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Seiberg-Witten Invariants of 4-Manifolds
With Circle Actions

presented by

Scott Jeremy Baldridge

has been accepted towards fulfillment
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SEIBERG-WITTEN INVARIANTS OF 4-MANIFOLDS
WITH CIRCLE ACTIONS

By

Scott Jeremy Baldridge

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2001

ABSTRACT

SEIBERG-WITTEN INVARIANTS OF 4-MANIFOLDS WITH CIRCLE
ACTIONS

By

Scott Jeremy Baldridge

The main result of this paper is a formula for calculating the Seiberg-Witten invari-
ants of 4—manifolds with fixed—point free circle actions. This is done by showing un-
der suitable conditions a diffeomorphism between the moduli space of the 4—manifold
and the moduli space of the quotient 3-orbifold. Two corollaries include b+>1 4-
manifolds with fixed-point free circle actions are simple type and a new proof that
SWy3x 31 = SWys. Using the formula, we show how to construct a nonsymplectic 4-
manifold with a free circle action whose orbit space fibers over circle. We also describe
a nontrivial 3—manifold which is not the orbit space of any symplectic 4—manifold with
a free circle action. An infinite number of b+:1 4-manifold where the Seiberg—Witten
invariants are still diffeomorphism invariants are constructed and studied. As an ap-
plication of the main results, we derive a formula for the 3-dimensional Seiberg-Witten

invariants of the total space of a circle bundle over a surface.

Copyright © by
Scott Jeremy Baldridge

2001

To my loving wife Lisa

iv

ACKNOWLEDGMENTS

I would like to give my heart-felt appreciation to the following individuals who
helped shape my life both mathematically and personally.

I am deeply grateful to Ronald Fintushel for introducing me to this field and for
his help, advice, and constant encouragement. I hope that someday I can be as good

a thesis advisor to my students as he was to me.

Professors Thomas H. Parker, Jon Wolfson, Selman Akbulut, Nikolai Ivanov, and
Peter Ozsvath deserve my gratitude for significantly contributing to my overall grad-
uate experience. In particular, I appreciate the time and effort Professors Parker and
Ozsvath took to discuss many of the salient issues related to my thesis.

I am in humble awe of my wife, parents, and brothers. You were always there for

me when I needed you most.

Professor Wei-Eihn Kuan deserves a special note of appreciation. Since my un-
dergraduate degree had little to do with pure mathematics, I will always be grateful
to him for giving me a chance to succeed.

TABLE OF CONTENTS

Introduction 1

Seiberg-Witten on 3-orbifolds

2.1 Definitions ................................. 6
2.2 Orbifold line bundles ........................... 8
2.3 Spine Structures on 3-orbifolds ...................... 12
2.4 Seiberg-Witten Equations on 3-orbifolds ................ 16
4-Manifolds with fixed point free circle actions 18
3.1 Homology ................................. 18
3.2 Line bundles over X ........................... 20
3.3 Seiberg-Witten Equations of Smooth 4-manifolds ........... 21
SpinC structures and SW solutions 25
4.1 Restrictions on Spine structures ..................... 25
4.2 Solutions to the SW equations ...................... 28
Diffeomorphic moduli spaces 34
5.1 7r* is injective ............................... 36
5.2 7r* is surjective .............................. 37
5.3 The kernels are isomorphic ........................ 38
Results 48
An Alternate Proof 51
7.1 Classifying free circle actions ....................... 51
7.2 Gluing along T3 .............................. 52
7.3 Spine structures which are not pullbacks ................ 55
7.4 Identifying the set VYX81(€|X’) ...................... 56

vi

8 Examples

8.1 A construction and a calculation .....................

8.2 Example 1: Non symplectic X 4 whose quotient fibers over 81 .....

8.3 Example 2: Y3 which is not a quotient of a symplectic X 4 ......

8.4 Example 3: b+ = 1 diffeomorphism invariants

8.5 A Application: A Formula for circle bundles over surfaces ........

9 Final remarks

BIBLIOGRAPHY

vii

58
59
60
62
63
68

70

73

CHAPTER 1

Introduction

The main idea of this work is to systematically study 4-manifolds that admit an
S 1—action and clasSify them using Seiberg-Witten gauge theory. When the action on
X 4 is free, the quotient by the S 1-action is a smooth 3-manifold Y and the manifold
with given circle action is classified by the Euler class X E H 2(Y; Z). When the circle
action is not free there will be non-trivial isotropy groups, which forces the orbit space
to be an orbifold rather than a manifold. The main result of this paper is a formula
for calculating the Seiberg-Witten invariants of any 4-manifold with a fixed point free
circle action. We derive the formula by proving the existence of a diffeomorphism
between the moduli space of the 4-manifold with the moduli space of the quotient
3-orbifold.

A given manifold may admit more than one circle action. So while the 3—manifold
and Euler class are fully sufficient to classify a free circle action, the Seiberg—Witten
invariants are stronger in that they are invariant of the underlying space up to dif—
feomorphism regardless of the circle action.

The first theorem we prove puts a restriction on the set of SpinC structures with
nontrivial Seiberg-Witten invariants for manifolds which admit a fixed point free circle
action. (See Chapters 2 and 3 for descriptions of Spine structures and Seiberg-Witten

invariants.)

Theorem A. Let g be a SpinC structure on b+3£1 4-manifold X with a fixed point
free circle action such that SWX(§) 7E 0. Then the SpinC structure 6 is pulled back

from a SpinC structure on Y.

See section 4.1 for the statement when b+ (X) Z 1. This theorem is already enough
to imply that X is SW simple type — that the expected dimension of the moduli space
for all SpinC structures with nontrivial invariants is zero.

Let 7r : X —> Y be the projection map from a smooth 4-manifold with a fixed
point free circle action to its quotient orbifold. The manifold X can be thought of
as a orbifold circle bundle over Y. If in is the connection 1-form of the circle bundle
and gy is any orbifold metric, we can form the metric gx = 77 ® 17 + 7r*(gy) on X.
After perturbing the Seiberg—Witten equations on Y by a closed orbifold 2-form 6
and on X by its self-dual pullback 7r*(6)+ = -;—(1 + *)7r*(6), there is a moduli space of
irreducible solutions to the Seiberg—Witten equations M*(X, gX, 7r*(6)+) associated
with X and M*(Y, 9y, 6) associated with Y (see sections 2.4 and 3.3 for definitions).
Let N*(X,gX,7r*(6)+) be the subcomponent of M*(X,gx,7r*(6)+) which are the
SpinC structures that are pulled back from Y. Theorem A tells us that these are the
only Spine structures that are useful to study. We can now state the main theorem

of this paper:

Theorem B. The pullback map 7r* induces a homeomorphism

71'* : M*(Y7 91/36) —> N*(X79X77r*(6)+)'

Furthermore, if either of the two moduli spaces is a smooth manifold, then both of

them are smooth, and 7r* is a difj‘eomorphism.

The approach to the proof of Theorem B was inspired by similar work done in

[MOY].

As in the free case, a manifold with a fixed point free S 1-action can still be
considered as a unit circle bundle, but now it is a unit circle bundle of an orbifold line
bundle over a 3-orbifold. In this setup, H 2(Y; Z) is replaced by a group called Pic‘(Y)
which records local data around the singular set (see section 2.2). Our main results

express the Seiberg—Witten invariants of X in terms of the Seiberg-Witten invariants

of the orbifold Y and the orbifold Euler class x:

Theorem C. Let X be a closed smooth 4-manifold with b+>1 and a fired point free
circle action. Let Y3 be the orbifold quotient space and suppose that X E Pict(Y) is
the orbifold Euler class of the circle action. Iffi is a SpinC structure over X with

SW§(€) aé 0, then 6 = 7r*(fo) for some SpinC structure on Y and

SWfilt) = Z swim,
€’E€o mod X
where 6’ — {0 is a well-defined element of Pict(Y). When b+=1, the formula holds for

all SpinC structures pulled back from Y.

This results produces two immediate corollaries. One is a corresponding formula
for manifolds with free circle actions. This corollary is useful for calculating exam-
ples. The second corollary is a proof of the well known fact that the Seiberg—Witten
invariants of Y3 x S1 are the same as the Seiberg-Witten invariants of Y3.

Theorem C, together with the conjectured formula when X contains fixed points
(see chapter 9), would completely calculate the Seiberg-Witten invariants for all
b+ > 1 4-manifolds with circle actions. These calculations are useful beyond just
distinguishing manifolds. When Seiberg—Witten invariants are combined with C.
Taubes’s results on symplectic manifolds (of. [T]), the formulas become an easy and
powerful way of calculating an obstruction for an S 1-manifold to admit a symplectic

structure.

As an example of the main result, we produce a nonsymplectic 4-manifold with
a free circle action whose orbit space is a 3-manifold which fibers over S 1. This
example runs counter to intuition since there is a well-known conjecture/ question of
Taubes that M 3 x S1 admits a symplectic structure if and only if M 3 fibers over S 1.
Furthermore, there is evidence [F GM] which suggests that many such 4-manifolds are,
in fact, symplectic. As another example of our formula, we construct a 3-manifold
which is not the orbit space of any symplectic 4—manifold with a free circle action.

Theorem B can also be used to study moduli spaces in the case when b+(X) : 1.
Normally when b+(X) : 1, the Seiberg—Witten invariant depends on the “chamber”
of the metric used to calculate it. A theorem of T. J. Li and A. Liu [LL] shows
how the numerical invariant changes when the metric moves from one chamber into
another. Under certain conditions, their theorem says. that the invariant does not
change (making it a diffeomorphism invariant again). We show how to construct an
infinite number of b+ = 1 manifolds with this property and study their moduli spaces
using Theorem B. This theorem provides a way to see explicitly why the invariants
do not change when a chamber wall is crossed.

Another application of the Theorem C is a formula for the Seiberg-Witten invari—
ant of the total space of a circle bundle over a surface. This formula can be thought

of as the 3 dimensional analog of the 4 dimensional formula.

This dissertation is organized as follows.

a In Chapter 2 we show how to define the Seiberg-Witten equations and invariant

on a 3-orbifold.

0 Chapter 3 shows the relationship between 4-manifolds with fixed point free

circle actions and orbifold line bundles over a 3-orbifold.

c We prove Theorem A in Chapter 4 by showing a relationship between solutions

and topology.

 

In Chapter 5 we prove Theorem B by showing that solutions to the Seiberg-

Witten equations on the 4-manifold are circle invariant.
Chapter 6 contains a proof of Theorem C and its corollaries.

Chapter 7 is an alternate proof of Theorem C in the case where the 4-manifold

admits a free circle action.

Chapter 8 describes some examples and applications of both Theorem B and

Theorem C.

In the last chapter, Chapter '9, we conjecture what the Seiberg-Witten invari-

ants are for a 4-manifold which admits a circle action that has fixed points.

CHAPTER 2

Seiberg—Witten on 3-orbifolds

We show that all of the usual notions of gauge theory hold for 3-dimensional real
orbifolds. Throughout, we assume that all orbifolds are oriented, connected, and

closed unless otherwise specified. We' start with the definition of orbifolds (of. [8]).

2. 1 Definitions

An n-dimensional orbifold Y is a Hausdorff space |Y| together with a system E3 =

({U,}, {90,-}, {(7,}, {0,}, {gb,;,-}) which satisfies
1. {U,-} is locally finite.
2. {U,-} is closed under finite intersections.

3. For each U,, there exist a finite group G,- acting smoothly and effectively on a

connected open subset U, of R" and a homeomorphism (p,- : (Ii/G,- —+ U,.

4. If U, C U], there exist a monomorphism fij : G,- ——> G, and a smooth embedding

95,-, : U,- —> (7,- such that for all g E 0,, a: 6 (7,, chm-(9 - as) = f,j(g) - (pm-(as) making

the following diagram commute:

 

 

U, W” U,
~ 1 (Pi; ~ i
Uz/Gz —‘* Uj/G]
901' 1901'
Ui j

where cpij are induced by the monomorphisms and the r,’s are the natural

projections.

The system E is called an atlas and each tp, o r,- : U,- —> U,- is called a local chart.
An orbifold Y is connected and closed if the underlying space |Y| is. Two atlases give
the same orbifold structure if there is a common refinement.

