becomes a

smooth, compact manifold.

Remark 2.1 For Y a closed manifold with b1(Y) 2 1, the moduli space
My of solutions to the Seiberg- Witten equations modulo gauge is, for generic
choice of metric and perturbation, a compact 0-dimensional manifold. The
Seiberg- Witten invariant of Y in the chosen spinc structure is then the signed
count of points in My. If b1(Y) > 1, the resulting number is independent
of the choice of generic metric and perturbation. The philosophy for the
situation above is that solutions on W that limit to the same solutions on
either end will give rise to a solution on the original closed manifold X; the
topological quantum field theory described in this section is meant to be an
algebraic way of counting these solutions. Some of these technical issues are

addressed in the Appendix.

On the ends of W, metrically the product 20 x Rik, these equations reduce
to the following: for a spin structure K i on 20 (recall that spin structures
on a Riemann surface are exactly the square roots of the canonical bundle
K), write the restriction S [20 as (K i 63 K ‘i) 8 E for some hermitian line
bundle E; note that E2 = ngo. Then the equations are for a t-dependent

section 05 = (01,16) 6 I‘((K% ®E) EB (K‘i ®E)) and a t-dependent connection

15

B in K i 8) E (compatible with the Levi-Civita connection on K), and read

iFB = %(|fi|2 - la|2)vol>:
—2z'53a = B mags = a B : 0.3.

The general setup for this problem, which is described in detail in the Ap-
pendix, implies that a finite-energy solution (that is, an H 3“ solution, in the
notation of the Appendix) on W must approach a constant solution near the
ends. For a constant solution either a = 0 or ,6 = 0; which of these holds is
determined entirely by the degree of E. Suppose m = degE < 0: then we

must have S = 0, and the equations are now

iFB = —%]a[2’U0lg

53C! = 0,

which are the Kahler vortex equations over 20 in the bundle K i ®E. In order
to obtain any solutions at all to these equations (in particular, for K % ® E
to admit any holomorphic sections) we must have 0 g deg(Ki 69 E), and
therefore —(g — 1) S degE < O. In case —(g -— 1) < deg E, we obtain a well-
behaved moduli space of solutions that can, according to [8], be identified
with the symmetric product Symnflo, where n = deg(Ki ® E) = g — 1 +

m. There is a symmetric statement for the case deg(E) > 0For notational

16

convenience, from now on we will write 23") for SymnEo.

Writing 20 = 3_W and 21 = 8+ W, we have restriction maps p,- : MW —>
2:") for i = 0,1 defined by following a finite-energy solution on W to its
limiting values as t —> 00 on )30 x {—t}, 21 x {t}. Note that we have an
identification between 20 and 21. Using Poincaré duality, we then get an

element

flm=(Po®P1)-lMWl e H.<23"))®H.<2§"’)

IIZ

Hom<H*(zi"’).H*<zi"’)).

(Here, as throughout the paper, we work with rational coefficients unless
otherwise specified.) This is the basis for our “TQFTz” to a surface 2 we
associate the cohomology of the symmetric product 23‘"), and to a cobordism

W between 20 and 21 we assign the homomorphism km.

2 +—> VE = H*(E("))
W (~—> km:Vgo—>V2,
From this point on, we will drop m from the notation, writing K, for the
map H‘(2[,9'1+’")) —> Hussy—1+”);
Gluing theory for Seiberg—Witten solutions (see Appendix) provides a

proof of the central property of TQFTs, namely that if W1, W2 are two

17

composable cobordisms then
EW1UW2 : NW2 0 KW]: (5)

Theorem 2.2 (Donaldson) The alternating trace of the map It calculates
the Seiberg- Witten invariant of X, in the following sense. If Sm denotes the

set of spin6 structures a on X that have c1(det a) - E = 2m, then

Trn = Z S'Wx(a),

ares...

where Tris = Z,(—1)‘tr(k|H.(2(n))). E]

The proof rests again on gluing theory for solutions; in the current sit-
uation one finds that ’Ii' k is calculating a coincidence number for the two
restriction maps p0, p1, and the theorem is a result that other gauge-theoretic
results would lead one to expect.

In order to calculate the invariant, then, we use the composition rule (5)
to reduce to the simplest possible situation, namely a cobordism W between
a surface 20 and a surface 21 of one higher genus. Equivalently, the height
function on W has a single index-1 critical point. It is a remarkable fact that
K. is determined in this situation by the formal properties of the theory; this

result will be described in the next section.

18

Recall (see MacDonald [10]) that the cohomology of the symmetric prod-

uct 2“" can be expressed as follows:
H()2(")— =€BA‘( ((H12 )® symn- (H°(2) ea H2(2)). (6)

The “interesting” part of this expression is the exterior power A‘(H1(E));
the symmetric part can be thought of as a fixed vector space of dimension
n — i + 1, independent of 2 (since this part of (6) has dimension independent
of the genus—see [3] for further discussion of this point). It turns out that k
is the natural extension of a map A‘(H1(20)) —> A“(H1(21)) that is defined

entirely topologically, a map we now describe.

19

3 The Pliicker Construction

In the following two paragraphs there is no reason to restrict the cobor-
dism W, so we return for now to the general situation of any cobordism
with boundary 6_W = 20, 6+W = 231. There are inclusion-induced maps
r,- : H1(W) —> H1(2,-) that we combine to produce a subspace FW 2
r0 6 r1(H1(W)) C H1030) 69 H 1(21). Recall the Pliicker construction: given
a subspace S C V0 EB V] of a sum of oriented vector spaces, we can form
a linear map [S I : A‘(Vo) —> A“(V1) by wedging together all the elements
of a basis for S, yielding an element of A‘(Vo 63 V1) 2 A‘(Vo) ® A‘(Vl) E“
Hom(A"(Vo), A‘(V1)) The second isomorphism uses the fact that the volume
form on V0 defines an isomorphism (A"V0)" 2’ Adim(v°)"°Vo. The result is a
linear map well-defined up to a multiplicative constant; we can reduce this
to an indeterminacy of :lzl in our situation (where V,- = H1(2,-)) by using the
integer lattices. Write pw for the Pliicker map associated to My.

This leads us, motivated by the constructions of the previous section, to

a “baby TQFT:”

surface 2 H V2 2 A'(H1(2))

cobordism W 4—) pw : V2,, —> V2,

20

Strictly speaking this is not a TQFT since the composition rule fails without
further assumptions on the cobordism. In the case that W is an “elementary”
cobordism, i.e., contains a single critical point, the map pw will be used to
find kw. We calculate this case now.

Suppose, then, that W connects 29 and 29,1 in the simplest way: there
is a unique critical point (of index 1) of the height function h : W —+ IR, and
the ascending manifold of this critical point intersects 29“ in an essential
curve that we will denote by c.

Now, c obviously bounds a disk D C W; the Poincaré dual of [D] E
H2(W,0W) is a 1-cocycle that we will denote £0 E H1(W). It is easy to
check that {o is in the kernel of the restriction r1 : H1(W) —> H1(Eg), so we
may complete {0 to a basis {0,61, . . . ,{29 of H1(W) with the property that
171 z: r1(§1), . . . ,7729 := r1(£2g) form a basis for H1(29). Since the restriction
r2 : H1(W) —> H1(Eg+1) is injective, we know 170 := r2(§0), . . . ,7729 := r2(§2g)
are linearly independent; note that r2 ((0) is just c", the Poincaré dual of c.

The choice of basis 5,- with its restrictions 77,-, 17,- gives rise to an inclusion
L : H1(29) —> H1(Zg+1) in the obvious way, namely ((17,) = 17,-. One may
check that this map is independent of the choice of basis {5,} for H1(W)

having £0 as above.

21

With this understood, it is clear that F = r1 EB r2(H1(W)) C H1(29) EB
H1(29+1) is spanned by {0 QB 6,771 65 771,...,1729 EB 1729}. The reader may
easily check that the Pliicker map A‘Hl(29) —> A‘H 1(29+1) associated to

*

this subspace is a v—-> (.(a) /\ c .

Theorem 3.1 (Donaldson) In the case that W consists of a single handle-
addition as above, the map K. arising from Seiberg- Witten theory agrees with
the Pliicker map above on the exterior powers of H1(Eo) and is the obvious

“identity” map on symmetric powers of H 0(20) EB H 2(230).

For proof, see [3]. It is remarkable that the proof of this theorem is entirely
formal, and in particular does not rely on any analysis of the Seiberg-Witten
equations. El

Using the composition rule (5), we see that the map K. associated to
the full cobordism arising from our original Morse function is given by the
composition of the Pliicker maps associated to each handle addition in the

cobordism. Note that here we are using the following facts:

1. The Seiberg-Witten equations give rise to a TQFT, satisfying (5) in

particular, and

2. The maps arising in this TQFT corresponding to elementary cobor-

22

disms agree with the maps arising from the topological construction

above.

This means that if W = W1 U2, - - - U2,,,-, Wm is a decomposition of W into
elementary cobordisms and fiw, is the extension of pw, to H129”), then kw
is given by the composition ,6", o - - - 0 m.

