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THE DONALDSON INVARIANT AND EMBEDDED 2-SPHERES
presented by

WOJCIECH WIECZOREK

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mm.

THE DONALDSON INVARIANT AND EMBEDDED 2—SPHERES

By

W0 j ciech Wieczorek

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1995

ABSTRACT
THE DONALDSON INVARIANT AND EMBEDDED 2—SPHERES
By

W0 j ciech Wieczorek

In this thesis we study the Donaldson invariants for smooth 4—manifolds containing
an embedded 2-sphere with an arbitrary negative self—intersection. In [F S3] Fintushel
and Stern have described the relation between the Donaldson invariant of the manifold
X and its blowup X #W. We prove the existence of a similar relation for X =
Y UL N, where N is a neighborhood of the sphere with self—intersection —p, and L
is a lens space L(p,1). We describe the most efficient way of covering by Taubes
neighborhoods the points that contribute to Donaldson invariant. The technique we
develop allows to compute explicitly the coefficients in this formula in some particular
cases, which we ilustrate at the end of the thesis.

ACKNOWLEDGMENTS

I would like to thank to my advisor, Ronald F intushel for the support during
the whole process of studying. His guidance and enthusiasm were very important in
the process of writing this thesis. Also, I am grateful for extensive discussions and
interest from Gordana Matié at the Park City Summer School.

TABLE OF CONTENTS

List of Figures
1 Introduction.

2 Construction of the Donaldson Invariants.

2.1 Donaldson polynomials on manifolds with cylindrical ends ......
2.2 The Gluing Theorem ...........................

3 The description of instantons on the cylindrical end.
4 Dividing the moduli space into open sets.

5 Computing intersection numbers in open sets UL”
5.1 Description of the neighborhoods of the reducible connections . .
5.2 Desigularization of Q x K ........................
1

5.3 Closing neighborhoods of reducible connections ............
5.4 Compactifications of the UL" ......................
5.5 The relative Mayer-Victoris sequence ..................

6 The proof of Theorem 1.3

7 Examples of computations.
7.1 Second stratum connections over CP2 .................
7.2 Second stratum connections over the orbifold N ............

Bibliography

iv

pl:-

11
14

20
20
23

23
28
31

33

36
38
40

44

LIST OF FIGURES

The splitting of the manifold X .................... 17
The graph .7 for a - a = -6 ...................... 19

1 Introduction.

For a long time Donaldson invariants were the most powerful tools for studying
smooth 4-dimensional manifolds. For a simply connected 4—manifold with b+ odd
and bigger or equal to 3, these invariants are defined as linear maps Dd : Ad(X) —* R
where Ad (X) is the set of elements of A(X) = Sym.(Ho(X) EB H2(X)) having degree
d. We provide the elements of H,-(X) with the degree %(4 — i). Donaldson’s original
definition of these invariants is subject to the restriction

3(b;(X) + 1)
2

and d > =3(b'{(X) + 1). The first version of the blowup formula in [FM 2] allowed the
removal of the second restriction. For a special class of manifolds (called simple type)

 

d (mod 4) (1.1)

one can define Donaldson’s invariants for d E w (mod 2), but in our paper
we shall stick to the original restriction. Since the discovery of Donaldson invariants
there arose a question about computing them. There are many elliptic surfaces for
which these invariants have been completely computed. Except for computing specific
examples one would like to develop general techniques, like the Mayer-Vietcris type
of argument: having the decomposition of the manifold X onto X1 9X2 and knowing

the invariant for the components X1, X2 and some properties of Y (which in fact
are related to the invariant of the 4—manifold R x Y), determine the invariant for
the manifold X. When X2 is a neighborhood of an embedded sphere with negative
self-intersection, then R. Fintushel and R. Stern provided the following answer in.