Let a: E |Y| and U, —> U be a local chart containing 51:. The local group at :13,
denoted G3,, is the isotropy group of G of any point in U corresponding to :1: (well-
defined up to isomorphism). Set EY 2 {cc 6 |Y| | Gm yé 1}. This set is closed and
nowhere dense, and in fact it is easily shown that dim BY 3 n — 2. After removing
the singular set, Y \ EY becomes a manifold.

All theorems henceforth will be stated and proved for 3-dimensional orbifolds Y
where ZY is a finite disjoint set of smooth circles l1, . . . , ln that are assigned integral
multiplicities a1, . . . , an given by their local isotropy group Z0, 2 Z/a,Z. Let D be
the standard complex disk and consider a Z0, action on it by rotation. We will take
a convenient atlas in all of the atlases which give the same orbifold structure. Equip

|Y| with an atlas of coordinate charts

¢iS(SIXD,SlX0)—>(Ui,li) i=1,...,n

<15,,:D;°‘,—>U;r xEY\{l1,...,ln},

where the (b,- induce homeomorphisms from (S 1 x D/Zm, S1 x 0) to (U,, l,), the $2: are
homeomorphisms, the U,- are all pairwise disjoint, U3, (1 BY : Q), and the transition

functions are all diffeomorphisms.

Example 2.1 The triple Y 2 (S3, K, n) where K is a knot in 53, K is the singular

locus EY : K, and the isotropy group around K is Zn, is an example of a 3-orbifold.

Define an n-dimensional orbifold bundle over Y in the following manner. Set
Um x V" over each U1. for an n—dimensional vector space V". Over U,- the vector bundle
is given by the quotient (S 1 x D x V")/Za, where (S 1 x D x V") is a Zai-equivariant
vector bundle specified up to isometry by giving a representation 0,- : Z0, —> GLn(V).
The vector bundle over Y is then Specified by a 1-cocycle of transition functions over

the overlaps.

2.2 Orbifold line bundles

Under tensor product the topological isomorphism classes of orbifold line bundles
form a group Pict(Y) called the topological Picard group. We describe this group in
this section.

We can record the information in Pict(Y) by using a generalization of equivariant
cohomology. Think of Y as the union of Y\ {l1, . . . , Zn} and H (l,- x D / Zai). Define YV
to be the union of Y\ {l1, . . . , ln} and H (l, x (D Xzai EZa,)) glued using sections of

l,- x (D \ {0} XZa, EZai) —> U,- \ l,. These sections are unique up to homotopy because
the fibers of the bundle are contractible.

The following theorem is contained in [FuS].

Theorem 2.2 The following groups are isomorphic:
1. H1(YV;Z) "é H1(|Y|;Z),

2. H2(YV;Z) g Pict(Y).

Remark 2.3 In the literature, the group Hf/(Y) :2 H*(YV) is often called the V-

cohomology ring of Y.

Here is another way to describe Pict(Y). Define an orbifold line bundle over Y to
be a trivial line bundle A : (Y\l,-) x (C and over U,- it is given by B : (S1 x D XC€)/Za,

where a E Za, acts using the standard representation

27ria 21ria

a-(y,w,z) +—> (meat w, e a,- z).

The bundle is glued together using a transition function goBA(y, w) = w on the overlap
S1 x (D \ {0}). For each l,, create such a line bundle called E,.

Let L be an orbifold line bundle over Y. There is a collection of integers 61, . . . 6,,
satisfying

OSfii<C¥t

SUCh that the bundle L <8> Effil <8) - - ~ (8) Egfi" is a trivial orbifold line bundle over each
neighborhood of the li’s. By forgetting the orbifold structure, it can be naturally

identified with a smooth line bundle (denoted by |L|) over the smooth manifold |Y|.

Theorem 2.4 The isomorphism classes of orbifold line bundles on Y with specified
isotropy representations {1%, . . . ,53" along l1, . . . , ln respectively are in bijectiue corre-

spondence with X E H2(|Y|; Z).

 

The proof below generalizes [F2] to the case of an arbitrary orbifold line bundle.

Proof: Given L E Pict(Y), we construct L<§§JE1—fil ®~ - -®E;Bn and its desingu-
larization |L| explicitly. Let 7r : X -—> Y be the unit circle bundle of L. Set Q = U U,-
in Y and P = 7r‘1(Q) with P,- = 7r‘1(U,-). Then X’ = X \ P is a principal Sl-bundle
over Y’ = Y \ Q.

In general, the unit circle bundle X is an orbifold rather than a manifold. P,- is a
quotient of l,- X D2 x S1 by the action of Za defined by f : (y, w, t) i—> (y, éw, fflt). It
follows that the isotropy group of a point in the quotient of l,- X {0} x S1 is {6 E Z, :
£5 2 1} for all points p E l. When the isotropy group is trivial (gcd(a, 6) = 1) the
quotient is smooth. In the case that 6 E 0 moda, L is a usual line bundle around that
loop, but the 4-manifold still has a nontrivial orbifold structure. Set at = gcd(a,-, 6,).

Let m,- = 8({0} x D x {1}) be the meridian loop of l,- before the quotient is
taken. Denote the class it represents in the quotient by 771,-. The homeomorphism
go,- : 8P,- —> BX’ determines a section 3 : BY’ —> 8X0 which is specified up to homology

by the relation:
saint] = (%) 84mg] + (%) lf’l.

where m;- is the meridian of l,- in Y’, f’ is a fiber of BX’, and 0 S 6,- S 0,. The local
invariants (cm, 6,) specify P,- up to orientation-preserving equivariant homeomorphism.

The bundle X’ can be extended to the unit circle bundle of [L] by equivariantly
attaching l,- X D x S1 with a bundle isomorphism (bi. Bundle isomorphisms covering the
identity are classified up to vertical equivariant isotOpy by homotopy classes of maps
in [8(S1 x D), S 1] = Z 69 Z. However we can change 45,- by a bundle automorphism
classified by [S1 x D,Sl] = H1(S1 x D;Z) = Z; these maps change (¢,)*([li]) by

a multiple of the fiber. Therefore the resulting bundle X’ U¢i (l,- x D x S 1) can be

10

 

completely specified by the map
(olez-l = 8*lm'l + 7“[f'l

for some r E Z. Thus we determine the principal S1 bundle of |L| by specifying that
r = 0.

In summary:

1. The unit circle bundle of L is obtained by gluing the quotient P using maps

s 012' , 52' I
(rdlmll — (g) simi + (d) [f 1.
Note that this bundle depends only on the section s,[m;] as well because bundle

automorphisms of P,- correspond to

[U,-, 51] = H1(l,- x (D xza, Ezag; Z) : H1(l,- x BZa,; Z) = z.

2. The unit circle bundle of L (8 Effil ® - - ' (8) Egg" is obtained by gluing in the

quotient H,- l, x (D/Za,) x S1 into X’ using maps (¢,-),[m,-] : 3*[mg].

3. The unit circle bundle of the desingularization |L| is obtained by gluing in

11,- l, x D x S1 using maps (¢,)*[m,] : s...[m;].

Next we show that two orbifold line bundles L1 and L2 with the same isotropy
representations and equivalent desingularizations |L1| 2 [L2] are equivalent as orbifold
line bundles.

Construct two principal S 1-bundles X1 and X2 from X’ to form unit circle bun-
dles |L1] and [L2]. The construction depends on choices of the class 2;; sj[m;] E

H1(8X’;Z) coming from sections 8, : BY’ —> BX;- forj = 1,2. Let 0,- E H2(Y’,8Y’)

11

 

be the obstruction to extending these sections over XJ’. Let r E H1(8Y’) be the

primary difference of 31 and 82. A diagram chase

H1(I_I,-l,- x D; Z) —‘“—> H2(Y, an, x D; Z) i» 1120/; Z)

[ A]o_.

H1(oY'; Z) 62 H2(Y’, or; Z)

6)

 

shows that ij’162(7-) = ij‘1(01 — 62) : c1(|L1|) — c1(|L2|) : 0. Thus there is an
element 7’ E H1(II,-l,- x D) such that 61r’ = A‘ldgr, and (53(r — i*r’) = 0. Therefore
7 E i*(H1(II,-l,; X D; Z)) implying that (31).,[mg] is homotopic to (32),.[mfi] through a
homotopy in l, X D. Since the construction of the unit circle bundle of the orbifold

line bundle in (1) depended only on these sections, L1 and L2 are equivalent. El

The above theorem means that a given orbifold line bundle L over Y is specified

by the data

(61011071817 - - - afln)

called the Seifert invariant of L over Y. (This data, of course, is not unique).

2.3 SpinC Structures on 3-orbifolds

The Spine structures on a 3—orbifold Y are defined by a pair 5 = (W, p) consisting of
a rank 2 complex orbifold bundle W with a hermitian metric (the spinor bundle) and

an action ,0 of orbifold 1-forms on spinors,

p: T*Y —> End(W),

12

 

which satisfies the property that, if e1, e2, e3 are an orthonormal coframe at a point in

Y, then the endomorphisms p(ei) are skew-adjoint and satisfy the Clifford relations
p(€i)p(€’) t p(6’)/)(€’) = 4527'-
We also require that the volume form e1 /\ e2 /\ e3 acts by
,o(e1 /\ e2 /\ e3) 2 —IdW.

We will write c1(§) for the first Chern class of det W.

Theorem 2.5 The tangent bundle T*Y of an orbifold always lifts to an orbifold
Spinc(3)—bundle.

Proof: If we can split TY into a 1-dimensional real line bundle and a complex
orbifold line bundle L, then w2(TY) = 102 (R69 L) : w2(L) is just the mod 2 reduction
of c1(L) for some orbifold line bundle L. Hence TY lifts.

We need to find a nowhere zero section of TY. Note that each l, x D/Za, comes
with an Zai-invariant oriented nonzero vector field that is tangent to l,- at each point
in D / Z. This vector field induces a nonzero section 3 : 8Y’ —> TY|3. Remove an extra
S1 x D from the interior of Y’ and put a similar nonzero section on the boundary.

The obstruction to extending the section into the interior of
Y” = Y\ ((S1 x D) u H, z,- X D/Za,)
is an element of H 3(Y” , BY” ; 7T2(S2)) = Z. Using the homology relation

[51KSl >< 19)] = - Zlafll >< D/Za.)l,

i

13

 

the obstruction can be removed by changing the framing on the boundary of S1 x D.

Thus TY admits a nowhere zero vector field. El

Remark 2.6 In [S], I. Satake treated the V-Euler class as the index of a unit vec-
tor field on TY with singularities and showed that XV(Y) = 0 for odd dimensional

orbifolds. Thus it is not surprising that nonzero vector fields exists on 3—orbifolds.

Theorem 2.7 The set of Spinc structures lifting the frame bundle of a 3-orbifold Y
is a principal homogeneous space over Pict(Y): The difi‘erence of two SpinC structures

{hég is an orbifold line bundle.

Proof: Let .5, and {2 be two Spine structures which are lifts of the frame bundle.
Away from the l,’s, the difference of two Spine structures is a complex line bundle as

in the smooth case. Because

(C1(€1)— 61652)) laili X D/Zai)i : 0

for all 1,, we can extend the complex line bundle over the desingularization |Y| using
techniques in Theorem 2.4. Thus we can investigate locally to show that any two lifts
of isotropy representations into Spinc(3) differ by a representation into S 1. Note that
this is not immediately obvious because there are many different representations of
Z, into Spinc(3) = U(2).

Let O, be the unit vector field on l, x D/Za which is tangent to the circle l, at
each point. We use the fact that p : Z, —> 80(3) is a rotation which leaves the
nonzero vector field 9, invariant. Identify SU (2) with the unit quaternions. The

map Ad: SU(2) —-> 50(3) is given by

gthfi

14

 

for all h E Imll-ll and is the double cover of 5 0(3). Thus 50(3) can be thought of as
the unit quaternions modulo the equivalence h ~ —h. Without loss of generality, we
may assume that the invariant vector field O, is generated by i E ImlHl at each point
in l,.

It is easy to see that elements of 50(3) which rotate the second two components
while leaving i invariant are of the form em E H. Hence p(1) : X’ where A is a
2a-root of unity in (C and 0 g r < a.