We have now seen that the 1-handle additions in W give rise to the map
above that is essentially wedging with the cocore of the new handle. The
2-handles give the transpose of that simple map in a way we now describe.

Suppose V1 and V2 are oriented vector spaces with fixed volume forms and
I‘ C V; GB Vz is a subspace with Pliicker map [F] : A‘Vl —> A’Vg. The obvious
isomorphism Vl EB V2 —> Vg EB V1 gives rise to a map [I‘| : A'Vg —> A’Vl, and

the relationship between the two maps is given in the following

Lemma 3.2 Whenever a E A‘Vl and b E Ang are such that |I‘|(a) and b

have complementary degree in A’V2, we have
a /\ [f[(b) = (_1)(dim(F)+dim(V1))(i+dim(V1))|F|(a) /\ b,

using the volumes to identify Amka, k = 1, 2, and in particular to resolve

the :l:1 indeterminacy in the definition of [I‘|.

The proof is straightforward, and we leave it to the reader. Cl

23

We are now in a position to obtain‘a formula for Tr It. Here and in what
follows we will write 20 for 8_W and 21 for 6+ W. We will also assume
that we have arranged W so that under the height function h induced from
the original Morse function 45, the index-1 critical points occur “below” the
index-2 critical points—i.e., W consists of a certain number, say N, of l-
handle additions, followed by exactly N 2-handle additions. Write 2 for the
“middle” of W, i.e., 2 = 6+(20U (all 1-handles)). There are N distinguished
curves on 2, the ascending manifolds of the critical points, that we denote
by c,-.

Note that if W0 is the “first half” of the cobordism, i.e., W0 = 20 U
(all l-handles), and W1 is the “second half,” then W0 and W1 are topologi-
cally identical. It will be convenient to assume that W0 and W1 are in fact
two copies of the same space, with a fixed identification between them, and
that the original cobordism W is obtained from these two by means of an
orientation-reversing diffeomorphism A : E = 0+ W0 —+ 2 = 6- W1. (See
Figure 1.)

A word about orientations. We should think of W1 as having the opposite
orientation from W0; loosely if W0 is written together with an orientation

00 as (Wo,oo), then (W1,01) is taken to be (W0, —oo). Now, 00 induces an

24

Figure 1: Decomposition of W

orientation on 0+Wo = E, which we will also write 00: 0+Wo = (E, 00). In the
same way —00 induces an orientation on 6_ W1 = 2, namely 6_ W1 = (2, —00)
(here 6+Wo and 6_W1 are taken to be orented as boundary components of
their respective manifolds). Thus the orientation-reversing map 8+ W0 —>
6- W1 in fact preserves the orientation 00 of )3 induced by the identification
2 = 6+ W0.

Clearly A contains all information about W; therefore A will play a cen-
tral role in the calculations that follow. We note that we may make certain
transversality assumptions on A (for example, that A(c,) meets c,- trans-
versely for all i and 3'), but as these issues will come up again later we do not
dwell on them here.

To avoid confusion, when referring to 2 we will always mean 6+Wo, so

that {c,-} are parts of the ascending manifolds of the index-1 critical points.

25

In this scheme, then, the descending manifolds of the index-2 critical points
intersect 2 in the curves A‘1(c,-).

With the situation now standardized, let us also fix an ordering of the
critical points in W0, and hence also an ordering of the curves c,. Each
critical point gives rise to a map P,- : H 12:1:1-115 —> H1233) where 29+,-
is the genus g + i surface obtained as 6,.(20 U {the first i 1-handles}). We
will denote by P the composition of Pliicker maps arising from the first N
handle additions: that is P = PN o 0 P1 : H128”) ——> H12("+N)) is
given by P(a) = L(a) /\ c”; A - - - /\ cfv where L is the inclusion induced in the
cohomology of the symmetric products from the composition of inclusions
H129) —> H129“) defined previously. In particular we extend the inclusion
to the exterior algebras of H1(-), then to the full cohomology of the symmetric
product via the identity on the Sym" (H OeBH 2) factor (compare equation (6)).

Simlarly, we write P for the composition of “transposes” of the Pliicker
maps, in the notation of Lemma 3.2; that is, I3 = P1 0 0 EN. In what
follows we will omit the map I. from the notation, implicitly using it to identify
H128”) as a subspace of H12("+Nl). We can write it as the composition
A = 13 o A-1* o P : H*(2§,"’) —> H‘(2("+N)) —> H*(2(“+N)) —+ H129”) =

H128”); our objective is to compute the alternating trace of k.

26

In what follows it will be convenient to fix a basis {a} for H123”) and
its cup dual basis {a°}——so for basis elements a and 0 we have a° U B = (Sag.
MacDonald [10] shows that we may take the or and a° to be “monomials,”

i.e., each (1 lies in a single summand in expression (6).

Proposition 3.3 We have

Tm = Z(—1)'°I+N|“'+N(N'1)/2(A‘(a)AA'CFVA- - -AA‘c;)U(aAc;v/\- - -/\c;),

(7)
where [(1] denotes the degree of a, and c; is the Poincare’ dual of the curve c,-

on 2.

For the proof, note that repeated application of Lemma 3.2 shows that if

(.21 and P(w2) are of complementary degree in A1H1(20)) then

LU] /\ 130.02) = (—l)le1|+N(N—l)/2P(w1) /\ (4)2

: (_1)NIWII+N(N-1)/2(wl /\ CI /\ . . . /\ Clv) A012.

In the calculation that follows we will apply the above fact to elements £01,022
in the cohomology of a symmetric product, and the relevant degree [call is not
the degree of wl as a cohomology class, but rather the degree of the part of (.01

that lies in the exterior algebra (we will sum only over a basis of monomials,

27

so the latter degree is defined in that case). However, these two are clearly
equal modulo two, so we are justified in using the above calculation in our

situation. To calculate the trace we write

Tm = Z(—l)'°"a° U P o A‘” o P(a)
= Z(-1)'°'+N'°'+fli—_—Q(a° A c; A - - - A c;,,) U A‘1‘(a A c; A - -- A cyv)

= Z(—1)IGI+NI°'+WA10° A c; A - .. A cjv) U (a A c} /\ - - - /\ Clvl-

a

[3

Remark 3.4 Our ordering of the critical points of 03 determines an ordering

of the c,- used above. However, Tr(k) is independent of this ordering.

Our objective is to see that (7) agrees with the invariant defined by Hutch-

ings and Lee. We next recall the definition of that invariant.

28

4 Spin“ Structures and Gradient Flows

We begin by recalling an association between spin“ structures on the closed
3—manifold X and elements of the first homology. Following [6], let X’ denote
X \(critical points), and write H (45) C H1(X’, BX’) for those 1-cycles whose
boundary consists of the sum of all index 2 critical points minus the sum of

all index 1 critical points.

Proposition 4.1 (Hutchings—Lee) The set of spin“ structures on X is in
one-to-one correspondence with H (¢) by the map sending a spin“ structure

a to the dual of the first Chem class of the determinant line of a.

For proof we refer to [6]. [I]

More explicitly, given a we form c1(det(a)), restrict this to X’, then apply
Poincaré duality H2(X’) —> H1(X’,6X’).

The invariant we are interested in will be an element of (a subring of) the
ring of power series with variables from H1(X’, BX’), and will have nonzero
coeficients only for elements of H ((1)) To be more precise, let A be the
Novikov ring of functions é : H1(X’, BX’) —> Z satisfying the condition that
the set {a E H1(X’,BX’)|§(a) 91$ 0 and a.2 g k} is finite for every k. We

may think of elements of A as series in H 1 (X ’ , 6X ’ ) (in particular the product

29

is the usual product of series), and will write them as such. To write down
the invariant I of Hutchings and Lee, define an analogue B of the Morse
boundary as follows: let Q1 be the rational vector space spanned by the
index 1 critical points and Q2 that spanned by the index 2 critical points.
Then define the linear map B : Q1 —> Q2 8) A on a basis {2:} by

3(3) = EN) 2 [7]

76.7-12.1!)

where .7-"(x, y) denotes the space of flows between the index 1 critical point
a: and the index 2 critical point y. Letting 0 denote the set of closed orbits
of the gradient flow of (t, we set

I = Ha — my“) det B.

760

Here 6(7) = :tl is a sign to be defined below. One checks that I has nonzero
coeficients only for elements of H (45); it is Hutchings and Lee’s conjecture
that as a function on spin“ structures, I agrees with the Seiberg—Witten
invariant.

As mentioned in the introduction, we will prove an “averaged” version
of this statement, in the following sense. Let p : H1(X’, BX’) —) Z[t] be the

map that sends a E H1(X’, aX’) to ta'z; then we consider the invariant p(I).