[FS2]:

Theorem 1.2 Let X be an oriented simply connected 4-manifold, which contains an
embedded 2-sphere 5 representing a homology class a with self-intersection 02 _<_ —2.
Then there are polynomials BM = Bj’k($) and AM depending only on 02, such that
for z E A(o"')

D(02'°‘lz) = Bo,kDa(z) + 23:} D(02j‘lB,-,k(:c)z) if 0'2 = —(2k + 1),
and

D(ozkz) = Ao,kDa(z) + 229:1 D(0'2j-2Aj,k($)2) if 02 = —2k.

Here the degrees of the polynomials BM are equal to k — j, whereas the degrees of AM
are k — j + 1. The A0,): and Boy: are constants.

The proof of [F32] uses the compactifications of certain open sets in the moduli spaces
provided by C. Taubes [T1] and does not show how to compute Bj,k(:c) for specific
examples. In this paper we would like to provide an alternative proof of a theorem
that generalizes the above result. Namely we prove that for an arbitrary number of
classes 0 representing an embedded sphere with negative self—intersection we have
the following result:

Theorem 1.3 Let X be as above and suppose that X contains an embedded 32 with
self-intersection 02 = —p S —1. T hen there is a formal power series B(p);n(:c,t) =

1

Ego 3‘1IB(p);n,j($)tj, SUCh that:

D(exp(o - t) - z) = D(Zp B(p)m(:c, t)o" - 2)

n=1

There are few comments necessary about this formula:
1. Each B(p);n.j is a polynomial in a: of a degree d satisfying the inequality 2d + n S j.

2. For j _<_ p this formula provides nothing but the trivial statement D(o-’) = D(oj).
Later though we shall define the relative Donaldson invariants, for which a similar
formula will become nontrivial.

3. For given 2 E H2(X) only half of the terms in the above formula are nontrivial.
Due to the restriction (1.1), either D(o°""" - z) or D(o°dd - z) is always zero.

4. When p = 1 or 2, X admits an orientation preserving diffeomorphism that induces
a map on H2(X) sending 0 onto -0 and which is identity on 0*. Thus in this case,

D(o°ddz) = 0 for all z.

5. In the case p = 1, Fintushel and Stern [F33] have actually computed the function
B(:c,t) = 8(1);0° Using their result and Theorem 1.3 one can compute the functions
8“,)” for higher p.

In the proof of the above theorem we split the manifold X into N, the neigh-
borhood of the embedded sphere with negative self—intersection, and the remaining
part Y. These spaces are connected along the lens space L = L(p,1). Section 2 pro-
vides a review of the results on gluing. Roughly speaking by stretching the cylinder
L x (-—e, e) we obtain open sets U,- in the moduli space for the manifold X that can
be described in terms of moduli spaces of X1 and X2. In order to combine these open
sets back into a global one, we need to understand how these sets overlap. Section
3 describes the moduli spaces of connections on the cylinders R x L(p,1), which are
essential component of the sets U.- H Uj. In section 4 we describe the most efficient
way of chosing the sets Ug, which cover the points of the moduli space over X which
contribute nontrivially to the Donaldson invariant. We introduce a partial order in
the set {U.-} that will enable us to make inductive argument in the proof of the main
theorem. The next section is devoted to computations inside the sets Ug. In the same
section, by using relative Mayer—Vietoris sequence, we also show that the computa-
tions in Ug’S done separately suffice for proving a global result. Finally, in section 6 we
prove the main result. The techniques developed in this paper have wider application
than just proving the existence of a general formula. In the final section we compute
three examples of particular formulas. One of such formulas has been used by [F33]
as a necessary initial condition for their differential equation defining the function
Bj’k(X) in the blowup formula.

The author would like to thank Ronald Fintushel for introducing him to gauge
theory, as well as for guidance and help while working on this problem.