The Spine representation a : Spine —> End(ll-ll) is given by U(g, ei0)h = ghew for
all h E H. Here we have used the fact Spinc(3) = 5U(2) >< 51/(—1,—1). Using this

identification, Spine projects to 50(3) by the adjoint map as well,

(9, 6’6) H 9h?

for all h E ImlHl. Thus the representation p lifts to p:

81

l

’0 Spine

/l

Za—7>50(3)

given by 6(1) : (AC/3(1)) (or equivalently (—)\T, —6(1))) for some representation
6 : Z0, —4 51. The representation 6 is given by 6(1) : A“ for some 0 _<_ k < Oz. Hence
the difference of two Spine structures 6, — £2 locally is a representation (61 — 62) :
Z0, —> 51.

Globally, £1 — £2 differs by a complex line bundle over |Y| and local isotropy

representations into 51, i.e., an element in Pic’(Y) as described in Theorem 2.4. E]

15

 

2.4 Seiberg—Witten Equations on 3-orbifolds

Fix an orbifold 50(3)-connection 0V on the cotangent bundle T*Y and a Spine struc—
ture g = (W, p).

Definition 2.8 A Hermitian connection V on W is called spinorial with respect to

°V if it is compatible with Clifiord multiplication, i.e.,

WWW) = WWW + p(v)(V¢)- (2-1)

The set of all spinorial connections will be denoted by .A(W).

Given a trivialization for W, the connection matrix of any °V-spinorial connection

V can be written with respect to this trivialization as
1 i i j .
aZwJ-(Xwfle /\e )+1bIdW

where w; are the connection matrices for °V, and b E {21(Y, R) is an orbifold 1-form.
We will often think of spinorial connections on a Spine structure as U (1) connections
on det W coupled with the Levi-Civita connection Vy on T*Y. A spinorial connection
V defines a Dirac Operator D A : P(Y, W) —> P(Y, W) on the space of orbifold sections
of W which is self-adjoint. The perturbed Seiberg-Witten equations are the following
pair of equations for (A, lII) where A is a U (1) orbifold connection on det W and \II

is an orbifold section of W:

FA + i6 — MOI!) = 0
0,011) = 0.

16

 

 

 

 

 

 

 

 

 

 

 

Here r : P(Y, W) —> {21(Y; HR) is the adjoint to Clifford multiplication, defined by

(p(ib)\II, WW 2 2<iba7(‘1’)>A1,

for all orbifold 1-forms b and all \I/ E P (Y, W). The 6 is a closed orbifold 2-form used
to perturb the equations.

For a fixed metric gy and perturbation term 6, the moduli space M(Y, {, gy, 6) is
the space of solutions to (2.2) modulo the action of the gauge group 9 = M ap(Y , 51).
Let M*(Y, é, gy, 6) denote the set of irreducible solutions (i.e., where \II 5:3 0). For a
generic perturbation, the moduli space is a compact, smooth manifold containing no
reducible solutions. In that case, the fundamental class [M(Y, 5, gy, 6)] is essentially
the Seiberg-Witten invariant. Evaluating it against some universal classes defines a
map 5W3(§) E Z which is independent of the Riemannian metric and perturbation
when b1(Y) > 1 (of, [M]). Denote the union over all distinct Spine structures by
M(Y, gy, 6).

17

CHAPTER 3

4-Manifolds with fixed point free

circle actions

In this chapter we study manifolds with fixed point free circle actions. We describe the
cohomology of these manifolds and show that under some circumstances, line bundles
with connections can be pushed forward to orbifold line bundles with connection
on the quotient. Finally, we describe how to pullback SpinC structures from Spine

structures on the quotient.

3. 1 Homology

A 4-manifold with fixed point free 51-action can be viewed as the boundary of a
disk bundle or the unit circle bundle of an orbifold line bundle L over a 3-orbifold
Y. Henceforth, we will assume that X is a unit circle orbifold line bundle L over Y
where each local invariant 6, is relatively prime to a,. Denote 7r : X —> Y for the
projection map.

When X is smooth, then X v —+ Yv is an honest 51-bundle and we have the Gysin

18

sequence:

0—er(Y) —>Hb(X) ——>H,0,(Y) ——>Hi(Y)

H H l] l

0—>H1(|Y|)—>H1(X) Z Pict(Y)

 

 

 

 

[Ll

—’H\2/(X) ——>Hb(Y) —>H{°}(Y)

—>H2<X> ——>H1(|Yl) —>Hi(Y)
Theorem 3.1 IfX is a 4—manifold with a fired-point free circle action over Y given

by the sphere bundle of a line bundle L over Y, then

H1(X Z) H1(|YiiZ)a [L] is not torsion

IIZ

H1(|Y|,Z)EBZ, [L] is torsion
H2(X;Z) a (Pic‘(Y)/<[L]>)®ker(-U[L]):H1([Y|;Z)—>H,3/(Y;Z).

In particular, since the kernel of (-U [L]) : H1 —é H3 is torsion free, all torsion classes

must come from pullbacks in 7r*(Pic’(Y)).

When [L] is not torsion, the rank of Pict(Y)/ < [L] > and ker(' U [L]) are both
equal to b1(|Y|) — 1. A basis for the former space can be represented by the Poincaré
duals of tori of the form 7r_1(loop) for smooth loops in Y \ BY. A basis for the later
space can be represented by surfaces in X, which, after integrating over the fiber, are
the Poincaré duals of surfaces in [Y] The simple intersection relationship between
loops and surfaces in [Y] implies that the intersection form 0;; should be simple as
well.

In fact, since the signature is zero (of. [HP]), the classification of intersection

forms says that Q X is equivalent to the direct sum of matrices of the form (where d

19

 

an integer)

01
1d

with respect to a basis {A, B} where A E 7r*(Pic”(Y)) is a class pulled back from Y.
Pulled back classes always have square zero by the naturality of the cup product and

the fact that the product of 2-forms on Y is always zero.

3.2 Line bundles over X

Orbifold line bundles E over Y pullback to usual line bundles 7r*(E) over X. Except
for the case X = [Y] x 51, this is a many to one correspondence. Nonetheless, it can
be made faithful in the following way. Given a line bundle E with connection A over

X with the following two properties:

1. The curvature two form of A pulls up from Y, i.e.,

LTFA : 0)

where T is the everywhere non-zero vector field generated by the circle action

onX.

2. There exists a point a: E Y \ ZY such that holonomy of A around 7r‘1(.v) is

trivial.

Then (E, A) can be pushed forward to an orbifold line bundle with connection on Y
(up to gauge equivalence). If one such point a: E Y\EY satisfies the second condition,
then all points outside the critical set do. Such connections are said to have trivial
fiberwise holonomy.

We state Proposition 5.1.3 from [MOY].

20

Proposition 3.2 There is a natural one-to-one correspondence between orbifold line
bundles with connection over Y and usual line bundles with connection over X, whose
curvature forms pull up from Y and whose fiberwise holonomy is trivial. Furthermore,
this correspondence induces an identification between orbifold sections of the orbifold

bundle over Y with fiberwise constant sections of its pullback over X.

Pull back connections 7r*A are characterized by V$‘A\II = 0 for all pulled back

sections \11.

3.3 Seiberg-Witten Equations of Smooth 4-
manifolds

A SpinC structure 6 = (W, a) on an oriented 4-manifold X is a hermitian vector bundle
W of rank 4, together with a Clifford multiplication a : T*X —> End(W). The bundle
W decomposes into two bundles of rank 2, W+ EB W‘, with det W+ = det W“. The
bundle W‘ is the subspace annihilated by the action of self-dual 2-forms. We set
c1(f) to be the first Chern class of det W+.

There is a natural way to pullback a Spine structure from Y to X. Let 7} denote
the connection l-form of the circle bundle 7r : X ——> Y, and let gy be a metric on Y,
then endow X with the metric gX = 77 ® 77 + 7r*(gy). Using this metric, there is an
orthogonal splitting

T*X 2’ R77 69 7r*(T*Y).

If§ : (W, ,0) is a Spine structure over Y, define the pullback offi to be 7r*(f,) :

(WU/V) EB 7r*(W), o) where the action

a : T*X —> End(7r*(W) EB 7r*(W))

21

is given by

0 7T*(P(a)) + bIdrr*(W)
7r*(p(a)) — bIdfi-(W) 0

U(bn + 7r*(a)) :

This defines a Spinc structure on X.

Choosing a Spine structure {0 2 (W0, p) on Y gives rise to a one-to—one corre-
spondence between Hermitian orbifold line bundles and SpinC structures on Y via
E I—> W0 (8) E. Likewise, the pullback Spine structure 6 : 7r*(§0) induces a one-to-one

correspondence between Hermitian line bundles and SpinC structures on X.

Remark 3.3 In this way we can think of a Spinc structure with respect to {0 or g
as a choice of line bundle on Y or X respectively. This allows us to push-forward
a SpinC structure with a trivial fiberwise connection on det W+ from X to Y via

Proposition 3. 2.

There is a natural connection on X which is compatible with the reduction T*X :
R77 69 7r*(T*Y). Let VY denote the Levi—Civita connection on Y and set °V = d EB

7r*(VY). This is a compatible connection which satisfies

°V77 : 0, and °V(7r*(6)) : 7r*(VY6). (3.1)

It is more convenient to use this reducible 5 0(4)-connection instead of the Levi-
Civita connection. By coupling it with a U (1)-connection A on det W+ we can define a
spinorial connection on W+. Define a Dirac operator DZ : PX(W+) —> PX(W‘) from
the space of smooth sections of W+ to W“. The 4-dimensional perturbed Seiberg—

Witten equations for a section \II E FX(W+) and a U (1)-connection A on det W+

22

F]; +i6 — q(\Il) = 0,
PH‘I’)

(3.2)

II
9

Here F; is the projection of the curvature onto the self-dual two forms, 6 is self—
dual 2—form used to perturb the equations, and q : FX(W+) —> 9+ (X, ilR) defined by

q(\II) 2 Won \11* — %|\II|2 is the adjoint of Clifford multiplication by self—dual 2—forms,i.e,
(U(ifi)‘1',\1’>w+ = 465, q(‘1’))m+ (3-3)

for all self—dual 2—forms 6 and all sections \II.

Similar to the 3—dimensional case, the moduli space M(X,§,gx,6) is the space
of solutions (A, \II) modulo the action of the gauge group. We are using a reducible
connection °V instead of the Levi—Civita connection on T*X, but this alternative com—
patible connection is an allowable perturbation of the usual Seiberg—Witten equations
and can be used to calculate the Seiberg—Witten invariants (see section 4 of [OS2]).
Under suitable generic conditions the moduli space is a compact, oriented, smooth
manifold of dimension
dc) = i (one? — 2X(X) — 3a<X>> (3,4)
which is independent of metric and perturbation when b+ (X) > 1.

The Seiberg—Witten invariant 5WX(§) is a suitable count of solutions. Fix a base
point in M and let 9° C Map(X, 51) denote the group of maps which map that point
to 1. The base moduli space, denoted by M0, is the quotient of the space of solutions
by go. When the moduli space M(X, E, gX, 6) is smooth, M0 is a principle 51-bundle

over M(X, é, gx, 6). For a given Spine structure 5, the 4-dimensional Seiberg—Witten

23

 

 

invariant 5WX(€) is defined to be 0 when d(§) < 0, the sum of signed points when
d(€) = 0, or if d({f) > 0, it is the pairing of the fundamental class of M(X,§, gX, 6)
with the maximal cup product of the Euler class of the 51-bundle M0.

The dimension formula (3.4) simplifies when the manifold has a fixed point free
circle action. Because X has a nonzero vector field T, the Euler class is zero. As

mentioned previously, the signature of X is also zero.

Proposition 3.4 Suppose that X is a I-manifold with a fixed point free circle action.

The expected dimension of the moduli space for a SpinC structure 5 is

61(5) = ’Cl(f)2-

24

 

CHAPTER 4

Spine structures and SW solutions

We continue to work with a circle bundle 7r : X ——> Y with an 51-invariant metric
gx = 772 -l- 7r*(gy). The perturbation 6 E 92(Y, HR) is a closed orbifold 2-form used to
perturb the 3-dimensional equations which is then pulled back and projected on to

the self-dual 2-forms of X to perturb the 4-dimensional equations.