30

To give an explicit computation of p(I), we recall some further definitions.
To the Morse function qb : X —> 5'1 there is associated a “Morse complex”
(M ’, d) in the following way: choose an orientation on each ascending and
descending manifold of each critical point, in such a way that the ascending
manifold A(m) and descending manifold D(:c) of a critical point a: have inter—
section number A(a:).D(:c) = +1. Then M i is defined to be the Q(t)-vector
space spanned by the index i critical points of o, and d : M i -—> M ”I is

defined by
dz,- = Z “HUM/2'
y
where a,,-(t) is a polynomial whose n-th coefficient is the number (counted
with the sign determined by the intersection of the relevant ascending and
descending manifolds) of gradient flows between 2:,- and yj.

The Morse function 03 also determines a return map f : 2 —-> 2 via
the upward gradient flow, defined away from the descending manifolds of the
critical points. Similarly, the iterates of f are defined away from codimension-
1 subsets of 2, though we will not keep track of this in the notation. In order
to count the closed orbits of d), we introduce the zeta function C (f) of f:

. ,. 1:"
<0) = exp (2 Fun )-k—) .

Ic>0

31

where F1x( f") is the number of fixed points of the iterate f’“, counted with
the sign associated to the corresponding intersection point of the graph of f "
with the diagonal in 2 x 2.

It can be seen (see [7, 14]) that

C(f) = 110 — ,..2,_.(.)

760

where, as above, 0 denotes the collection of closed orbits, and 6(7) is the
sign of the fixed point of the iterate of f corresponding to 7.

It is not hard to see that p(I), introduced above, is simply C ( f) - det(d :
M1 —> M 2). In the current situation, where the Morse complex M “ has
nonzero terms only in degree 1 and 2, the determinant of the differential is
exactly the torsion of the complex (c.f. equation (1.3)). Our task, then, has
come down to explicitly calculating the n-th coefficient of C (f) det(d) and
comparing this to Tm. That calculation and comparison is the content of

the next section.

32

5 Combinatorial Calculations

Our proof that Tr K. is the nth coefficient of C ( f) det(d) will consist in showing
that both quantities are given by a Lefschetz-like intersection of the graph
of the map induced by A on 2("+N) with a cycle D that can be thought of
as a diagonal, modified to include information about the c,. Indeed, in the
case that our Morse function has no critical points (i.e., gt : X —> .S'1 is a
fiber bundle), D will be exactly the diagonal and the intersection gives the
Lefschetz number of A on 2("). This can be seen immediately to agree with
our formula for Tr k in this case.

Returning to the general case, we can describe the diagonal-like class D

as follows. Write points of symmetric products as sums of points on the

original surface; then D E H2,,+2N(2("+N) x 2("+N)) will be defined by

D = [{21. p.- + 22.. q,, 2.1.10: + 2;. q.|p.-,p: e c.- for each 0] . (8)

Thus D consists of diagonal points of 2M x 2(") together with pairs of points
on the c,-, one pair for each i. To obtain a well-defined homology class, we take
D to be the image under the symmetrization map a : (2” x 2“) X (2" x 2”) ->
20””) x 20””) of the class cN x >< c1 x An x c1 x x CA), where An is

the diagonal in 2" x 2". Then the Poincaré dual of D is given by the image

33

of
c}, x x c; x (23-1850 x3) xc; x x c},
13
under symmetrization, where {B} is a basis for H12") and {5°} its cup dual
basis. It is not hard to see that this is

PD(D) = Z(—1)I°I(c;, A . . . A c; A 62’) x (a A c; A - .. A c;,,)

a

for {d} a basis for H120”), {d°} the cup dual basis. In fact we may take the
sum to run over only the basis {a} for H128”) C H12(")) C H12("+N))
we used previously, since if some it is complementary to the subspace spanned
by {a} then either c'i or (32° contains at least one c; (we take our basis elements
to be monomials in a basis for H°(2), H12), and H2(2) as in MacDonald
[10]) and hence the corresponding term in the sum vanishes.

If {g}, {€°} are cup dual bases for H12("+N)) extending {5:}, {51°} above,
we can write the Poincaré dual of the graph I‘A(..+~) in cohomology as

PD<P..(.+~)) = Z<-1>"'5° x we).
6

We also assume that {5} includes all elements of the form a /\ c; /\ - - - /\ c7

8;"

i1 < < it. Finally, we can calculate:

D.FA(n+N) = PD(FA(n+N)) U PD(D)

34

= (2W...)
6

N(N-1)
(Z(—1)IOI+N|GI+ 2 (0° A c; A A cyv) x (a A c; A . . . A c;,,)

a

Z(—1)6(g° 0 (a° A c; A - . . A cm) x (A15) 0 a A c; A - . . A Cy),
0.5

where e = [E] + [a] + Nlal + |§|(|a| + N) + N(N — 1)/2. The first term in
the last cross product is zero unless g = a° /\ c; /\ - - - /\ cjv, and in that case it
is 1. Carrying out the sum over g and comparing to Proposition 7 therefore

gives the following:
Proposition 5.1 Tm = D.I‘A(..+~). E]
On the other hand, we have:

Proposition 5.2 The intersection from the previous proposition can be cal-
culated as
D.I‘A(~+..) = V: Fix(A("))(A;“"(c1 A - .. A cN), A:1c1 A . . - A A:1cN), (9)
k=0
where A, is the action of A on H1(2) extended as a derivation to a map on
the exterior algebra, and the angle brackets refer to a certain determinant-like
product of pairwise intersections of A§(c,-) with A’1(cJ-). We take A2 = I and

Fix(A(°)) = 1.

35

l

To be more explicit, for {a,, b,-}[°=l E H1(2) we define the notation (a1 /\

.../\ak,b1/\-~/\b,,)tomean

Z(—1lsigna(al-bo(1))°“ (ale-ba(k))a

085;.

where a,.b,,(,-) is the intersection pairing on 2. The expression in the proposi-
tion is to be evaluated by distributing AQ‘“ across c1/\- - -/\cN as an (unsigned)
derivation, then using the above definition for each of the resulting terms.

To prove the proposition, we will count in a combinatorial fashion the
points in the intersection D H I‘ 110”"): then see that the points are assigned
the appropriate orientation. In what follows we will write points of symmetric
products 20‘) as sums of points of 2.

3111313059: then, that P = (2.111% + 22:1 qj’ 21:1 A(Pi) + 22:1 AM»)
is a point of I‘A(~+..) that also lies on D. In this expression we take p,- E c,-
for i = 1,...,N; compare (8). It does not follow that A(p,) E c,-; rather,
we know merely that N of the points on the “vertical” side of P (that
is, the points making up the second coordinate of P, namely EA(p,-) +
ZA(q,-)) are on the ch, in fact precisely one on each c), by the definition
of D. Once we have determined, given P, which N members of the list
A(pl), . . . ,A(pN),A(q1), . . . ,A(q,,) are on Uck, we know that the remaining

n points, together with the q,- on the “horizontal” side form a point on the

36

diagonal of 2‘") x 20').

Suppose to begin with that each A(p,) is on Ur. ck: then the remaining
points q,- together with their images A(qj), lying as they do on A”, form a
fixed point of Am. If M0 is the (unsigned) number of points Z,p,~ E 2“”)
such that p,- E c,- and A(p,) E Uck, we see that the number of points P of

this form is

IFixI(A‘">) - Mo (10)

where by [Fix] we mean the “raw” unsigned count of fixed points. To calcu-
late Mo, proceed as follows. For the first point p1, we have 28:, #(A(cl)flc,-)
possibilities, since A(pl) may be on any of the ck. Given a choice for
p1, suppose that A(pl) E c1. Then the number of possibilities for p2 is
28:2 #(A(c2) fl cj)—note that A(pz) may not be on c1 since P must be on
D. And so forth: let us write 1,,- for #A(c,-) an; then for the final calculation
of Mo we get
Mo = Z Il,a’(1)I2,a(2) ' “IMO-(N)-

OESN
Note that we could just as well set 1,,- = #(c, n A‘1(c,-)).

Remark 5.3 This assumes that c,-flA‘1(c,-) consists of finitely many points.
In Section 6 below we will prove a transversality result that justifies that

assumption.

37

Suppose now that P is as above, but on the “vertical” side the points
A(pg), . . . ,A(pN),A(q1) are the ones that lie on Uck. Then A(pl) must be
in the diagonal part of P, and hence A(pl) must appear in the list ql, . . . , q,,
on the horizontal side. If A(pl) = q1, then A2 (p1) E U ck, and the remaining
diagonal points (Q2, . . . ,q,,, A(Qg), . . . , A(q,,)) form a fixed point of A(”‘1). To
count points of this form, write 18’ = #(A2(c,-) F] c,) = #(A(c,~) fl A‘1(cj));
then we see that the number of points P with A(pg), . . . ,A(pN),A(q1) on
Uc,c and A2(p) = q] is given by

IFix|(A‘"‘”) Z 1820,1242) ...1,.,,(,.,.

UESN

If we wish to allow any of the p,- to play the role of p1 above (note that
though they are constituents of a point in a symmetric product, the p,- are not
interchangeable since we have assumed p,- E c,-), we must allow the superscript
(1) to appear on any of the I in the above expression and take another sum.