2 Construction of the Donaldson Invariants.

In this chapter we will recall some main points of the theory of Donaldson invariants.
The details can be found in [D1] and in [DK]. Let X be a simply connected four
dimensional manifold and P —) X be an SU (2) bundle over X. These bundles are
classified by their second Chern class c2(P) = It. For a generic Riemannian metric on
X, the space of gauge equivalence classes of anti—self-dual (ASD) connections on P
is a manifold of dimension 8]: — 3(1 + b+). We will denote this manifold by Mk(X).
Over the product M k(X ) x X there is a universal 5 0(3) bundle P which gives rise
to a homomorphism p : H;(X) —> H4“(Mk(X)) obtained by assigning to a E H;(X)
the class — ip1(P)/o. To each cohomology class p(a) corresponds a codimension 4 —i
variety V, contained in M k(X ) Let [:0] denote the generator of H0(X), and u = ”(3)
with V” its dual divisor. The class V can be also realized as —%p1(M£(X)), where
Mj',(X) is the moduli space of based anti—self—dual connections. The Donaldson
invariant is a linear function

D : A(X) = Sym,,(Ho(X) EB H2(X) ) —+ R

which assigns to the classes a], . . . ,0, E H2(X) and 3 copies of the generator [1] E
H0(X) the intersection number

#(Mk(X) n v,, n---v..n:=1vu) (2.1)

where the numbers k,r and s are related through 2r + 43 = 8k — 3(1 + b+). In
order to make sense of this intersection number, one has to compactify the mani-
fold Mk(X). For that purpose we need to define ideal connections on X as a pair
([w],(xl,:c2,.. .,:z:,)), where [w] is a point in Mk-_,(X) and ($1,232,. . . ,x,) is an un-
ordered s—tuple of points in X. To an ideal connection A = ([01], ($1,232, . . . ,x,)) we
associate its measure (or curvature density) pm 2 ||F,,,,||2 + 87r2 ‘ 63,.

r=l

Definition 2.2 (Compare [DK]) We say that a sequence of connections {can} in
Mk(X) converges to an ideal connection A = ([w],($1,a:2,.. .,a:,)) if the following
conditions are satisfied:

1. The measures pwn converge to [IA-

2. There are bundle maps pn : P’|x\{,, ,,,,, 3,} —v P|x\{,,.1 ,,,,, ,3} such that p;(w,,)
converges on compact subsets of punctured manifold to w.

Let us denote by s"(X) the nth symmetric product of X. Then the above definition
of convergence provides the set

1M,.(X) = Mk(X) u Mk-1(X) x X u Mk_2(X) x 32(X) u ...

with a topology. The Uhlenbeck compactification MAX) of M k(X ) is the closure
of M k(X ) in I Mk(X ) This is a stratified space with singularities. We can define a

fundamental class of a singular space (and thus introduce an intersection theory) if
the singular set is of codimension at least 2. This is the case when d > %(1 + b+). In
[FM 2] it is shown that the classes p(o) extend over the compactified moduli space,
whereas the class V extends only away from the lowest stratum, corresponding to a
completely concentrated connection. Under the above restrictions on the dimension
of the moduli space, the intersection of divisors V, takes place away from completely
concentrated connections, thus the intersection (2.1) is well defined. One can extend
the definition of the Donaldson invariants beyond the dimension restrictions by con-
sidering connections on S U (2) bundles over X #CW and using the first version of
the blowup formula from [FM 2].

2.1 Donaldson polynomials on manifolds with cylindrical
ends

In order to prove Theorem 1.3 we need to separate the sphere S with its tubular
neighborhood N from the rest of manifold X. As 6N = L(p, -1) (which we abbreviate

as L), we can represent X as Y LLJ N. From this point on, one could try to study spaces

of connections on the manifolds Y and N with boundary L. However for boundaries
as simple as lens spaces, it is more convenient to stretch the neck L X (—e, e) to an
infinite length and study spaces of connections on manifolds with cylindrical ends (see
the definition below). The advantage of this approach is that the restrictions over L
of connections with finite energy on bundles over X may be almost arbitrary, whereas
the connections with finite energy on bundles over a manifold with a cylindrical end
limit to a flat connection. The whole set of flat connections is a finite dimensional
variety, thus we have very limited set of possible “boundary values”.