4.1 Restrictions on Spine structures

First, we make some basic observations. If SWX(§) 75 0 for some Spinc structure 6,

then the expected dimension of the moduli space is nonnegative, implying

If b+(X) = 1, then the metric gX induces a splitting H2(X;R) 2 21+ 619 ’H— where
70‘ is one dimensional. Let c1(§)+ be the L2 projection onto the self-dual subspace
”H+. When c1(£)+ is nonzero, it provides an orientation for 71+. In this situation
the Seiberg—Witten invariant depends on the chamber of 27rc1(§) + 7r*(6). We will
say that or E H 2(X ;R) lies in the positive chamber if 05+ - c1(§)+ > 0. Denote the

Seiberg—Witten invariant calculated for a = (27rc1(€) + 7r*(6)) in this chamber by

25

SW;(§) and denote the invariant of the other chamber by SW); (5).
When c1(§)+ = 0 there no distinguished chamber. However, if SWX(§) 31$ 0 in

either chamber, the dimension of the moduli is nonzero and

0 _<_ 61(6)? =(c1(t)‘)2 : 0.

Since the intersection form on 71‘ is definite, c1(§) is a torsion class and pulled back
from Y by Theorem 3.1.

With this as background, we can state:

Theorem A. Let 5 be a SpinC structure on 4-manifold X with a fixed point free circle
action such that 5Wx(f) # 0 (in either chamber when b+ : 1).

1. If b+(X)>1 or b+(X)———0, then c1(§) is pulled back from Y.

2. If b+(X)=1, then either c1(€) is pulled back from Y, or 5W;(f) = 0.

Remark 4.1 In case 2b, the Seiberg-Witten invariant of the other chamber can be

calculated using the wall crossing formula of [LL].
Corollary 4.2 If b+(X) > 1 then c1(f)2 = 0 and X is SW simple type.

Recall that a 4-manifold is SW simple type if the dimension of the moduli space
is 0 for all SpinC structures with nonzero Seiberg-Witten invariants.
Theorem A follows easily from the following formula about Seiberg-Witten solu-

tions.

Theorem 4.3 Let (A, \I/) be any solution in M(X,§,gx, 7r*(6)+). Then
acre-«16> = / IVT‘I'|2+|D+‘1’|2+ ILTFAl2+27r2cl(€)2-
X

The vector field T is the everywhere nonzero vector field generated by the circle action.

26

Remark 4.4 The equation in Theorem 4.3 only holds for perturbations which are

pulled back from Y. It does not hold for a general self-dual 2—form on X.

The rest of this section contains a proof of Theorem A assuming Theorem 4.3
above. We will then come back and prove Theorem 4.3 in the next section. We prove
eaCh case separately.

Proof of case 1: When b+(X) > 1, the moduli space is nonempty for all generic
metric and perturbation pairs. Since generic pairs are dense in the space of metrics
and self-dual 2-forms, we can take a sequence of generic pairs which converge to the

pair (9X, 0). By compactness, solutions of the generic pairs converge to a solution

(A, \I') E M(X,€,gx,0) and it satisfies
0 = / [VT\I’]2 + [ID—hp]2 + [LTFA]2 + 2W2C1(€)2 (4.2)
X

by Theorem 4.3. Using equation (4.1) we conclude that all terms in equation (4.2)

vanish; in particular, c1(€)2 : 0 and
LTFA Z 0.

Since dFA : 0, this equation implies LTFA = 0 by Cartan’s formula. Together the
equations LTFA = LTFA = 0 imply that FA is pulled back from Y. Since 01 (g) = fiFA,
case 1 follows.

When b+(X) = 0 we have that b2(X) = 0 is also zero because the signature

is zero. Thus c1(€) is always a torsion class and this is pulled back by Theorem 3.1. D

Proof of case 2: Assume that c1(§) is not pulled back. By the argument
proceeding the statement of Theorem A, c1(§)Jr aé 0.

We proceed by contradiction. Suppose that SW;(§) 75 0. In this chamber, the

27

moduli space will be nonempty for all generic pairs of metrics and perturbations.
Note that the unperturbed Seiberg-Witten equations (6 = 0) are in this chamber
because (c1(§) — 0)+ - c1(§)+ > 0. Hence we can use the same argument as in case
1 to Show that c1(€) is pulled back from Y — contradicting our assumption. Thus

SW§(§) : 0. E]

4.2 Solutions to the SW equations

In this section we prove Theorem 4.3. The idea is to prove a Weitzenbfick-type
decomposition for the Dirac operator we constructed in section 3.3. Before we prove
this decomposition, however, we need to show that the full Dirac operator IDA :
PX(WJr EB W‘) ——> I‘X(W+ 63 W") is self-adjoint. The following technical lemma

accomplishes this.

Lemma 4.5 Let E = (W, a) be a SpinC structure over X. Let V be a spinorial con-
nection created by coupling a connection A E A(det W+) with the 50(3)-connection
C’V defined in section 3.3. Similarly, let VL‘C' be the spinorial connection created by

coupling the same connection A with the Levi-Civita connection °VL'C‘. Then

5.0- = n. — gem A dm. (4.3)

Since DL'C' and Clifi’ord multiplication by 3—forms are both self-adjoint operators, IDA

is self-adjoint.

0,e1,ez,e3} on a patch

Proof: Extend 77 to an orthonormal coframe {77 = e
of X so that e0 = n, and {e1, e2, e3} are horizontal lifts of an orthonormal coframe
{61, 52, e3} on Y. Let {e0 : T, e1, e2, e3} be the dual vector fields with respect to the

metric g X.

28

 

 

The difference 1-form w = VL'C' — 0V E 91(50(T*X)) can be thought of as an

element in 91(A2T*X) via the vector space isomorphism
i : 50(T*X) —> A2(T*X)
defined by

1 . .
i(a§) = iZajceJ /\ ek.

j<lc
The action of 50(T*X) on the bundle W is modeled on 0A2 oi. Thus we can Clifford

multiply the A2 component of w E {21(A2T’iX) to get

510 = 10A t Owe/12(0))

where 0A1®A2 : A1 (X) A2 —> End(W) is a linear map defined by
(Mia/WU)l ‘3 5) = “600(5)
for a basis element or (X) 6 E A1 (8) A2. This map can be conveniently reformulated as
(Mica/WW ‘59 5) : —0(Lab5) 'l' 0(a A 5),

where Lab is contraction with the vector field which is gX-dual to a.

Let {(12, (13, (23} be the functions defined by
an = 2(12e1 /\ e2 + 2q13e1 /\ e3 + 2C23€2 /\ e3. (4.4)

We can use equation (4.4) and the first Cartan Structure equation

de’zZejAw;

j

29

to calculate the connection matrix for oVL'C'. For example, we can write dn as
d1] 2 81 /\ ((1262 'l' (1363) + 62 /\(—<1281+C13€3)+ 63 /\ (—C1381 — C2362)

to get the top row of the connection matrix

 

 

( 0 C1262 + C13633 —C1261 + C2363 "C1361 — C2382 \
—C12€2 — (1363 0 —§1260 + w; —(1360 + to; (4 5)
C1281 — C2383 C1280 — 60% 0 _<23f30 + ”6

\ C1361 + C2362 C1380 — 62% C2360 — 00:3,? 0 f

The wj’s in the second, third, and forth row are pulled-back from the connection

1-form for the Levi-Civita connection on Y. The connection matrix for 0V is

{0 0 0 0 )
0 0 (12% ca, (4 6)
0 —w% 0 a;

(0 —w, —w§ 0 )

 

 

Using the isomorphism i, the difference VL'C' — V can be written as

3
1 , 1
(0:52 e®n/\ie,(dn)+§n®d77.

i=1

A straight forward calculation gives 0A1®A2(w) = —%o(n /\ dn).

Remark 4.6 The operators D510 and IDA have the same index.

30

Lemma 4.7 The square of the Dirac operator decomposes into
1
(mm: = —(vT>2 + (19+)“?+ + 50m A war). (4.7)

where ( )+ is the projection onto self-dual 2-forms.

Proof: We work with the full Dirac operator first. By using the definition of

°V from equation (3.1), we see from

(Offllvr‘ljiq’l = (‘1’7VT(0(77)CD)>
= (‘1’,°VT(77)<1>>+<‘1’,0(77)VT<1>>

: OI]? 0(77)VT(I)>

that 0(77)VT is L2 self-adjoint.

The Dirac operator decomposes into a sum of two self-adjoint operators:

PA = U(UWT t D»

where D = 23:, o(e’)Ve,.

Squaring and noting that VTn : 0 and o(77)o(77) = —Id yields
$31. = *(Vrl2 t D2 + {0(77)VT, D}-
The last term simplifies using Clifford relations and the equations (2.1) and (3.1):

{0(77)VT7 D} = Z 0(77 A 6i)[VT, Veii'

31

One can use the connection matrix (4.5) to calculate that
[T, a] = °v!,< -'-eL —°vL- C T: 0 (4.8)
for i = 1, 2, 3. In this situation the curvature reduces to
FV(T,e,-) = [VT,Ve,], (4.9)
and we can see that

{a(n)VT,D} = Za<nAei>Fv(T,e.->=0<n>o<LTFvi (4.10)

i=1

By the definition of °V, the action of [VT, V6,] for i = 1, 2, 3 commutes with Clifford

multiplication. Therefore FV(T, e,) is a scalar endomorphism, so
1 .
FV(T,€z') = §FA(Taei)' (4-11)

Restricting attention to W+ gives the formula. Cl

Proof of Theorem 4.3: Take a solution (A,\II) E M(X,gx,7r*(6)+), apply
(IDZY jg, and take the inner product with \11. After applying Lemma 4.7 and inte-

grating over X we get

0 : ((lDA) MEN ‘1’)

w \II>+ +<<D+>*D+\II. .1.) + §<a<m A warm. .1.)

||
><\><2

= /{@071me|D+w12+2<<anFA> ml». (4.12)

32

 

In the last step we used the adjoint property of q(\I!) from equation (3.3) and the fact
that q(\II) is self-dual.

Substituting the q(\II) = F; + 7r*(i6)+ from (3.2), we get

2<(77/\LTFA)aLI(\I’)> : 2((77ALTFAlaF/i +7T*(i‘5)+)>
Z (LTFA, LT (FA+*FA+7T*(1(S)+*7F*(i6))>
= [LTFA]2+ (LTFA,LT* (FA+*7T*(i(i))>

1
= ILTFAl2-l-éiFAAiFA—IFAAW*(6) (4.13)

The last equality is true by the following calculation. Let F,, be the functions defined

by
FA 2 Z iF,jei /\ ej.
ogi<j
Then
Ar, = iFOlel + name2 + iF03e3
and

LT * FA 2 IF1283 — iF13€2 + iF23e1.

Taking their inner product gives
1. .
(LTFAa LT * FA) 2 F01F23 — F02F13 + F03F12 = 21FA /\ IFA-

A similar calculation shows that (LTFA, LT * 7r*(6)) = ——iFA /\ 7r*(6).

Integrating equation (4.13) over X gives the lemma. El

33

CHAPTER 5

Diffeomorphic moduli spaces

In this chapter we prove Theorem B. We continue to work with the circle bundle
7r : X —> Y with the 51-invariant metric gX = 77 ® 77 + 7r*(gy) as in 3.3. Fix a closed
orbifold 2-form 6 to perturb the 3-dimensional equations and pull it back to get an
51-invariant 2—form on X. Perturb the 4-dimensional equations by projecting 7r*(6)
onto the self-dual 2—forms to get 7r*(6)+.

The total moduli space M(X, g X, 7r*(6)+) is a disjoint collection of moduli spaces,

one component for each SpinC structure 6 on X. Define
N(X, 9X7 7T*(6)_+-)

to be the components of the total moduli space whose cohomology class c1(£) is pulled
back from Y.

Theorem A implies that we need only look at these components to calculate the
Seiberg—Witten invariants when b+ > 1. This restriction on the total moduli space is
done to rule out SpinC structures covered in Theorem A case 2b.