Then we would obtain the number M1 of points 2, p,- E 2‘”) satisfying
- p,- E c,- for each i;
- there is a unique index j for which A2 (17,-) is on U ck;

— for all i # j we have A(p,) E U01.-

38

Finally we have an expression analogous to (10) for the number of points of

the form under consideration, namely
(Fix|(A("-1)) - M1.

One could continue in this vein, but instead of doing so we reorganize
our count slightly. Instead of counting based on the number of A(p,) that
are on U ck, we note that by arguments similar to the elementary one above,
for each p,- there exists a nonnegative integer a,- such that A“‘+1(p,-) E U ck.

We arrange our count according to the quantity a = a1 + + cm; above
we have calculated the number of P with a = 0 and with or = 1. Note that
since there are exactly N + n points on the vertical side of P, N of which
must be on the c), and not contribute to a diagonal point, we must have
a S n. Likewise, given a, we see that there are exactly n - a of the q,- on the
horizontal side not “used up” as the images of p,- under iterates of A—hence

the remaining q,- and their images A(qj) form a fixed point of A("‘°') Writing
15,? := #(A’+‘(c.~) 0 c.) = #(A'(c.> 0 Are».

we have that the total number of points P with fixed a = n — k, say, is given

by
(mu/4‘“) E: Ii:‘(’.)---Ii:t:2~, . <11)
a1+---+0N=n—l¢
GESN

39

Finally, to complete the count of #(DflI‘Am...) ), we sum the above expression
over all k from 0 to n.

It is now clear that the proposition will hold so long as the sign attached to
P as an intersection point of D and I‘A(~+..) is the following one: each p, gives
rise to an intersection point of A“‘(c,) with A‘1(c.,(,)) for some permutation
o, and this intersection point has an associated sign 6,. The permutation has
a sign of its own, (—1)"", and the fixed point of A(""‘") arising as above has a
sign €N+1 in the usual way. Associating the sign (-1)""61 - - - €N+1 to each of
the points counted by (11) gives exactly the quantity on the right hand side
of (9) (after performing the sum over k). We leave it to the reader to check
that the sign of P that arises by considering P as an intersection point of D

and PAW”) is the one described here. D
Proposition 5.4 The n-th coefficient of p(I) = C ( f) det(d) is given by

[p(I)]n = ZFiX(f(“))(fl‘-"(Cl /\ ° ' ° /\ 61v), A"l(61) /\ ° ' ' /\ A’1(CN)>- (12)

16:0
Here f(") is the induced action of f on the symmetric product 2““), and the

angle brackets are defined as in proposition 5.2.

The proof is another direct calculation. First, it is not hard to calculate

40

from the definition
40) = exp (2 mung)
n=1

that the kth coefficient [C ( f )] k is given by

2 % (fiFixv’v) (iFixv’v) .

“Lit???" '
One can see by an easy combinatorial argument that the above expression is
simply F1x( f‘kl).

Next we must calculate an arbitrary coefficient of the differential d : M, —>
M2 of the Morse complex associated to (25. This differential is represented
by a matrix of rank N, and the entry in the (i, j)-th position is the series
2,2,1 f"(c,).A‘1(c,-))t" (note that the coefficient of the polynomial a,,-(t)

defined in section 4 is simply the intersection number used here). Some

algebra shows that the k-th coefficient of the determinant of d is given by

Z (—1)'°" (f"‘(cl)-(A‘1(c.(1))) (f"”(c~)-A‘1(c.uv))) .

06$"
h1+mkN=h

which is precisely equal to (f:(c1 /\ - - - /\ cN), A'l(c1) /\ - - - /\ A‘1(cN)). Since

[C(f) det(d)ln = Z[C(f)]k[det(d)],,_k, we are done. Cl

41

6 Final Results

We have now proved the bulk of the main theorem (restated below); together
with the TQFT results from section 2, in particular Theorem 2.2, the follow-
ing result completes the proof of (the averaged version of ) Hutchings’ and

Lee’s conjecture.
Theorem 6.1 Trn : [p(I)]n.

From the work of previous sections we have:

Tm = Z:(—1)""'+""°"(A‘c;V A - - - A A‘c; A A*(a)) U (c)v A - . - A c; A a)
= D.FA(N+n)

= Z Fix(A("’)<A2“’°(c. A - - - A cN).A:1cl A ° - - A AIICN>
le=0

= ZFiX(f"‘))(f;‘—’°(cl A ...A.,),A—1(.,) A ...AA—1(.,.)) (13)
16:0

= [p(Illn
The only part of this left to prove is (13). We will see that the replacing of A
by f is legal in two steps: first we show that the right hand side of formula
(9) is unchanged if we use a certain non-generic flow map A : 2 —+ 2 in place
of the diffeomorphism A. That is, we will see that I‘A(N+..).D = [p(IA)],,,

where [p(IA)],, is given by a formula analogous to (12) above.

42

Second, we show that the invariant p(I) computed using the flow A is the
same as that computed using a generically perturbed flow—then the fact,
whose proof was sketched by Hutchings and Lee and set down in final form
in [4], that I is a topological invariant will complete the proof.

We define A as follows. Our cobordism W is made up of two (identical)
parts W, U A W2; on W1 we take the map F0 : 20 —> 2 induced by gradient
flow of the original Morse function 0’) : Y ——> S1 (we are being sloppy here:
F0 is only defined away from the descending manifolds of the index 1 critical
points). On W2 we take the identical flow—that is, we consider W2 simply as
another copy of W1 and use the same flow we used in that case. This latter
flow map we denote by F1 : 2 —> 21; note that after identifying W, = W; we
have F1 0 F0 = id = F0 0 P, where the compositions are defined. We then set
A = FoFlA : 2 —> 2; apparently A agrees with the restriction of A to the
complement of the collection of curves A‘1(c,), the descending manifolds of
index-2 critical points intersected with 2.

To see now that I‘A(N+..).S' = éFix(A("))(A2"‘(cl /\ /\ cN),A:1c1 /\

- - - /\ A: 1c”), it will suffice to show that
i. Fix(A("‘)) = Fix(A("‘)) for all m S n, and

ii. Am(c,-).A‘1(c,-) = Am(c,).A‘1(c,-) for all i and j and all m, 0 S m S n.

43

We will in fact show that these conditions are true after perturbation of

A:
Lemma 6.2 We can adjust A so that

a) All fixed points of A", 1 S k g n, are isolated and all intersections

A"(c,) fl A”1(cj), 1 S k g n, are transverse.
b) No fixed point of A" occurs on Uc, for any 1 S k S n.
c) All fixed points of A’“ occur in the domain of A“, 1 g k g n.
d) All intersections c,- 0 A‘““1(c,-) occur in the domain of A“, for k S n.

From this, it is clear that (i) and (ii) above hold.

Proof: That we can arrange for part (a) to hold we take as obvious: this is
a standard transversality assumption.

The remaining arguments are all similar in flavor; each involves consider-
ing a point p that fails to meet the criterion in question and giving an explicit
modification of A to repair the defect. At each stage, we must check that the
modifications we make to A preserve the properties we obtained in previous
steps.

For part (b), consider a point p that is a fixed point of A", with p E c,.

We assume that k is minimal, i.e., p is not fixed by any smaller power of A,

44

and therefore we may find neighborhoods U,- of the iterates Aj (p) that are
all disjoint from one another. The curve A’“(c,) passes through p, and by
(a) we may take this intersection to be transverse. Let X be a vector field
that is supported in a small neighborhood V C U0 of p and whose time—1
flow 4px moves p ofl' of c,- parallel to A‘“(c,): in particular, we can arrange
that cpx(c,) fl A‘“(c,) consists of the single point p(p). We replace A by
A’ = A(px; clearly (since (px is supported on U0) we have A’“ = A"<p on U0.

It is clear that the perturbation A —> A’ cannot introduce new fixed points
of order less than k by choice of the neighborhood in which the perturbation
is supported. We must see, then, that A”‘ has no fixed points on c,. By
construction of cpx, we have #(A”‘(c,)flc,) = #(Akpx(c,)flc,) = #(cpx(c,)fl
A"‘(c,)) = 1; suppose this intersection q is in fact a fixed point of A’“.
Then 4 = A”‘(c.~) 0 c.- = 2‘1’°(90x(c.-)n A"°(c.')) = A"(<.0xtp)). so that p =
(Akcpx)‘1(q) = q. This is a contradiction, for p is clearly not fixed by A’“.
Hence there are no fixed points of A’“ on c, 0 U0.

By repeating this argument for all fixed points of A" in succession (these
are isolated and therefore the above argument may be carried out indepen-
dently for each), we conclude that A may be modified so that all fixed points

of A" occur off of the c,-, without introducing new fixed points of lower or-

45

der. Hence the same argument shows that we can modify A to have no fixed
points of order less than n + 1 that lie on U c,.

Claim (c) is immediate from (b): if p is a fixed point of A", then since
the domain of A is the complement of Uc, we see that for p to fail to be
in the domain of A“ it must be the case that Aj (p) E Uc, for some j < k.
Naturally q = Aj (p) is fixed under A", and hence is a fixed point of A" lying
on U c,, in contradiction to part (b).