In our formal description of this procedure we shall follow the notation of [MMR].
We are going to apply this theory to both N and Y; thus for the presentation of
general theory let us fix a closed, oriented, Riemannian 4-manifold (Z, gz) whose
boundary is a 3-manifold (L, g).

Definition 2.3 A Riemannian 4-manifold (Z, gz) is called a manifold with cylin-
drical end if Z has a subset isometric to [—1,oo) X L. This subset we will call the
cylindrical end.

We do not assume that the above is the only end of Z. We shall refer to manifolds
with two cylindrical ends as tubes.

Definition 2.4 On the manifold Z with cylindrical end let rz : Z -+ R be a function
which maps a cylindrical end onto {—1, 00) and is less than -1 on the complement of
the cylindrical end.

Define Z, to be 751((-oo,a]) and Zla.b] = Z1, \ Z, = T‘1([a,b]).

There is a corresponding theory of ASD connections on bundles over Z. Let E —) Z
be a C°° principal S U (2) bundle. For any Lilac connection w and any real number

6 Z 0 define the Li); norm on C§°(Z, Ap(ad E)) by

1:
“sun; = (2 /, wares-Tr” (2.5)

i=0

Set Li“); to be the completion of Cg°(Z,A”(ad E)) with respect to this norm. By
A” we denote the set of all connections with the norm defined above. When 6 = 0
we abbreviate Lips by Li”. Let 93 be the group of L3 gauge transformations of E .
Then BE denotes the quotient AE/QE . Define also 92; to be a group of those gauge
transformations which are the identity on a fixed fiber Ex of E. If we divide .43 by
gg, then we get the space of based connections denoted by 873, or 3,”; if we want to
indicate the point over which we fix the fiber. The same space can be also obtained
as

3;; = A}; X E;- (2.6)
91-:

Every element of the space Bf; can be represented as a pair [00, p,], where w is a gauge
equivalence of connection in BE, and p, is a framing of E at 1:. For any complete
Riemannian metric on Z, the energy k of an ASD connection w is:

_1 2
kzgfi‘éterA’FFw: leng

—1
Q3“
(the last equality holds only when w is ASD). Note that, unlike in the case of closed
manifolds — the number k need not be integer. In fact we shall show in the next
section that k is of the form %, where n is an integer and p is the self-intersection of
5'. With this understood, denote by M k(Z ) the space of gauge equivalences classes
of g—ASD connections on E with the energy 1:. Again, if we divide the space of ASD—
connections by g0 then we obtain the space of based connections denoted Mz(Z), or
M}: (Z) when we will need to indicate the point at which the connections are based.

Let x(L) denote the space of gauge equivalences of flat connections in B E( L). This
space is called the character variety of L and is identified with

Hom(7r1(L), SU(2) )/AdSU(2)

Similarly, the flat connections modulo based gauge transformations are identified with
’R(L) = Hom(7r1(L), S U (2)), the representation variety. According to Theorem 4.6.1
of [MMR] there is a well-defined map 3° : MflZ) —) ’R(L). This map descends to
0 : Mk(Z) —+ x(L). These maps associate to every ASD connection w (or gauge
equivalence class [w]) the limit lirntnoohulth]. This limit is (the gauge equivalence
class of) a flat connection on L. In the particular situation when L = L(p, 1) the
character variety is just [’23] + 1 isolated points. Let us denote the elements of the

character variety x(L) by numbers {0,1,2, . . . [g] }. Notice that x(L) is a quotient
of Z, by the Z2 action given by multiplying by -1. Thus for given p there is a map
Z —> x(L) that is a composition of the modulo p quotient of integers with the above
Z2 quotient.

Definition 2.7 We will refer to the elements 0 and p/ 2 E x(L) as trivial elements of
the character variety. Thus for p odd there is only one trivial element. The remaining
elements of x(L) we will call nontrivial.