Note that for any Spinc structure 5 whose c1(§) class is pulled back and for any

34

2-f0rm 7r*(6),

Cl(€)2 = 0

ci(€)'7T*(5) = 0. (5-1)

In particular, the expected dimension of the moduli space is 0 by Proposition 3.4.
A pullback of a solution (A0, \IIO) to (2.2) on Y is the solution (A, \II) : 7r*(A0, \110)

to (3.2) on X. Pick an orthonormal coframe on a patch of X

{e0,e1,e2,e3}

0 = n, and {e1,e2,e3} are horizontal lifts of an orthonormal coframe

so that e
{€1,62,e3} on Y. Let {e0 2 T,e1,e2,e3} be the dual vector fields with respect to

the metric gX. In this case the Dirac operator can be written as

A :- 0(77)VT t DA (5-2)

where V is a connection on W+ created by coupling A with the reducible connection
°V (see section 3.3) and D = Zo(e‘)V,, for i = 1,2,3. From the construction of
the pulled back Spine structure, it is immediately clear that 7r*(\I’) is harmonic since
it is constant along the fiber and comes from a harmonic spinor on Y. The first
equation of (3.2) is satisfied by pulling up the first equation and projecting each term
onto the self-dual 2-forms. Since a gauge transformation on 6 pulls back to a gauge

transformation of 7r*(§), we get a well defined map on the level of moduli spaces.

35

Theorem B. The pullback map 7r* induces a homeomorphism
7r* : M*(Y,gy, 6) —> N*(X,gx,7r*(6)+).

Furthermore, if either of the two moduli spaces is a smooth manifold, then both of

them are smooth, and 7r* is a difi‘eomorphism.

One remark: There is no restriction on b+ (X) in the above theorem.

The next three sections contains the proof of this theorem. We show that 7r* is
a homeomorphism in the first two sections. In the final section we show that d7r* is
an isomorphism on the kernel of the linearizations. This is sufficient to prove that
the moduli spaces are diffeomorphic because the expected dimension of each is zero.

Bochner vanishing arguments are used to prove that 7r* and d7r* are surjective.

5.1 7r* is injective

Suppose we have two irreducible solutions to the 3-dimensional equations whose pull-

backs (A, \II) and (A’, \Il’) differ by a gauge transformation g E Map(X, 51),
g(A, \II) = (A’, \IJ’).

We wish to show that g is in fact pulled back from Map(Y, 51). We think of g as
a section of End(det W+) : End(7r*(det W)) = 7r*(End(det W)). Use A to create a

connection VEnd on End(7r*(W+)) which has trivial fiberwise holonomy. Then

(andgl‘l' = Wham—WW

: 0

36

because \I1’ = g\II are pulled back sections. By the unique continuation theorem for

elliptic operators, \11 7E 0 on a dense open set, hence

Vgndg E 0
on X. Thus 9 is a fiberwise constant section of the line bundle 7r*(End(det W)) and
by Proposition 3.2, it can be pushed forward to a section of End(det W) on Y, i.e., a

gauge transformation on Y.

5.2 7r* is surjective

Take a solution (A, \II) E N*(X,gX,7r*(6)+) to the Seiberg-Witten equations (3.2).
We will show that the solution is pulled up from a solution (A0, \110) on Y.
Combining the formula in Theorem 4.3 with the fact that Spinc structures from

N*(X,gx, 7r*(6)+) satisfy equations (5.1), we get

2W61(€)‘7T*(5l 2 / [VT‘I’lgtlD+‘I’l2+ILTFA|2+27T201(€)2- (5-3)
\ V J X "

0

 

and that the following terms must be identically zero:

0 = VTQI, (5.4)
0 2 DW, (5.5)
0 = LTFAa (5-6)

Equation (5.6) implies [.TFA = 0 and together the equations imply that FA is
circle invariant and pulled up from Y. Equation (5.4) and the fact that \11 i 0 means

that A has trivial fiberwise holonomy. Therefore we can apply Proposition 3.2 and

37

 

Remark 3.3 to 6 with connection A to conclude that \II corresponds to an orbifold
section \Ilo on a Spine structure {0 with connection A0 on Y.

In this situation, D+ is the Dirac Operator on the orbifold Y, so by equations (5.5)
and (5.4), the second Seiberg—Witten equation of (2.2) is satisfied for (A0, \IIO). It is
easy to check that (A0, \110) also satisfies the first Seiberg-Witten equation.

Therefore the map 7r* is a homeomorphism of moduli spaces. E]

5.3 The kernels are isomorphic

Consider an irreducible solution 5 = (A,\II) to the 4-dimensional Seiberg-Witten
equations for a fixed metric and perturbation (9x, 77*(6)+) in the Spine structure {. In
the previous section we saw that (A, \II) was pulled back from a solution $0 = (A0, \110)
to the 3-dimensional equations on Y in the Spine structure {0.

We now describe the tangent space at the solution S. The following sequence of

operators (for a fixed k 2 5)
0 —> TlLi+2(Y, 51) A TsLi+1(iT*Y 69 Wt) fl: Lg(iA+T*Y e W‘) ——> 0

is called the deformation complex at S. The map CS is the infinitesimal action of the

gauge group at S described by its differential at the identity:
Ls : if H (—2ialf. ifor).

and LSW4 is the linearization of the 4-dimensional Seiberg-Witten equations with

fixed perturbation 7r*(6)+. We can wrap LSW4 and £5 into one operator

T5 : Li+,(iT*X e W+) —> Lfi(iA+T*X e W‘ e iA°T*Y)

38

by setting 7}, : LSW4 + cg. Then the kerTs is the set of (a, it) which satisfy

d+a — 4W, ‘1!) _ g(ql, 1?) : 0)
PAIL + %0(a)‘1’ = 0, (5-7)
—2d*a+iIm(v,b,\I/) ~_—_ 0.

The last equation is a slice condition for the gauge group action.
Let ”H; denote the cohomology of the deformation complex at S. We can now

state Lemma 2.2.11 from [N, page 129]:

Lemma 5.1 The deformation complex at S is Fredholm, that is, the coboundary maps

have closed ranges and the cohomology spaces are finite dimensional. Moreover,
H2 2 kercs, H; e kerrs
and
coker 7g ’5 Hg EB Hg.

In particular, the expected dimension of the moduli space for the Spinc structure 6 is

d(§) : —XR(7{§) = —— dimng + dim it; — dimug.

A metric and perturbation (g, 6) is called a good pair if H0 = H2 = 0 for every
solution to the Seiberg—Witten equations in the SpinC structure g. If (9x, 7r*(6)+) is
good, then the moduli space M(X, f, gX, 7r*(6)+) is a smooth manifold of dimension
d(§), its formal tangent space can be identified with H; at the point S, and we can
use it to calculate the Seiberg—Witten invariants of g. For a careful treatment of these

ideas, see pages 127-135 of [N].

39

There is a similar complex for the solution SO on Y. It too can be described by

an operator
7'50 : Li+1(iT*Y 63 W) —+ Li(i/\2T*Y EB W EB iA0T*Y)

given by the map

d+a0 — 7(1/5 KP) — T(\I’7 77b)
0.0 TS

1D —9 PAOTPO + %P(00)‘I’0
0 —2d*a0 +IIII1<160,\I’0>

 

 

It also has a complex at S0, and an associated cohomology denoted by ”Hgo which can
be described using T50 and a similar statement as Lemma 5.1 above.
By definition S is irreducible if and only if ’Hg 2 0 (and likewise for SO). Hence

solutions in N*(X, g, gX, 7r*(6)+) satisfy
0 = d(§) = dim it; — dim Hg (5.8)

by Proposition 3.4, equation (5.1), and the previous lemma. Therefore ”Hg vanishes
for these solutions precisely when dim 7i; : 0. We will use this fact and the following

theorem to show when dim ”Hg 2 0.

Theorem 5.2 Let S = (A, III) E N*(X,.f,gX, 7r*(6)+) be a irreducible solution to the
Seiberg- Witten equations and let SO 2 (A0, \Ilo) E M*(Y, £0, gy, 6) be the solution such
that S = 7r*(SO). Then

amt.) = Hg,

i.e., the kernels of 750 and Ts are naturally isomorphic via 7r*.

Because the expected dimension of the moduli space on the 3-manifold is always

40

zero, when there is a good pair (9y, 6) such that dim ’Hgo : dim H30 = dim ’Héo : 0
for all solutions in M(Y, gy, 6) we get by Theorem 5.2 that the dimension of ”H; will
be zero for the pulled back solutions as well. Hence ”Hg 2 0 by equation (5.8) for all
irreducible solutions S implying that N * (X , gx, 7r*(6)+) is a smooth manifold. Thus
Theorem 5.2 finishes the proof of Theorem B. If, in addition, N (X , gX, 7r*(6)+) does
not contain any reducible solutions, then (9x,7r*(6)+) will be a good pair for any
Spinc structure pulled back from Y.

The rest of this section contains the proof of Theorem 5.2. We use a Bochner
vanishing argument similar to equation (5.3).

Certainly a solution to 7§0(a0, w0)=0 pulls back to a solution of
7'5(7r*(a0), 7r*(w0)) = 0. We need to show that 7r* is surjective, i.e., for each

solution (a, w) of the equations (5.7), we will prove that
VTw = 0 and a E 7r*(Ql(Y; iR)).

Use 77 to decompose a into a = fn + c where f E 90(X; HR) and c E {21(X; iR).

Since (a, it) satisfies D16) + %o(a)\II = 0, we have

0 = /X llDir+-;—0(a)‘11l2
1 + 1 2
Z /X ((a(n)(va+,f\I/)) +<D ¢+,0<0>L>|

: / lva + —1—fo1|2 + |D+v + 1a(c)\I/|2 + (5-9)
X 2 2
2Re(o(n)VTw, D+tb) -l- Re(o(n)VT1/2, o(c)‘Il) +

Renown. mm + Beacon. goon.

Two of the cross terms in equation (5.9) are zero as follows. First, since VT?) 2 0

we have

41

2/XR8<0(77)VW,D+1A> = (U(n)VTib,D+w>+<D+w,0(n)VTi/J>
<10, 0(77)VT(D+¢)> + (Tl), D- (U(nlvwll

(d): {0(77)VT7 D}l,b>

II
><\><\><\

But by equations (4.10), (4.11), and (5.6),
1
{0(77)VT7 D} Z 577 /\ LTFA Z 0.
Similarly, we can use the fact that fn and c are both self-adjoint to show

2 /X Re<a<fn>w<c>w> ——— /X<<a<c>a<fn>+a<fn>a<c>w>
= —2/X<c.fn>lw = 0-

The remaining two cross terms in equation (5.9) are analyzed in the following

lemma.

Lemma 5.3 In the situation above,

/XRe(o(n)VTi,b,a(c)\Il)+Re(fo(n)\P,D+w) 2/ ILTdal2. (5.10)

X

Proof: Let {n = e0,e1,e2,e3} be a local coframe where e1,e2,e3 are pulled
back from the base. First we take the adjoints of both terms on the left hand side of

equation (5.10).

42

Applying the adjoint of o(77)VT in the first term of equation (5.10) gives

a<n)vT(a(c)\IJ) = Z 0(n)a(°vT(c.-ei>>v + a<n>o(c)(VTL>
: Z (7(77)0(°VT(c,-ezl))\II
.4. :0 ((Tc,)77/\ei) ‘1’ +0(77)0(070VT(8i))‘I’

i=1

.: Za((Tc,)77/\ei) \I/ (5.11)

We used equation (2.1) in the first line, and (5.4) in the second. We also used the
definition of °V from equation (3.1).

Similarly, we take the adjoint of D in the second term of equation (5.10) to find
Devon) = U(df A or + comm = 0(df A my. (512)

Next we show that the sum of the right hand sides of equations (5.11) and (5.12)
is equal to

0(77 /\ iT(da))\Il.