Finally, we must prove that (d) can be arranged without disturbing the
previous work. Another way to state the proposition is that we would like

to ensure that all the intersections

1.6 = (U c,) 0 (A°‘(U c.-))0(z‘1"(Uc.-))

are empty, for a < E S n. By (a), we may assume that all the [QB consist of
a finite number of points. We proceed inductively, ordering the 105 first by
,8, then by a: that is, 10,131 < Imp, if ,6, < ,82 or ,8, = ,82 and a, < (1;. Begin,
then, with [1,2, and suppose p is in this intersection—then p = A2(q2) =
A(ql), with, say, q, E c,,, g; E c,,, and p E c,,, We take a neighborhood U of

p such that:

1. If A(p) E c,3 for some i3 then the configuration of curves A2(c,,), A(c,,),

46

c,,,, and A‘1(c,,) appear in U as four arcs intersecting transversely at

the single point p:

A‘1(c.-.) A28”) c,.,

 

 

2. If A(p) E Uc, then the curves A2(c,,), A(c,,), and c,,, appear in U as

three arcs intersecting transversely at p, as above.

Perturb A in a small neighborhood of q, using a flow parallel to A(c,,)
so that under the perturbed map A’ the point ql misses p. The sequence of

neighborhoods A‘2(U), A‘1(U), U now appear as

 

Cg: A(ciz)
<12 <11 6“
A_2(U) 14‘1“] )

Apparently p is no longer in 11,2, but if case (1) above holds then A(p) is in
11,3. This is not a problem, since [13 is still to be dealt with in our inductive

procedure (if case (2) holds, then p can at worst give rise to a point of It,”

47

for r > 1, in the case that A'(p) is on U c,—this set will also be dealt with
later). Similarly, we have two more points p’ = A’ (q) and p” = A’ (c,,) 0 c,,,
as labeled above that may give rise to points in 10,5. Indeed, if A" (p’ ) E U c,-
then A"(p’) E I,+1,,+2, while if A“(p”) E Uc, then A“(p”) E I,,,+1. Note,
however, that by our choice of U we must have s, t _>_ 2, so these points are
also in lag that will be considered later in the inductive process. Finally, if
case (1) above holds, then we have a third new point p’” = A’2(c,,) fl c,3 that
can at worst give rise to a point of IN” for some q 2 1, in the case that
A”(p’") E U c,. Again this set will be addressed later.

The above argument shows that if I13 is nonempty, we may perturb A
in a neighborhood of a preimage of each of its points and by doing so empty
out the set 11,2, at the cost of possibly adding points to other 1a,, that will
be considered later. Note that this process cannot interfere with previous
work since we are working near points in the orbit of a point on U c,, which
cannot be the same as points in the orbit of a fixed point by part (b).

The inductive step is essentially the same: suppose 103 is empty for all
[3 < jg, and let p E 13-”, with p = Aj’(q2) = Aj1(q1) for q, E c,,, (12 E c,,
and p E c,,,. It may be the case that some image A“(p) E c,, for some i3,

but since if this were true then A“(p) E Ik,),+J-,, we must have k 2 jg — jl

48

by the inductive hypothesis. Cases (1) and (2) above for the choice of the

neighborhood U of p now read

1. If A35”.l (p) E 0,3 for some i3 then the configuration of curves Aj’(c,-,),
A"1 (Ci: ), 6:0, and A"1 ‘j’ (c,,) appear in U as four arcs intersecting trans-
versely at the single point p:

, , A"’(c.-.)
A’"" (ea) c,,,

A” (6..)

 

 

2. If A013 (p) E Uc, then the curves Aj’(c,,), A].1 (c,,), and c,,, appear in

U as three arcs intersecting transversely at p, as above.

Again we perturb A to A’ in a neighborhood of q, to obtain a similar

perturbed picture as above:

 

Ci: Ah _j1(ci2)
c,,
(12 <11
A-j2(U) A-jl (U)

49

In the worst cases, then, we see that we will have A"(p) E I,,,+j, for
some 3 2 jg — jl + 1, or A"(p’) E I,+,,,,+j, for some r 2 jg — jl + 1, or
A"(p”) E I,,,+J-, for some t 2 jg -j1 + 1. Finally, in case (1), we may also
have a point p’” = A’j’(c,,) fl c,3 with A”(p’”) E 1“,”. All four of these
possibilities are addressed later in the inductive process, so again we may
empty I j, 3, without disturbing the status of Iafi previously emptied, nor the
work of part (b) and (c). D

Finally, we must show that we can perturb A to a generic flow and obtain
the same invariant I. It is apparent that the reason A is non-generic is that
it connects the upward flows from the index-2 critical points, A(y), to the
downward flows from the index-1 critical points ’D(x). To make it generic,
therefore, we need only perturb the flow in a neighborhood of 20 = 21, in
such a way that these two types of flow miss one another. In fact, we may
choose to modify the flow in a small neighborhood of the N points in which
A(y) intersect 20. Such a perturbation, however, cannot affect either the
number of fixed points of A" for k g n, or the number of intersections between
A“(c,) and cj—simply because we can take the support of the perturbation
to be away from any of the corresponding flow lines of A, there being finitely

many such flows. Therefore, [p(Ig)],,, which is calculated as a combination

50

of these two quantities, is unchanged by this perturbation. This completes

the last step in the proof of (13).

51

APPENDIX

A Appendix: Technical Results

It is the purpose of this final section to provide some justification for some
of the results used in previous parts of the work. In particular, the results of
this section should suffice to prove both the fact that the Seiberg—Witten map
kw satisfies the composition law (5) for topological quantum field theories,
and Theorem 2.2 showing the trace of kw gives the Seiberg-Witten invariant
for the closed manifold X. We do not, however, provide complete proofs of
these facts here.

To begin with, we review the Seiberg—Witten equations on an oriented
Riemannian 3-manifold X. We consider the Seiberg-Witten equations per-

turbed by a 2-form 17 E 92(R x 2):

DA‘II=0

*(FA—i-in) = ir(\II,\II) (14)

where the notation is the same as in Section 2. For convenience, let us fix a

base connection A0 on the line bundle L. We can think of the solutions of

(14) as 6‘1(0,0) where 6 is defined by

s:r(5)sol(x;in) —+ r(3)onl(x;z‘1a)

(\II, a) +——> (Donli + a.\II, *(FAO + da +i17) — ir(\Il,\Il))

53

By way of notation, we will sometimes write C = C (X) = 115') ED 91(X; ilR)
for the space of Seiberg-Witten configurations on X, and C“ = C1X) for
those configurations (\II,a) with ‘1! not identically zero—that is, C“ is the
space of irreducible configurations. The linearization L(‘P.a) of G at a point

(\II, a) is given by

L(\I’.a) : [(3) EB 91(X;ilR) —> F(S) EB 91(X; ilR)

L(q,,a)(¢, b) = (DAOQD + a.¢ + 031’, *6”) — 2iRe T(\I’, 05))

where Re r(\IJ, ()5) denotes the element -;-(r(\II, qt) +r(¢, \II)) of 91(X)CQ&(X).
It can be shown that for generic metric and n the moduli space of solutions
modulo gauge is smoothly cut out by the equations—that is, the linearization
is always onto.

When we consider the Seiberg—Witten equations on a cylinder Rx 2 and in
appropriate gauge, the above linearization becomes a 0-th order perturbation
of an operator of Atiyah-Patodi-Singer (APS) type. Recall that a linear first
order elliptic differential operator D : I‘(X; E) ——> 11X; E) for X = R x 2 is
said to be of APS type if it can be written as % + A for a self-adjoint elliptic

operator A : I‘(2; E) —> 112; E). We now recall the theory we will need.

54

A.1 Cylinders and Cylindrical Ends: Linear Theory

To begin with, we work on R x 2 with a product metric. We are in the
situation of the previous work, i.e., we have a hermitian bundle E0 over 2
whose pullback over R x 2 will be denoted by E, and an operator D on
sections of E over IR x 2 that is of APS type. Assume E is also endowed
with a hermitian connection V. Fix once and for all a 0“0 function r(t, x)
that agrees with It] for It] > 1, and is within some small distance of [t] in
the 0°-norm on [—1, 1] x 2. Since A is elliptic and self-adjoint, its spectrum
is discrete. Letting {A} denote the set of eigenvalues of A, there exists a
number A0 = min,\¢o(|/\|). Fix a real number 0 < 6 < 2A0. For 0 _<_ k E Z,

define the weighted Lz-norm || - II)”; on sections of E by

I:
”out. = f 2 ( Giver) ei'deozexe.
x =

We then let H 3° denote the closure of the space of smooth, compactly sup-
ported sections of E under this norm.
We say a section a of E is an extended H f section if it is locally in H j“,

and if for |t| sufficiently large a can be written in the form
o(t, x) = 00(t, x) + 7r"&(t, x)

where 0’0 E H}, 7r : R x 2 —> 2 is the projection, and {7 E ker(A). We denote

55

the space of extended Hf sections by H fie. The following result is due to

Atiyah—Patodi-Singer [1].