The representation variety consists of an isolated point or points corresponding to
trivial elements in x(L), and a copy of 32 corresponding to every nontrivial element
of x(L). For given m E x(L) define Mk(X,m) as 0'1(m). As we have seen before,
the number lc is of the form 3. If the moduli space M k(X , m) is nonempty, then the
numbers m and k are not arbitrary. In order to describe their relation we shall define
the Chern—Simons function. Every connection w on the trivial bundle 0 = S U (2) x Y
we can write as w = O + a, where O is the trivial connection and a E 01(Y,ad0).
Then define:
CS(w) = Atr(a/\da+ gaAaAa)

The gauge transformation may change the value of the function CS by an integer,
thus Chern—Simons function descends to a function CS : B (Y) —) R/ Z. According
to the Chern—Simons theorem we have

k E C' S (m) mod Z
The formal dimension of Mk(X, m) is given by the formula:

1 0
_ h_'1_+_h_m + BLT.) (2.8)

3
8k — §'(0(Z)+X(Z)) 2 2

Here o(Z) is the signature and x(Z) is the Euler number of Z. The hi” are dimensions
of cohomology groups H‘(L, ad m) and p(m) is the Atiyah—Patodi—Singer p invariant
for the signature complex twisted by ad(m) <8) C. The invariant hm = hln + hg, is

equal to:
1 when m is nontrivial (2.9)

hm = { . . .
3 when m 18 triwal.

It is important to notice that the invariant hm does not depend on an orientation of
L, whereas p(L) = -—p(L). We will give more explicit formula for p(m) in Lemma 3.8.
For the sake of later gluing theorems, we need some facts about ASD connections on
Z in local coordinates.

Definition 2.10 Let I‘ be a flat connection on a SU (2) bundle 17 over L, and Up be
an open neighborhood of I‘ in the space of gauge equivalence classes of connections on
17. Let I be a subinterval of [0,00). Following [MMR] we say that an L3 connection
w = I‘ + A(t) + a(t) dt on I x 17 is in standard form with respect to Up iffor all t E I

O A(t) 6 Up
e a(t) E Ker(A1~)l
where Ar is the Laplacian on 0°(L,ad (I‘)).

Theorem 2.11 (Corollary 4.3.3 in [MMR]) There is a constant 50, depending
only on L, such that for a generic metric g on Z and for all 1 S a S b S 00 and any
Lilac g—ASD connection w of finite energy satisfying

b+1 2
/ lat<ah
a-l

there is a neighborhood Up and an L3,“, bundle isomorphism d) : [a, b) x 17 ——> E lla.b)x L
such that 45%..) = F + A(t) + a(t) dt is in standard form with respect to Up.

An important consequence of the previous theorem is:

Lemma 2.12 Let Z be a Riemannian manifold containing a submanifold isometric
to C1 = (—l, l) x L for some 3—manifold L whose character variety x(L) consists of
finitely many points. Then there is an e > 0 such that for every ASD-connection w
on an S U (2) bundle E over Z satisfying

.LHRWSC (am)

there is a unique gauge equivalence class of flat connections [F] E x(L) which is closest
to the restriction w|{o}xL in the L3,)2 norm.

Proof: Let 6 be the minimum distance between any two points in the character
variety x(L) in the L3” norm. Consider the set Uz of all ASD connections on Z
satisfying (2.13), and the set U0, of corresponding restrictions to 01. Because of
the energy bound, any sequence of connections on E Ia, cannot bubble on C1. Thus
the set U0, is compact as is the corresponding set Uz in the Uhlenbeck compact-
ification. Hence by Theorem 2.11 we can cover Uz with finite number of the sets
U1! = {wlwkqu 6 Up for all t E (—l, 1)} Now we want to estimate the distance
llwl{t}xL — FHLi/z by the curvature of w on BIC;-