First, note that for i = 0, 1, 2, 3,
de77 = 0 and deei : 0. (5.13)

This holds for e0 = 77 since d77 is the curvature of a principal orbifold circle bundle

so is pulled back from Y; it holds for the remaining i since e1, e2, e3 are pulled back

43

from Y. Hence,

nAbrlda) = nALT(d(f77+C)) (5-14)

3
2 77 /\ LT ((df /\ 77 + fd77) + Z(dc, /\ ei + c, /\ (16))

i=1

3
2 df /\ 77 + Z(Tc,)77 /\ ei

i=1

Combining equations (5.11), (5.12), and (5.14) and projecting onto the self-dual

2—forms we get:

/X Re<0(77)Vw,0(C)‘P>+Re<fa(n)\1’,D+w> = fRelw,(nMTda)+‘1’>

Using equation (5.7), we can reduce further

/X Re<¢,(77/\vrda)+\1’> = /X %<¢,(n/\era)+‘1’>+%<(77/\era)+‘1’,ib>
: A2<(T]/\LTda)+aq(\Ilahb)+q(w7\11)>

= /2((77Adea),d+a)
X
1
: fldeal2+—/idaf\ida.
X 2 X

The last equality is the same calculation as in equation (4.13). El

Combining equations (59-510), gives the sum of non-negative terms. Hence we

conclude that the following terms are identically zero:

VT7/2+%f\11 = 0, (5.15)
D+w+%o(c)\ll = 0, (5.16)
dea = 0. (5.17)

44

Notice that the equation (5.17) is equivalent to
OVTa = (If. (5.18)

We investigate equation (5.15) more carefully in the next lemma.

Lemma 5.4

1 I
/ IVT¢+§le2 = / IVTw|2+Zf2|WI2+2ldfl2-
X X

Since f = LTCL and \II 76 0 almost everywhere, we conclude that

szi = 0, (519)

iTa : 0. (5.20)

Equation (5.19) implies that the spinor is circle invariant while equations (5.17)
and (5.20) imply that a is pulled back from Y. These two facts together imply that
(a, 7b) is pulled back from some (a0,w0) on Y. Equation (5.16) shows that (a0,w0)
satisfies the last equation of 750. It is easy to verify that (a0, wo) satisfies the other

two equations of Tso. Hence (a, w) is in 7r*(ker 7E0) and this completes the proof of

Theorem B.

Proof of Lemma 5.4: We must show that the cross term satisfies

foewTwm: /X 2W.

45

Integrating by parts and noting that VT\Il : 0,

foewTwm = [few-warlo-

Pulling out the imaginary valued function °VT f , using equation (5.7), and integrating

by parts again,

fXReW—wmm = —2/X<°w.d*a>
= 2/X(f,°VTd*a). (5.21)

The results follows once we show °VTd*a : A f . We first calculate d*a at a point
p E X over p0 : 7r(p) E Y. Choose a coframe {e0 = r7,e1,e2,e3} at p such that the
{e1, e2, e3} are pulled back from a coframe on Y chosen such that the pull back of the

connection 1—forms satisfy adj-(p) = 0 in the matrix (4.5). Then

3

3
d*a :. _ 2 :beiovL’CLb : _ E :<ové.C.a,ez>.
i=0

i=0
Differentiating this with respect to °VT,
°de*a : _ ZovchL-Ga, eL)

: _ Z<ovTovL'.C.a,ei>. (522)

Next we will show using the connection matrices (4.5) and (4.6), and equation

(4.8) that

Z<[°VT, °vg-C-]a, e’) = 0. (5.23)

By setting a = Z akek and using the fact that °VTei = 0,

3 3
Z<[OVT, ové.C.]a/’ei> : Z<OVT ((61; . ak)ek + akové.C.ek) _ ové.C. ((T . ak)ek) ,€i>
i=0 i=0

3
= Z T - A. + (T - anew-Lek. .2) + WWW-06k, .2)
i=0

—e, - T - a, — (T - ak)(°Vfi‘C'e’°, e").

The first and fourth term cancel because [T, e,] = 0 by equation (4.8). The second

and last term also cancel. The third term is equal to
akT - (°Vfi'C'e", e2) (5.24)

because °VT is compatible with the metric and °VTei : 0. But

3 3
Z<0V6Li.c.€k’ei> : _ Z<€k70Véaeil
i=0 i=0
Using the fact that wj-(p) = 0 for i,j : 1,2,3, we can see that °Vfi'C'ei = 0 by
inspecting the connection matrix (4.5). Since this term vanishes, equation (5.24)
vanishes giving equation (5.23).
Therefore we can commute °VT with °VeLi'C' in equation (5.22), and apply equation
(5.18) to get: 3
°de*a. = — gem-Cape) : A f.
7120
This statement is independent of frame, so we can substitute it into equation (5.21).

The lemma now follows by integration by parts.

47

CHAPTER 6

Results

We are now ready to prove the formula for calculating the Seiberg—Witten invariants of

a 4—manifold with a fixed point free circle action and state some immediate corollaries.

Theorem C. Let X be a closed smooth 4-manifold with b+ > 1 and a fixed point
free circle action. Let Y3 be the orbifold quotient space and suppose that X E Pic‘(Y)

is the orbifold Euler class of the circle action. Iffi is a Spinc structure over X with

SW35“) # 0, then g = 7r*(€0) for some SpinC structure on Y and
SW§<€> = Z SW68),
E’Eéo mde

where 5’ — 60 is a well-defined element of Pic’(Y). When b+ : 1, the formula holds

for all SpinC structures which are pulled back from Y.

Remark 6.1 In the b+(X) : 1 case, the numerical invariant may still depend on the

chamber structure on if b1(Y) = 1.

Proof: Recall that-for a generic choice of metric and perturbation (gy, 6) the
moduli space satisfies ’Hgo = 71;, = H20 2 0 for all solutions SO 2 (A0, \110) to the 3

dimensional Seiberg-Witten equations (see section 5.3 for more details). For this good

48

 

pair the moduli space M(Y, gy, 6) is a smooth manifold with no reducible solutions.
Since we can choose a perturbation generically such that the projection of F A0 + 6
onto the harmonic 2-forms is not a multiple of the harmonic representative of X for

all solutions in M(Y, 9y, 6), we have that
(”*(FAO) + 76(5))I 7L 0

on X as well, hence N (X, gX,7r*(6)+) does not contain reducible solutions either.
By Theorem B, N (X, gX,7r*(6)+) is diffeomorphic to a smooth manifold without
reducible solutions. We have in effect shown that (gX, 7r*(6)+) is a good pair and that
this moduli space can be used to calculate the SW invariant.

Choose a specific Spine structure f on X such that c1(§) is pulled back and
5WX(€) gé 0. There exists a Spine structure {0 on Y such that g : 7r*(€0) by
Theorem B, and

N(X7€79X7W*(6)+) : H M(Y,€’,gy,6).

6’ Ego mOd X

From this the formula follows. Cl

When the action is free, the theorem above reduces to the formula:

Corollary 6.2 Let X be a closed smooth 4-manifold with b+ > 1 and a free circle
action. Then the orbit space Y3 is a smooth 3-manifold and suppose that X E H2(Y; Z)
is the first Chern class of the circle action on X. If f is a SpinC structure over X

with 5W§(§) 75 0, then 6 = 7r*(€0) for some Spine structure on Y and

SW§(€) = Z SW68).

{’Efo mod X

49

 

where 5’ -— {0 is a well-defined element of H2(Y; Z).

Because of this formula, it is particularly easy to calculate the Seiberg—Witten
invariants for manifolds with free circle actions. We will use this version of Theorem
C in Chapter 7. In the next chapter we give an alternative proof of this corollary

using a gluing theorem and assuming the next corollary, which is also a consequence

of Theorem C:

Corollary 6.3 (c.f. Donaldson [D]) Let X E“ Y3 X 51 with b+(X) > 1. If a Spine
structure 6 has 5WX(§) # 0, then there is one SpinC structure £0 on Y such that
5 = 7r"‘(50) (Md

5Wi(€) = SW3(€0)-

The usual route used to explain the corollary above is to consider the cyclic cover-
ing of X by Y3 x R. There is a natural way to pullback solutions of (3.2) to solutions
on Y3 X R for Spinc structures pulled up from Y3. After putting the solution in
temporal gauge it satisfies the 3-dimensional Seiberg-Witten equations because it is
a constant gradient-flow of the Chern-Simons-Dirac functional [CM]. Thus for each
£ on X such that SWX(€) 7t 0 there is a Spinc structure on Y whose moduli space
is nonempty for all generic metrics and perturbations. This corollary shows that this
moduli space can actually be identified with the moduli space of X and can be used

to calculate the Seiberg—Witten invariant.

50

CHAPTER 7

An Alternate Proof

In this chapter we prove Corollary 6.2 assuming Corollary 6.3. It shows that the
Seiberg—Witten invariants can be computed using the gluing formula in [MMS]. This
proof cannot be generalized to the fixed point free case because the argument breaks
down: the sum on the left hand side of equation (7.2) cannot be reduced to a single

term as in the free case.

7 .1 Classifying free circle actions

Let X be an oriented connected 4-manifold carrying a smooth free 5 1-action. Its
orbit space Y is a 3-manifold whose orientation is determined so that, followed by
the natural orientation on the orbits, the orientation of X is obtained. Choose a
smooth connected loop l representing the Poincare dual PD(X) E H1(Y; Z). Remove
a tubular neighborhood N E“ D2 x l ofl from Y, and set X’ = (Y \ N) x 51. View
X’ as an 51-manifold whose action is given by rotation in the last factor. Let m’ be

the meridian of l in X’, and let f’ be an orbit in X’. We then have:

Lemma 7.1 The manifoldX is diffeomorphic {by a bundle isomorphism) to the man-

51

ifold
X(l) =X’U,, D2 XT2 (7.1)

where (,o : T3 —> BX’ is an equivariant diffeomorphism which evaluates go,..([(9(D2 X

pt)] 2 [m’ + f’] in homology.

When gluing D2 X T2 into the boundary of a manifold, the resulting closed manifold
is determined up to diffeomorphism by the image in homology of [8(D2 X pt)]. (For
example, see [MMS].)

The proof follows immediately from construction (3) in Theorem 2.4 where the
section on the boundary 3 : 8(Y\N) —> BX’ is given by 3*[8D2] = m’ + f’ . Henceforth,

we shall work with X (l) and refer to it as X.

7.2 Gluing along T3

Since we have X z X’ up (D2 X T2) we may apply the gluing theorem of Morgan,
Mrowka, and Szabt’) [MMS].

Theorem 7 .2 (Morgan, Mrowka, and Szabé) If the Spine structure 6 over X
restricts nontrivially to D2 X T2, then SWX(§) = 0. For each SpinC structure £0 —> X’
that restricts trivially to 0X’, let VX(€0) denote the set of isomorphism classes of Spinc

structures over X whose restriction to X’ is equal to £0. Then we have

2 SWx(o= Z SWstx(o+ Z SWX0,.(o, (7.2)

€€VX(€O) EEVYXSMEO) EEVXO/,(€0)
where the manifold X071 2 X’ Um, D2 X T2 is defined by the map 900,, which maps

[8(D2 X 1 X 1)] +—> [f’] in homology.

In our situation, this formula simplifies significantly. Let i denote the inclusion

of BX’ into X0. A study of the long exact sequences in homology shows that the

52

left hand side consists of a single term when i...[m’ + f’ ] is indivisible. Since i...[f’] is
independent of i...[m’] and i,.[f’] is a primitive class in H1(X0; Z), i...[m’ + f’] is such
a class. Therefore, the formula enables the calculation of the SW invariants of X in
terms of the SW invariants of Y X S1 and a manifold X071.

The manifold X071 admits a semi-free 5 1-action whose fixed point set is a torus.

Its orbit space is Y \ N, and 8(Y \ N) = 8N is the image of the fixed point set. The

condition b+(X) Z 2 of the Corollary 6.2 implies that b+(X0/1) > 1 and that
rank H1(Y \ N, 8(Y \ N); Z) > 1.
The two statements are proved as follows. The Gysin sequence
H2(Y;Z) —”—'>H2(X;Z) ——>~H1(Y;Z) imp/n) (7.3)
implies
H2(X;Z) ’5 (H2(Y;Z)/ < X >) GE ker (UX : H1(Y;Z) —> H3(Y;Z)). (7.4)

Each component of the direct sum above has rank b1(Y) — 1. The bilinear form of X
is the direct sum of hyperbolic pairs which implies that b+(X) = b, (Y) — 1. Since [l] is
not a torsion element, removing N from Y implies the rank of H1(Y\N, 8(Y \ N); Z)
is also b1 (Y) — 1. The second statement now follows because b1(Y) — 1 = b+(X) > 1.

The first statement requires the following Mayer-Vietoris sequence
H3(T3; Z) —> H2(X’;Z) e 1572(1)2 x T2; Z) —> H2(X0/,; Z) —"> H1(T3;Z).