Lemma A.1 1. The operator D above induces a Fiedholm map ng ->
HP.

2. D admits a right inverse P : Hf-l —-> Hie.

Sketch of Proof: We begin by applying the spectral theorem to A to
obtain a complete orthonormal basis of eigenfunctions $3 E I‘(2; E0). Then
any section a of E can be written as o(t, x) = Z; og(t)¢,\(x) and the operator
D= g,-+Aappearsas

WEE-0e-

A

Following [1], we solve the equation D0 = p by the explicit formulas

t
/ e’\("’)p,\(s)ds (A Z 0)
PAPAU) =-‘ _°°

—/ e“: t) p(s)ds (A<0)
t
Equivalently, if we write 6(t) == x[0,oo)(t) for the characteristic function of the

right half-line, and set

56

we define ng), = f), at p,\ where * denotes convolution:

(9 * ’0“) = /‘°° g(t — s)h(s)ds.

—oo

To invert D, define Pp = 2(ngg)¢g. One easily checks that o = Pp
A

formally solves Do = p; it remains to determine the mapping properties of

P. Now for A > 0,

I3... = f If err-"e0 — enemas

where r(t) z |t|. Ignoring a small error factor, we replace r(t) by |t|, and

2
e“T(’)dt,

 

”PAPA

 

2

write
/ egt+’\“A’6(t—s)pg(s)ds dt

“smut. = f
0 —oo
0 oo 6
+/ / €—§t+M—M6(t-S)p,\(8)d8
—m —m
= f / eIg—mt—‘ldt—s)e%‘p,\(s)ds
0 -—oo

0 co 6 6
+/ / e'(5+)‘)(”‘)6(t — s)e"5‘pg(s)ds

2
e(%"\)‘6(t) at eitp)((t)l[L2 + [[e_(%+)‘)’6(t) * e‘itpg(t)

 

 

2

dt

2
dt

 

 

 

 

2
dt

 

 

2
S ]

 

 

 

L2

Applying Young’s inequality || f * g“ [,2 3 N f || [,1 || 9]] L2, we see that the above

is no more than

(ngpllpxllgs + Wilmllis

57

which finally gives the bound

“PAPA“(2),6 S fifllmllgs (15)

for A > 0. An entirely analogous calculation gives an identical bound for

||PApA||0,5 with A < O. For the case A = O, we have

P,\p,\(t) = foo 6(t — s)p;\(s)ds.

Since p; E H g , this integral has a well-defined limiting value k as t —> 00. In
fact, if we choose a smooth cutoff function 0 g V(t) S l with V(t) = 0 for

t < —T << 0 and U(t) = 1 for t > T >> 0, the section

PAPA¢A = (PAPA — Vk)¢A + Vk¢A

is in H 2‘. Moreover, another calculation of the type made above leads to
estimates for the “non-extended part” of P,\p,\ near the ends of the cylinder,

of the form
”PAPA — Vklng([T,oo)) S ill/Allan (16)

and similar on (—00, —T].

Remark A.2 It is at this point that the use of the weighted Sobolev norm

|| - ”N is required: without the exponential decay guaranteed by the condition

58

p; 6 H2, we do not obtain a bound even on the non-extended part of PAp),

for A = 0, as is apparent by letting 6 —> 0 in the above estimate.

Our object is to see that P maps H 2 —-) H i”:- First, we note that since D
is a first order elliptic operator, we may define the H} norm using D rather

than V: explicitly,

llfl|1,6- _” ”fllo,6 + “a— dtf||0,6 + ”AfHOp6

Then, for a section p = 2“,, pm); that does not involve the O-eigenvectors,

we have

“Ppuia = “Emma,”Ilzgzamnaw”Ema/wont
< _2__

- (g—W

+ Z Azllpxpxllgs

1+A2
E 2Z(——%_ _le 2HPAHO,6

S C(6v A(1)2(1 + XENIpAllgfii

llpxllgs + Z (ll/Allis + A2||PAPAI|3,5)

for a constant C(6, A0) depending only on 6 and the number A0, being the
minimum of |A| for nonzero eigenvalues A. The sum converges since p E H g
and the eigenvalues A of a self-adjoint elliptic operator A may not accumulate

at 0.

59

One makes a similar calculation using the estimate (16) to bound the H g
norm of (PApA —- Vk)¢,\ With A = 0 on the ends of the cylinder, and we infer
that

P:H§—+H§,e

as required. Given the estimates above, we can use the same argument as in
[1] to see that P extends to a continuous map H f‘1 —> H is for general It. The
existence of the right inverse P and the observation that ker(D) '-‘_-’ ker(A) is
finite-dimensional completes the proof of the lemma. C]

We turn now to the case of a cylindrical-end manifold. Explicitly, let W
be a compact Riemannian manifold with boundary 8W = 2, and complete
W by adding a cylinder [0, 00) x 2 to the boundary. We denote the completed
space by W. For convenience, we will assume that 2 has a collar neighbor-
hood [—1,0] x E C W, although this restriction is not really necessary (see
[1]). We suppose that there is a bundle E over W restricting to E0 —> 2
over 8W = E, and extend E over W in the natural manner. We consider
an operator D on sections of E that is of APS type over [—1,oo) x 2. The
function r(t) from above is defined on [0, 00) x 2 and we may assume that
defining r(w) = 0 for w E W extends 7' smoothly to a function on W, which

we also denote by 7'. Using this function, we may define the H g norm on W

60

just as we did for the cylinder and again consider the space H £6 of extended

H g sections of E.
Lemma A.3 0n W, D induces a Fredholm operator H 3‘ —> H 2:1.

Sketch of Proof We must produce a parametrix for D on W; that is an
operator P so that DP and PD differ from the identity by a compact op-
erator. The argument is a fairly standard “patching” technique, as follows.
First, note that DIW extends naturally to an operator D1 on the double
W = W U; W. Since D1 is an elliptic operator on a closed manifold, it
admits a parametrix P1. Let us write D2 for the operator D on R x )3; it was
proved above that D2 has a parametrix (indeed, a right inverse) P2. Now
choose a partition of unity {(351, 452} subordinate to the covering {U1, U2} of
W given by U1 2 W, U2 = [—1,00) x E. For notational convenience, we

choose 451, (b2 in such a way that (I)? + 45% E 1. Now define

P = ¢1P1¢1 + ¢2P2¢m

thinking of ¢,- as multiplication operators and omitting the obvious restric-
tions, extensions, and identifications necessary for this statement to make
sense—e.g., for a section a on W, (p10 vanishes on [0, 00) x E and therefore

may be thought of as a section on the double W by identifying W with one

61

side of W and extending (1510 by zero to all of W. Likewise, we can “transfer”
$1 to W by extension by O, and then for a section p over W, (1)1 p vanishes on
one half of W and thus can be “transferred” back to W in the obvious way.

Now we calculate, writing an : 91(W) —> End(E) for the symbol of D:

DP = D(¢1P1¢1+¢2P2¢2)
= 6D(d¢1)PI¢1 + ¢ID(P1¢1)

+ 00(d¢2)P2¢2 + ¢2D(P2¢2)

Since D agrees with D1 on W = supp(¢1), and with D2 on [—1,oo) x E =
supp(¢2), the second term becomes ¢1(II + K )¢1 where K is a compact op-

erator, and the fourth term is just 63. Thus
DP = ll + O’D(d¢1)P1¢1 + O'D(d¢2)P2¢2 + ¢1(K¢1). (17)

We must check that the terms following ll form a compact operator. Now,

we may assume that |d¢1| g 1, so we calculate for a section p E H ,9 over W,

HUD(d¢1)P1¢1PHLg(W) S CllP1(¢1P)lng(W)
S CH¢1PHL2(W)
= CH¢1PHHg(W)

S CllpllH§(W),

62

for universal constants c. Here we have used that P1 : L2(W) —> L'flW)
is continuous. Therefore, the map p v—-> aD(d¢1)P1¢1p is a bounded map
H g —> H g, and hence induces a compact operator H g —) H g. The remaining
terms in (17) are dealt with in a similar manner, and an analogous calculation
also shows that PD is the identity modulo compact operators. This gives
the lemma. CI

Finally, we consider the case of two cylindrical end manifolds, and the
linear gluing problem associated to our elliptic operators. Let W1, W2 be

compact Riemannian manifolds with 6W1 = 0W2 = 2, and form

W, = W1 U ([0,00) x 2)

W, = ((—oo,0] x :3) u W2.
Assume we have bundles E1 and E2 over W1 and W2 respectively, that each
restrict to E0 over 2 and extend in the usual way to bundles over W1 and
W2, also denoted E,, i = 1, 2. Let D,, i = 1, 2, be operators on sections of E,
that each restrict to the ends of VI", to 2% + A for fixed A as above. For T a
nonnegative real number, we can form WT = W1 U [—T, T] x E U W2. Then
WT has a natural bundle ET that is isomorphic to E, over W, and to 7r‘Eo
over [—T, T] x 23 (recall 7r : [—T, T] x 2 —> 2 is the projection). Likewise, we

obtain an operator DT on sections of ET. To fit into our previous weighted

63

Sobolev picture, we define the function TT on WT using the functions r, on
the cylindrical-end manifolds W, as follows:

rT(w,) = r,(w,) = O for w, E W,

rT(t,:c) = r1(t + T, 2:) for t E [—T,O], a: E E

TT(t,:c) = 72(t — T, at) for t e [0,T], a: E 2
In fact we take 77» to be a C°° function that is C’O-close to the one described
above, which is only continuous on {O} x 2. Thus rT E 0 on W,, rT(t, as) 2
T +t for t E [-T,O], and rT(t,:c) z T — t for t E [0, T]. Using rT we can

define the H 3’ norm || - “M on WT, by the formula

1:
Hana = f 2 Iver e5"dvol.
WT i=0

We define the space of extended H g sections, H g“, as follows. Fix a cutoff
function V1 on W,, where u = 0 on W1\([—1,0] x 2) and V1 = 1 on [0, 00) x 23.
For any element 1b 6 ker(A), there is a natural extension of mr‘ib to W1 which
will also be written mfg/A). A section 4) of E1 is said to be an extended Hf

section, «[2 E H 63¢: if w can be written in the form
«r = w' + urn/3 (18)

for some 1/3 E ker(A), and some w’ E HflWl). Note that we have a natural

map H g, —) ker(A) by w »—> ab. A similar definition using a cutoff function

64

V2 holds for ng(W2). We define the HgJWl) norm of w E HgJWl) by

III/Jill»,e = lid/Ilia + llwlliz 2»
.( )

and similarly for W2.

We are interested in “gluing” H 51,8 solutions of D,1/J, = 0 over W, that have
the same limiting value 2b,, := 1b, = 2b; to form solutions of DTwT = 0 over WT
for sufficiently large T. By way of notation, we will write V for the subspace
of ker(Dl) EB ker(Dg) consisting of pairs of sections having the same limiting

value. Explicitly, V is the kernel of the map ker(Dl) EB ker(Dg) —> ker(A)
given by ($1, z/J2) H 1131 — 1/32-

Lemma AA 1. If coker(D,) = coker(Dg) = 0, then there exists a right
inverse PT for DT for all sufficiently large T. Furthermore, the operator
norm ||PT|| defined using the H 3 and H}; norms is bounded independent of

T.

2. In the situation of part (1) above, there exists an isomorphism

ker(DT) g V C ker(Dl) $ ker(Dg).

Proof Under the assumption coker(D,) = O, the paramatrices P, constructed

in Lemma A.3 may be taken to be right inverses. We apply another patching

65

argument. Let (f), be a cutoff function on WT, with gt, E 1 on W1, 65, E 0

on W2. We may assume that the derivative dd) is bounded in norm by a

constant times %. Likewise, let (p, be a cutoff function with 652 E 1 on W;

and (p2 E O on W1; we assume that d)? + 63 = 1 on WT.

Define a parametrix

QT = (151131451 + 452132452-

The same calculation as in Lemma A.3 shows that

DTQT = H + 001(d¢1)P1¢1 + OD: (d¢2)P2¢21

where there is no additional term involving a compact operator K since we

have assumed that P, is a right inverse of D,. Then we calculate

“0(d¢1)P1¢1¢”H2(Wr) = l|0(d¢l)Pl¢l¢HH?(W‘)
g |]d¢1IILwIIP1¢1¢|ng(W1)
g 0%IIPIH]]¢1¢”H2(W1)
g 0%IIP1IIII2/Jllugm)’
and obtain a similar bound on the third term in the exPI'eSSion for D TQT'

This tells us that ||lI —- DTQTII S 0% for a constant C independent of T,

and we recall that by a well-known geometric series argument whenever an

66

operator is within 1 of the identity in the operator norm it is invertible. Thus

for sufficiently large T, DTQT is invertible, and setting

Pr = QT(DTQT)—l

we obtain the desired right inverse of DT. To bound the norm of PT, we note
that from the formula (II — F)‘1 = 1 + F + F2 + ---, we obtain the bound

”(11— F)"|l s ,—_,‘,—F”. Taking F = II — DTQT, this gives
1

l
< —)
HII - DTQT“ — 1 -%

 

“(DTQT)_1” S 1 _

which is bounded as T ——> 00. It is easy to see that ||QT|| 5 ”Pl” + ||P2||,
and these bounds on QT and (DTQT)’l lead to a bound for PT that is
independent of T. This proves part 1 of the lemma.

For part 2, we construct a map f : ker(DT) —-> V as follows. For a section
¢T E ker(DT), we have an expansion ib|[—T, T] x 2 = 2px(t)¢x in terms
of eigenfunctions ¢A for A. In fact, since wTH—T, T] x E E ker(g; + A),
we have that p),(t) = p,\(—T)e"‘(‘+Tl. Now, the restriction ¢T|W1 solves
DT’l/JT = Dle = 0 over W1 since DT and D1 agree on W1. Furthermore,

lew, has a unique extension to W; as an element w, of ker(Dl), namely

¢TIW1 on W]
1/11 = (19)

ZPA(—T)€_M¢A on [0,00) X 2

67

Note that w] E Hg’c(W1), and $1 = VZA=0 p,\(—T)¢A. We make a similar

restriction and extension of wT to W2 to obtain $2 6 H 3,41%), and define

fwT) = ($1,102)-

It is a straightforward matter to check that f (th) is indeed in V.

For convenience below, we extend f to all of H §(WT) as follows: any
section 1,!) has an expansion of the form used above over [—-T, T] x 2. To
define wl, restrict 1,1) over W1, then use formula (19) to extend this section to

W1. A similar construction for W2 gives a map
f I H:(WT) _) ng(W1) e Hzea/V?)
1P H fltb) = (f1(¢),f2(¢)) = ($1,412)

Now define g : V —> ker(DT) as follows. For sections (1/11,¢2) E V we
have decompositons 1/2, = w; + WNW/30 coming from the definition (18), where
1/30 is the common limiting value of w, and 2,02. Using the partition functions

45, from above, we define

9(1/J1,¢2) = H(¢i¢i + 4531/); + Vlefiltllo):

where H : H ¢}(WT) ——> ker(DT) is orthogonal projection.

Claim: The linear maps f and g satisfy

”(H ‘ 9 0 f)¢rll1,6 S gll‘l’rllm

68

Hence, for sufficiently large T, g o f is an isomorphism.

Proof: First we note that by expanding wT in a series 2 pAqu, the 43, being
eigenfunctions for A, and where the expansion is valid over [—T — 1, T+ 1] x 2,
we can write wT = w} + VTr‘zbo where VT is a cutoff function supported in
[—T — 1, T + 1] x E and identically 1 on [—T, T] x 2 and where 1,50 is the part

of “PT involving O-eigenfunctions. Then we can write

is = ¢§<¢a~ + uric) + ¢§<¢a~ + wr‘z/Bo).

Ifinthermore, by definition of f we have that ¢§f(wT) = qbfwT either as
sections over WT or over W,. Observe that from part 1 of the lemma, DT
has no cokernel and therefore the projection II onto its kernel is given by

11 — PTDT. Thus for 1M 6 ker(DT),

“(H — g fW’TIlM = WT — (H — PTDT)(¢if1(1/JT)' + ¢§f2(¢T)' + Vflllfiolllm

 

WT — ( ileT)’ + 4532“”), + ”WV/30)

|/\

I1,
+ IIPTDT(¢¥f1(¢T)'+ ¢§f2(¢2~)’ + mutants

= IIPT(ao.(d¢i)¢’T + ao.(d¢§)¢a~)nl,a

s gnanuwnm.

In the last line above we have used that 2M 6 ker(DT) to obtain a bound

in terms of the 1,6 norm of wT rather than the 0,6 norm. Since ||PT|| is

69

bounded independent of T by part 1, the claim is proved.

Claim: We have

C
“(H — f 0 g)(¢11¢Z)HH61'e(W1)®H61’¢(W3) S Tllhpl)$2)lng.¢(W1)$Hg.e(W2)

The proof is similar to the above calculation; it is easiest in this case to
use the extension f.

We have shown that for sufficiently large T both g o f and f o g are
isomorphisms, from which it follows that each of f and g is an isomorphism.

This completes the proof of Lemma A.4.

A.2 Gluing Seiberg—Witten Solutions

We turn now to the non-linear situation of the Seiberg—Witten equations. As
in the linear setup, we will work with H 6:: configurations. More precisely, we
consider a cylindrical-end 3-manifold W with end isometric to [0, 00) x E for
)3 a genus g Riemann surface, and a line bundle L over W that restricts over
the end to the pullback of some bundle Lo over 2. As in Section 2, the spinor
bundle SW restricts to a bundle over 2 of the form (K 'i' GB K ‘%) 8) E0, with
E2 = L0. We assume once and for all that there are no reducible solutions
to the vortex equations on E, which is equivalent to the assumption that

deg E0 524 1 — g. In the situation that we consider, namely that 2 is a

70

regular level surface of an Sl-valued Morse function, we can always arrange
for this condition to hold by introducing cancelling critical points for the
Morse function and thereby artificially inflating the genus of 2.