The rank of H2(X’; Z) is 2b1(Y) — 1 and the rank of the image of the first map is 2.

Therefore b2(X0/1) = 2b1(Y) — 2. Since the bilinear form of X0), is also a direct sum

53

of hyperbolic pairs, b+(X0/1) > 1.
We can now apply the following general theorem about manifolds with semi-free
circle actions (whose local isotropy group at a point is either trivial or 51) to the

manifold X 0 )1.

Proposition 7.3 Let X be a smooth closed oriented 4-manifold with a smooth semi-
free circle action and b+(X) > 1. Let X* = X/S1 be its orbit space. Suppose that X*

has a nonempty boundary and rank H1(X*, OX”; Z) > 1. Then SWX E 0.

Proof: Let F denote the fixed point set of X and F * its image in X *. Then
BX’“ C F*. The restriction of the circle action to X \ F defines a principal 51-bundle
whose Euler class lies in H2(X* \ E“; Z). Let X’ E H1(X*, F*; Z) denote its Poincaré

dual. Consider the exact sequence

0 —> H1(X*,6X*;Z) 2; H1(X*,F*;Z) —> H0(F*,8X*;Z) —> H0(X*,6X*;Z).

Since the rank of H1(X*, 8X*; Z) is greater than 1, there is a class in
i...(H1 (X *,8X *; Z)) which is primitive and not a multiple of X’. This class may be
represented by a path or in X * which starts and ends on BX but is otherwise disjoint
from F *.

The preimage 5 = 7r‘1(oz) is a 2-sphere of self-intersection 0 in X. It has self-
intersection 0 because the path 07 can be perturbed slightly to another path a’ which
is disjoint from a; hence 5’ = 7r“1(a’) is homologous but disjoint from S. The Gysin

sequence gives:
H3(X*,F*,Z) ——> H1(X*, F*,Z) 3) H2(X,F,Z) ——> H2(X*,F*,Z)

where p...(i,..[a]) = [5]. The image of H3(X*, F*,Z) g Z in H1(X*, F*, Z) is generated

by X’. Since i...[a] is primitive and not a multiple of X’, the class [5] E Imp C

54

H2(X, F, Z) is not torsion; hence [5] is nontorsion as an element of H2(X; Z).

It now follows from [FSl] that SW X E 0. Cl

Proposition 7.3 implies that the formula (7.2) simplifies to

was = Z mask). (76)

é’EVyxsl (SIX’)

7 .3 Spine structures which are not pullbacks

There are Spine structures on X which do not arise from Spine structures that are
pulled up from Y. In this section we show that the Seiberg—Witten invariants vanish
for these Spinc structures.

Fix a SpinC structure {0 = (Vi/'0, p) on Y and consider its pullback 6 : 7r*(§0) over
X (see section 3.3). The other pulled back Spine structures are now obtained by the
addition of classes 7r*(e) for e E H 2(Y; Z).

Looking at the Gysin sequence (7.3), if a class e E H 2(X ; Z) is not in the image

of 7r*, then f + e is not a Spinc structure which is pulled back from Y.

Lemma 7.4 Ifg is a SpinC structure on X which is not pulled back from Y, then
SWX(§) = 0.

Proof: We claim that there exists an embedded torus with self-intersection 0
which pairs nontrivially with c1(§). Then by the adjunction inequality [KM] the

Spin0 structure 5 has Seiberg—Witten invariant equal to zero. Let
H : ker(- U X: H1(Y;Z) —> H3(Y;Z))

in equation (7.4), and consider for a moment the projection of c1(€) onto the

first factor of H 69 7r*(H2(Y; Z)) by changing the SpinC structure by an element of

55

 

7r*(H2(Y; Z)). Since 6 is not pulled back from Y, c1(6)|H 75 0, and since H1(Y; Z) is
a free abelian group, 01 (6 )IH is not a torsion class.

Examining the Gysin sequence, c1(6)|H E H2(X; Z) maps to a class 6 E H1(Y; Z),
6 U X = 0. Thus the Poincaré dual of 6 can be represented by a surface b, and there
is a 1-cycle A in Y \ N rel 8 such that [A] - [b] # 0. Since 8N is connected, [A] is
actually represented by a loop A in Y \ N. The preimage 7r‘1(A) = A X 51 in X is a
torus with self-intersection 0, and c1 (6 )]H - [7r'1(A)] : [b] - [A] # 0.

On the other hand, if A E 7r*H2(Y; Z) then its Poincare dual is represented by a
loop a in Y which may be chosen disjoint from A. Thus A- [n_1(A)] = 0. This means
that c, (6) - [7r‘1(A)] 7S 0, as required. [3

7 .4 Identifying the set VYX S1 (6 | X,)

We identify the Spine structures in the set VyX51(6|X7).
According to the previous lemma, the only nontrivial Seiberg-Witten Spine struc-
tures are those pulled up from Y. Thus far we have seen that for such a SpinC structure

6 = 7r*(6*) with 60 2 fly, we have

SWXié) = Z SWYxsl(€’)-

E’Evyxsl (60)

Let it : Y X S1 —> Y be the projection. We identify the set va 31 (60) of isomorphism

classes of SpinC structures over Y X 51 which restrict on X’ to 60.

Lemma 7.5 VyX51(60) = { ii" (6* +n - X) In E Z}.

56

Proof: The diagram

 

induces SpinC structures on X, X’, and Y X 51 which satisfy

’inc*(7T*(€))= {0: mm (6*)-

Recall that 6 is the only Spine structure induced on X by 60 since i...[m’ + f’] is
indivisible. Since 7i*(6*) E Vyxsl (60), the set of Spine structures on Y X 51 is {7i*(6*) +
e|e E H2(Y X 51; Z)}. Now fr*(6* )+e lies 1n VYx51(60) if and only if inc*‘7r( (6* )+e):

60, i.e. if and only if inc*(e) : 0. Therefore,

le(€o) = {6*(6) + 6 l z'77»C"‘(€) = 0}- (7-6)

The kernel of inc* is equal to the image of 7* in the diagram below.

   

 

H2(Y x SI, (Y \ N) x SI,Z) LHW x sl;Z) (xxx)

[p5 [pp [pp

772(1)2 x T2;Z) H2(Y x SI; Z) ——>H2(X’,8X’; Z)

 

 

 

V
O

W] ,_ > nu x 7']

However 7,. [pt X T2] = [l X f’], and since 7i*(X) = PD‘1[l X f’], the lemma follows. CI

Applying Corollary 6.3 completes the proof.

57

CHAPTER 8

Examples

In this chapter we discuss many interesting examples related to a question / conjecture

of C. Taubes:

IfY is a S-manifold such that Y X S1 admits a symplectic structure, must

Y fiber over the circle?

A well-known technique of Thurston shows that if Y is a 3—manifold which fibers
over the circle (with homologically essential fiber) then Y X S1 admits a symplectic
structure. One might ask whether one could use similar techniques to construct a
symplectic structure on X, a 4-manifold with a free Sl-action, whose orbit space is a
3-manifold Y which fibers over the circle with essential fibers. Partial positive results
to this question have been posted by Fernandez, Gray, and Morgan [F GM]. However,
using the formula of the above theorem, some knot theory, and work of Taubes [T],
we construct in Example 1 a 4-manifold with a free circle action whose quotient fibers
over the circle (with essential fiber) but which admits no symplectic structure.

Another example related to Taubes’ question is a 3-manifold which cannot be the
orbit space of any symplectic 4-manifold with a free circle action. In Example 2, we
demonstrate a 3-manifold which has the property that each 4-manifold which is a

principal Sl-fiber bundle (i.e. admits a free 51-action) over it has a SW polynomial

58

 

Figure 8.1. YK before surgery.

whose coefficients are all greater than 1 in absolute value. Since Taubes has shown
that the canonical class of a symplectic 4-manifold has Seiberg—Witten invariant equal
to 1, these 4—manifolds cannot admit a symplectic structure.

First we describe the main construction for all of our examples.

8.1 A construction and a calculation

The following construction is similar to but simpler than the main construction
in [FS2]. Let ZK denote the manifold resulting from 0-surgery on a knot K in 53.
Let m’ be a meridian of the knot in Z K. Let m1, mg, mg be loops that correspond to

the 51 factors of T3. Construct a new manifold

YK : T3#m1=mZK : [T3 \ (m1 X D2ll U [ZK \ (m X 192)]

by removing tubular neighborhoods of m and 777, and fiber summing the two manifolds
along the boundary such that m : m1 and (9D2 is sent to 8D2.

This is a familiar construction. If one forms a link L from the Borromean link by
taking the composite of the first component with the knot K (see Figure 1), then YK
is the result of surgery on L with each surgery coefficient equal to 0. If K is a fibered

knot, then the resulting manifold T3#m,:mYK is a fibered 3—manifold.

59

 

 

Consider the formal variables t5 = exp(PD(6)) for each 6 E H1(Y;Z) which
satisfies the relation ta+5 : tatfl. The Seiberg—Witten polynomial SW of X is a
Laurent polynomial with variables t5 and coefficients equal to the Seiberg-Witten

invariant of the Spinc structure defined by tfi.

Theorem 8.1 (Meng and Taubes [MT]) For a closed oriented 3—manifold Y with
b1 > 0, the Seiberg- Witten polynomial is given by the Milnor torsion of Y. In the

situation above we can simplify this to:
SW32, 2 AWE“) (8.1)

where AK is the symmetrized Alexander polynomial of K.

For example, the manifold YK in Figure 8.1 where K is the trefoil knot has Seiberg-
Witten polynomial

3Wi/K(tm1) = 4,7,3 +1 — 133,1.

8.2 Example 1: Non symplectic X 4 whose quotient
fibers over 51

We first produce an example of a nonsymplectic 4-manifold which admits a free
circle action whose orbit space is a 3-manifold which is fibered over the circle. Our
construction generalizes easily to produce a large class of such manifolds with this
property. Let K, and K 2 be any fibered knots. Form the fiber sum of the complements

of K1 and K2 with neighborhoods of the first and second meridians of T3, i.e.,

YK1K2 : (SB \ I{1l#m:m1:11316”:7Tt2=m(‘5’3 \ K2)

60

@2678)

Figure 8.2. YKlK2 before surgery

where m is the meridian of the corresponding knot. Since both K1 and K2 are
fibered, the manifold YK1K2 is a fibered 3-manifold. By the Meng—Taubes theorem,

the Seiberg-Witten polynomial of this manifold is

8W§K1K2(tmll tm2) : A[(1 (t’iznl )AK2 (21.72712 ) '

Let X K, K2(l) be the 4-manifold with free circle action that has YK1K2 for its orbit
space and PD[l] for the Euler class of the circle action. Taking both K1 and K2 to
be the figure eight knot (see Figure 8.2), we get a manifold with the Seiberg—Witten

polynomial:

SW3 2 2 25—2172 — 315;; +152 7‘2 — 357,? + 9 — 31:3,, + t‘2t2 — 32:3,, +152 t2

YKIK m1 m2 m1 M2 m1 m2 m1 m2“

The Seiberg-Witten polynomial of the manifold XK1K2(4m1) can be calculated from

Corollary 6.2,

4 _ —2
SWXK1K2(4m1) — 2t

m1. +m2

— 315;; + 9 — 673,, + 2112 — 37.2

m1+m2 mz’

where t5 = exp(7r*(PD(6))) is the pullback of the Spin6 structure on YK1K2.

61

 

A theorem of Taubes [T] implies that the first Chern class C, of a symplectic 4-
manifold must have a Seiberg—Witten invariant i1. We thus see that the manifold
XK1K,(4m1) admits no symplectic structure with either orientation. This is not the
only free 51—manifold over YK1K2 with this property. The manifolds X K1K,(—4m1),

XK1K2(4m2), and XK1K2(—4m2) also admit no symplectic structures.

8.3 Example 2: Y3 which is not a quotient of a
symplectic X 4

Next we produce an example of a 3-manifold which is not the orbit space of any
symplectic 4—manifold with a free circle action. Let K, = K 2 be the nonfibered knot

52 (see [R]). Note that H 2(YK1 K2; Z) has no torsion. The Seiberg—Witten polynomial

Of YKlKg IS
3 _ —2 —2 —2 2 —2 —2
isKle _ 4tm1tm2 — 67m, + 4tm1tm2 — 611m, + 9

— 6153,, + 4t‘2t2 — 6173,, + 4::2 t2

m1 m2 m1 "12'

By Corollary 6.3 we see that XK1K2 (0) : YmK2 X 51 does not admit a symplectic
structure. For all other principal 51-bundles above YK1K2, we need to calculate as in
Example 1. There are only finitely many free 51 manifolds XK1K2 (l) which need to be
checked because for all l : am1+ bmg with la], [bl > 4 the Seiberg—Witten polynomial
SW4 is equal to the 3—dimensional polynomial (only the meaning of the variables will

change). An example calculation of the Seiberg—Witten invariant for the t;,21t;,2 Spine

62

 

structure of XK1K,(6m1) is

4 _ _ n _
SWXK1K2(6m1)(tm?tm2) : Z SW3” 2(t (m21+6 tmi)
nz—oo
= ---+ 5ngle .35..) + SW3K1K,(t;fit;.i) +

SWSKIK (t4 r2 )+

m1 m2

= ...+0+4+0+...

The 6m, pairs the Spine structure tm2 t,“,,2 with Spine structures that are outside the set
of Spinc structures with nontrivial invariants. For principal 51—bundles with la], |b| g
4, calculations show that the Seiberg—Witten invariant for each Spine structure is
greater than one in absolute value. For instance, the Seiberg-Witten polynomial for

XK1K,(2m1) is

_ —2 2
.9wa K 2,“, _ 2th — 3 + 2tm2.

Hence for all l E H1(YK1 K2; Z) the principal 51-bundle XK1K2(l) does not admit
a symplectic structure. Therefore, YK,K2 is not the orbit space of any symplectic

4-manifold with a free circle action.

Remark 8.2 The above two examples show:
1. There exist nonsymplectic free 51-manifolds with fibered orbit space.

2. There exists a nontrivial 3—manifold which is not the orbit space of any symplectic

I-manifold with a free Sl—action.

8.4 Example 3: b+ = 1 diffeomorphism invariants

In this section we construct a b+(X)=1 4—manifold with free circle actions whose

Seiberg-Witten invariants are still diffeomorphism invariants. In this situation we can

63

 

 

 

use Theorem C to calculate its Seiberg—Witten polynomial. We then use Theorem B
to study the moduli spaces of X and its quotient Y and explain why the invariants
do not change when crossing a “wall.”

Recall the construction from section 8.1. Instead of the Borromean link, use the
Whitehead link in 53 and compose each component with the knots K1 and K2 (see
Figure 1). Then the 3-manifold YK1K2 is the result of surgery on this new link with
each surgery coefficient equal to 0. Because the Whitehead link is fibered, when and

K1 and K2 are fibered knots, the resulting 3-manifold fibers over the circle.

13/ — \K.
E

Figure 8.3. YK1K2 before surgery.

 

 

 

 

 

 

 

 

Define the XK1K,(L) to be the unit circle bundle of a line bundle L over YMKQ.

When c1(L) is nontorsion, we get the following facts:

1. 61(YK1K2) : 2, 51(XK1K2(L)) Z 2, and b+(XK1K2(L)) = 1.

2. The cup product pairing

U 2 H1(XK1K2(L)iZ) ® H1(XK1K2(L)iZ) —) H2(XK1K2(L)iZ)

is trivial. This can be computed from the cup product on YK1K2 using the

64

 

lSOIIlOI‘phISIn 7T* Z H1(YK1K2; Z) —> H1(XK1K2 (L); Z).

The two facts above are exactly the conditions needed to show that the wall
crossing number is zero for all Spinc structures [LL]. Hence Seiberg-Witten invariants
are still diffeomorphism invariants for these manifolds. In fact, any unit circle bundle
over a three manifold which satisfies the conditions above will be such an example.
The manifolds constructed above are also particularly easy for calculating the Seiberg-
Witten polynomial using Theorem C. We give one example.

Let Y = YK1K2 be the manifold where K, and K2 are the fibered 63 knot in [R]

(see Figure 2). Then the Seiberg—Witten polynomial

swine, y) = (x_4 — 33-2 + 5 — 3:1:2 + x4)(y_4 — 337-2 + 5 — 3y2 + 114)

is calculated using Milnor torsion (Theorem 8.1). In this setup x 2 exp(PD(m1))
and y = exp(PD(m2)) are formal variables where m1, m2 E H1(Y; Z) represent the
meridian loops of each component of the Whitehead link. Thus the term 9x2y2 in the
polynomial above means that the Seiberg-Witten invariant for the Spinc structure

identified with PD(2m1 + 2mg) is 9.

(2)1)

Q

Q

Figure 8.4. Y constructed out of 63 knots.

QC/Q

65

 

 

Let X = XK1K2(L) be the unit circle bundle of a line bundle L which satisfies
c1(L) = 4PD(m1). Since the Seiberg—Witten invariants for X are independent of the
wall crossing, we can use a similar argument as above to show that c, (6) is pulled back
from Y. Thus d(6) = 0 and condition (5.1) holds for Spine structures with nontrivial

SW invariants. We can apply Theorem C to get

SW‘} (x, y) 2 7y‘4 — 6x2y_4 — 21y“2 +18x2y_2+35 — 30x2 — 21y2 +18x2y2 +7y4 — 6x2y4

where the formal variables are defined by x : exp(7r*(PD(m1))) and y :
exp(n*(PD(m2))) and represent the pullback of Spine structures on Y. X is also
another example of a nonsymplectic 4-manifold with a circle action whose quotient
fibers over the circle.

The power of Theorem C is that one can actually visualize why the Seiberg-
Witten invariant does not change when crossing a wall. Let 0X be the product space
of metrics and P(A+), then (gX,6) E 0;, is called a good pair if the moduli space
M(X ,6 , 9X, 6) is a smooth manifold without reducible solutions. When b+ > 1 the
wall of bad pairs is at least codimension 2 and a cobordism can be constructed between
the two moduli spaces of good pairs. However, when b+(X) = 1 it is possible that
two good pairs cannot be connected through a generic smooth path in GX without
crossing a wall of bad pairs where reducible solutions occur. Passing through a bad
pair could cause a singularity to occur in the cobordism. For a general b+ = 1
manifold, this will often break the invariance of the Seiberg-Witten invariant.

Suppose that we had two good pairs that can not be connected without going
through a bad pair. Connect the two good pairs with smooth generic paths to good
pairs of the form (gX = 772 + gy,7r*(F i 77)+). Here gX is fixed, ||77|| is sufficiently
small, and F is the harmonic curvature form which represents 27ricl(60) for some Spine

structure on Y. Suppose for the sake of argument that {y(t) = (gX, 7r*(F +t77)+) | —

66

 

1 g t S 1} is a smooth generic path in GX connecting the good pairs. Then a bad
pair occurs in both 0;, and Gy precisely when t = 0. While the wall has codimension
b+(X) = 1 in GX and hence unavoidable, the wall in G,» has codimension b1(Y) : 2.
Thus it is possible to perturb the path in Gy to a smooth generic path which avoids

the bad pairs. The moduli spaces M(Y, 60, gy, F i 77) are then cobordant, and

SWY(£O:9Y:F_77):SWY(€019Y1F+77)-

This can be done for each Spinc structure 6’ on Y such that 6 : 7r*(6’), so by
Theorem C

SWX<€1gX17T*(F_n)+):SWX(€79X17T*(F+77)+)7

i.e., the Seiberg-Witten invariant is independent of metric and perturbation.

Note that the perturbed path in Gy will correspond to a perturbed path in 0;,
which will still go through a bad pair. The moduli space for X will have reducible
solutions at the bad pair, but they do not change the value of the Seiberg-Witten
invariant.

The same analysis holds for any b+:1 4-manifold with a fixed point free circle

action and b1(Y) = 2. Therefore we get the following corollary to Theorem C.

Corollary 8.3 Let X be a b+=1 4-manifold with a fixed point free circle action whose
quotient Y satisfies b1(Y) = 2. U6 2 7r*(60) is a SpinC structure which is pulled back

from a Spine structure 60 on Y, then

SW§<£> = Z SW38)

6’E6o mod X

and the numerical invariant does not depend on the chamber in which it was calcu-

lated.

67

 

8.5 Application: A Formula for circle bundles over
surfaces

A corollary to Theorem C is the calculation of the 3-dimensional Seiberg-Witten
invariants for the total space of a circle bundle over a surface. The following corollary

can also be derived from [MOY] using different techniques.

Corollary 8.4 Let it : Y —> 2,, be a smooth 3-manifold which is the total space of
a circle bundle over a surface of genus g > 0. Let c1(Y) : nA E [172(29; Z) where A
is the generator and n 75 0. The only invariants which are not zero on Y come from

Spinc structures which are pulled back 77 : Y —> 29. Hence,
syn/(wan): Z swggxsmrun)
tEs modn
where it : 239 X 51—) 29.
Proof: Let 7r : Y —> 29 be the total space of a circle bundle over 23 with Euler
class nA. Then the manifold Y X S1 can be thought of as a smooth 4-manifold with

a free circle action for which the orbit space is 29 X 51. The Euler class of the action

is 7i*(nA)). Applying the Corollary 6.2 gives

SW$xsl((r,id)*(7~T*(8/\))) = Z SW§st(€')

ir*(tA)—Eir*(sA) mod ir*(nA)

the right hand side of the equation. Applying Corollary 6.3 shows that SW4 2 SW3

in this case. C1

The Seiberg-Witten polynomial for the product of a surface with a circle,

swggxsla) = (t —t‘1)29‘2,

68

 

follows from the Seiberg—Witten invariants of 239 X T2 in [FM2]. Combining this with
the previous results gives a formula for the Seiberg—Witten polynomial in terms of

the Euler class and the genus of the surface.

Corollary 8.5 Let 7r : Y —> 29 be the total space of a circle bundle over surface with
g > 0. Assume c1(Y) = nA where A E H2(Eg;Z) is the generator and n is an even

number n : 2i 75 0, then the Seiberg- Witten polynomial of Y is

Ill—1 k22g—2 2g — 2
SWYU) Z sign(n) Z Z (—1)(9‘1)+L+k|1|( . )tfi
i=0 k=_(2g_2) (9 _ 1) "l" L ‘l’ kill
wheret = exp(7r*(A)) and defining the binomial cofiicient (Z) = 0 for q < 0 and q > p.

For the formula where n is odd, replace l by n and t2i by t".

This formula highlights the fact that principal 51-bundles over surfaces are simple
examples that illustrate the difference between Milnor torsion and Turaev torsion. If
one uses [MT] to calculate the Milnor torsion for a circle bundle Y over a surface, one
finds that the invariant is identically 0. This is because all SpinC structures on Y with
nontrivial invariants have torsion first Chern class. Turaev introduced another type
of torsion in [Tu1, T112] and a combinatorially defined function on the set of SpinC
structures T : 8(Y) —> Z derived from this torsion, and showed that this function
was the Seiberg—Witten polynomial up to sign. Therefore Turaev torsion is not 0 for
principal 51-bundles over surfaces. Note that this can also be seen by calculating

both Milnor torsion and Turaev torsion directly.

69

CHAPTER 9

Final remarks

Theorem C together with an affirmative answer to the following conjecture would
establish a way to calculate Seiberg—Witten invariants for any b+>1 4-manifold with

a circle action.

Conjecture 9.1 IfX is a b+ > 1 smooth closed 4-manifold with a circle action that

has fixed points, then SWX E 0.

There is already considerable evidence which suggest that this is true. For simply

connected 4—manifolds carrying a circle action, we can apply the classification result

of R. Fintushel [F1, F2].

Theorem 9.2 (Fintushel) Modulo the 3-dimensional Poincare’ conjecture, a simply

connected 4-manifold carrying a smooth S 1—action must be a connected sum of copies

of S4, (3113?, erg, and 52 x 52.

This classification result is enough to show that in the b+ > 1 case, X is the
connected sum of two b+ > 0 pieces, and hence SW X E O.

The conjecture also follows from Proposition 7.3 for 4-manifolds with smooth
semi-free actions whose orbit space Y has a nonempty boundary and the rank

H1(Y,8Y,Z)>1.

70

 

A counter example to the conjecture above would be just as interesting as the

proof.

71

 

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72

 

[13}

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75

 

 

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