As in the introduction to the Appendix, we will fix a background connec-
tion A0 on L in order to identify the space A(L) of connections on L with
01(W; ilR); we will assume that A0 is a product connection on the end of W.
Note that A0, together with the Levi-Civita connection, induces a connec-
tion on the spinor bundle 5]): = (K i 63 K ‘%) <8) E. We will not distinguish
A0 and this induced connection in the notation. As in the previous section,
we define a cutoff function V that is equal to 1 on [0, 00) x 2 and vanishes
away from [—1, 00) x 23 (using the collar neighborhood [—1,0] x 2 C W). A
solution (2,5,6) = ((a,fi), a) E I‘((K% 69 K‘%) ® E) 63 (21(2); ilR) of the vortex
equations

iFAo + id& = %(|fi]2 — ]O]2)d’l)0l2
5,40,.501 = O 5§o+afi = 0 a3 = 0
can be pulled back to W using V, to a configuration we denote (wr'zb, V7r‘6).
An extended H: configuration (‘11, a) on W is then one that can be written

in the form

(\Il,a) = (\II' + un‘zb,a' + mr‘c’i)

71

for (\II', a’) e H §(W) We consider the function G from the beginning of the
Appendix to map H :3 —> H 2’1 for appropriate choice of k.

Let W1 = W, U [0,oo) x 2 and W; = W2 U (—oo,0] x 2 be a pair of
cylindrical-end manifolds with spin‘ structures t,, spinor bundles 3,, and
determinant lines L,. We assume that over the ends of W, the L, are equal
to the pullback of a bundle Lo on 2, i.e., that the spine structures “agree
on the ends.” Let C: denote the space of irreducible H 26 configurations on
W}, and let us write 3, C C: for the space of solutions of the Seiberg—Witten
equations (14) on W, in the spin‘ structures on W,. Let M, = M,(t,) be
the moduli spaces of gauge equivalence classes of solutions, i.e., M, = B,/g,
where g, = Map(W,; $1) is the space of H g“ gauge transformations. Our
object is, given a spinc structure t on WT = W1 U [—T, T] x 2 U W; that has

tlw, = t,|W,, to construct a “gluing map” for large T:
'7 :M1(t1) X0 M202) -> MT(t)

where MT = BT/QT is the moduli space of Seiberg-Witten solutions on WT.
The notation x5 represents a fiber product: M1(t1) x5 M2(t2) stands for
those pairs of gauge equivalence classes [\Ill, a1], [‘Ilg,a2] of solutions on W,
and W2 whose limiting values (1b,,6,) agree. As a first step, we have the

following.

72

Proposition A.5 There exists an “approximate gluing map”
if : 81 X5 82 ——> C}
that has
6(‘7((\Ill,a1),(\112,a2))) = 6T = (chm?) E Hf'1(S EB A1(T‘WT;iR))
where 67 is a configuration approaching 0 as T —> 00.

Proof: We introduce cutoff functions qbl, 452 on W, and W2 as follows. On
W1, take 431 to be supported on W, U ([0, T] x )3), and to be identically 1 on
W, U ([O,T - 1] x 2). Similarly, 452 is supported on W2 U ([—T, O] x 2) and
equal to 1 on W2 U ([—T + 1,0] x 2). Let (\II,,a,), i = 1,2, be extended Hf
solutions over W, both having limiting values (the, 60). That is, on W, we can
write

‘11, = \II: + un‘ibo a, = a; + wr‘ao

for (\I':, a2) 6 H 5‘. Define multiplication operators 95,, i = 1, 2, by setting
$s(‘1’i,az‘) = ((Ei‘l'imgiai) = ($914+ V77.’¢Lo:¢iai + ”7.630)-

That is, 43, is a cutoff to the limiting value.
Clearly we may think of 43,011,, a,) as configurations on WT (after shifting

the origin in the identification of the ends of W, with IRi x 2), so for (\I',, a,) E

73

B, we define

., (131(‘1’1,01) on W, U ([—T,O] x 2)
7((‘111’ a1): (‘1'?) 02)) :
¢2(‘I’2,02) OD W2 U ([0, T] X 2)
By construction, this configuration is in H ¢]‘(WT) and solves the Seiberg—
Witten equations on W, U ([—T, —1] x )3) U ([1, T] x Z) UWz. For convenience,
we will write u, for the solutions (‘11,, a,). To estimate the error term 6T =

6(‘7(u1,u2)), then, we need only consider the region [—1, 1] x E C WT. On

[—1,0] x 2 we have

ll€Tllk-1,6 = “Emu" + ”77350, 0' + V’F‘aolllk—ix
< ||(DA,mr‘z/30 + unnamed, + d¢1.\II’ + 611),, \II’
+ afa'w' + anal/wt, + uvr’doJII')”
+ I] * (FA0 -+- dmr‘ao + in) — r(u7r‘1/30, mr‘zbo)

+ * (dgbl /\ a' + (blda') - 45911631") — 2451116701", 1177350)”

Now, since V E 1 on [T — 1,T] x E C W,, we have (wraiomrao) =
(#160, «‘60) is a constant Seiberg-Witten solution in this region. There-
fore the terms involving only 1230 and 60 drop out of the above expression.
Mthermore, we know that (‘II', a') is an H f configuration and therefore has

exponential decay on the cylinder. Since the remaining terms in the expres-

74

sion above can all be estimated in terms of constants times the norms of \II’
or a’, we infer that IIET]]},_1,5 —> 0 as T ——> 00. This proves the propositionfl

We now must consider the difference between the image of the approx-
imate gluing map 77 and the actual Seiberg-Witten solution space on WT.
First, let us suppose that (O,b) 6 CT is a configuration on WT. Using the
notation L(9,b) for the linearization of 6 at (O, b) a calculation shows that

for any variation (6, b) we have
6(6) + a, b +13) — e(e, b) = L(e,,,(9, 5) + (13.0, —T(o, 0)).

That is, 6 differs from its linearization by a term of the form q((0, b), (0, b)),

where

4((91,bi),(92,b2)) = (b1.02,—T(91,02)).

It is a straightforward matter to verify that q satisfies an inequality of the

form

||(1(u, u) - q(u, v)ll S Cllu - vll(llU|| + ||v||)- (20)

Our object is to use a quadratic contraction mapping principle to produce
an exact solution on WT from the approximate solution 7(u1,u2). First,
note that for generic metric and perturbation n, the conditions of Lemma

A.4 hold for the linearizations of the Seiberg—Witten maps 61, 62 on W,

75

and W2. Therefore, there is a right inverse P for the linearization Lima“),
whose norm is bounded independent of T (see the remark after Theorem A.7,
however). Writing ii = :y(u1, u2) for the approximate solution, we will look

for an exact solution of the form
u = i1 + Pu
for some configuration 2) on WT. This means that u must satisfy
0 = 6(u) = 6(21 + Pu) = 6(i2) + Lz,(,,,,,,,)(Pv) + q(Pv, Pu)

01‘

0 2 6T + v + q(Pv, Pu).

Lemma A.6 If Q is a quadratic operator on a Banach space B, Q : B EBB —->
B, satisfying an inequality of the form (20), then for all sufiiciently small 2

there exists a solution x to the equation

x = Q(x,x) + 2.

Note that since the norm of P is bounded independent of T, the map
Q : v r—> q(Pv, Pu) satisfies the condition of the lemma.

Thus we obtain:

76

Theorem A.7 For all sufi‘iciently large T, there exists a map

7TIM1X6M2—>MT

defined by

7(u1,u2) = Wuhuz) + P”

where P is the right inverse of the linearization L:,(,,,,,,,) and v is the solution

0f

0 = 6(’y(u1,u2)) + v + q(Pv, Pv)

provided by Lemma A.6. Furthermore, 7T is a difi'eomorphism onto its image,

which is an open subset of MT.

Remark A.8 Here we are suppressing the fact that the inverse P of the
linearization exists only “modulo gauge.” To be more honest, we should in-
troduce another component into the map 6 that fixes the gauge, in which
case the linearization will indeed be of APS type and the work of the previ-
ous section applies. This point is not particularly difficult to deal with, but
involves more notation than we care to use here. Note, however, that the
fact that the gauge action is free—i.e., we are dealing only with irreducible

configurations—enters in an essential manner.

77

We will end our discussion of gluing theory here, noting only that the
smoothness of '7 follows essentially from Lemma A.4: that 7y is smooth is
a routine check, and the fact that the passage from ’y(u1,u2) to 7(u1,u2)
is smooth can be seen roughly as follows. The derivative of ’7' provides an
isomorphism with the tangent space of the image of i with ker Lul X3 ker L,,a
(the space V of Lemma A.4), while the tangent space of 7(u1, U2) is identified
with ker L7(,,,,,,,). The map between the two can be seen to be essentially the

linear gluing construction of Lemma A.4.

78

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