According to Lemma 4.1.1 of [MMR] there is a 61 such that for every connection A
on the S' U (2) bundle over L and satisfying “FA“ Lg), < 61, there is some flat connection

I‘A such that
IM—Pmnn<fl2

which, because of our definition of 6, must be unique. Thus to conclude the lemma
we need to bound the energy of the curvature of calm“, by the curvature of the
connection w on the cylinder C1. This is done in Lemma 3.5.1 [MMR] which provides
a constant C1 > 0 such that the following estimate holds:

2
(wamla)scql um:

[t—l,t+l]xL

which combined with Sobolev inequality ”FllL'fl, S C; - ||F||L¥ gives that

2
2
(llFWI{¢}xLllL§/2) _<_. 0/01 lel

Thus when 6 < 5% every connection as satisfying (2.13) has a unique flat connection
IL. 0

Once we consider the based connections [0), pn] E M24, +(X1, Cdo), then we can
“recover” the flat connection in representation variety.

2.2 The Gluing Theorem

A goal of this subsection is to set up tools for covering the moduli space M k(X ) by
well—understood open sets. We next describe the gluing procedure which joins con-
nections on two manifolds with cylindrical ends. We follow [MMR] and [MM]. For any
m E x(L) and real positive numbers Ti let us define open sets M let (X *, [T*, e], m) C
M ks: (X *3, m) as containing those ASD connections wi on the bundles E*, for which

E, 2<
[WWII in. e

Whenever possible we are going to skip some of the indices in this notation. Thus
the lack of the e parameter means that eis equal to 50/ 2, where so is taken from
Theorem 2.11. Analogously we can define a based version M;(X, T, m) of the above
sets. It follows from [MMR] that for generic metrics on X i and any lei > 0 the
spaces M lea: (X ‘1‘, T*, m) are smooth manifolds. Consider the fibered product

0,31,“ = Mfi_(X_,T',m)§gM§+(X+,T+,m) (2.14)

We now describe how to glue together two connections [w+,w_] 6 0,31,“. Let E0 =
S'U (2) X L be the restriction of E3: to L. For a constant do chose the numbers
Ii 2 Ti: +d0 and construct a manifold X; by identifying the cylinders XIII —do.l++do] C
X(l++do) with X [7._ doJ- No] C X(1-+do). Let us denote by Cdo the cylinder on which the
identification takes place. Note that X; contains a subset isometric to a larger cylinder
C1 of the length I. This construction provides gluing of manifolds X f: and X f. . When
it is clear from the context, we are going to skip subindexes Ii indicating finiteness
of the cylindrical end, thus using the symbol X i for the spaces with finite or infinite
cylindrical ends. Theorem (2.11) gives bundle isomorphisms 17* : [T*,oo) x E0 —)
Eihrim)“ such that (17*)*(wi|[1¢_do,oo)) = I‘ + A*(t) + ai(t) dt, where I‘ is a flat
connection on E0. These bundle isomorphisms provide a clutching of the bundles Ed:
through an isomorphism 17’(17+)‘1 : E+|{z+}xL —-> E" |{¢-}xL. Also if both connections
wt are based at a point (l*,n) E {1*} X L, then the above isomorphism provides a
basing at a corresponding point in Cdo. Using the partition of unity {¢"', 43‘} on C4,,
we can define a glued connection by:

w+ on Z340;
’iz(w+,w") = W! = F + ¢+(A+(t) + 0+(t) dt) + ¢’(A'(t) + 0"(t) dt) 01! Cato;
w" on Z,‘__do;

When the connections w'flw" vary in some open sets U 1', U ', then '71(U+, U ‘) de-
scribes an open set in the space 35;. The goal of the gluing theorems is to project

this set onto the space of ASD connections. The main result of [MM] answers when
can we do it:

Theorem 2.15 (Propositions 4.1.1 and 4.2.1 in [MM]) T here is a constant Io
depending only on the sets U31,“ such that for every ([w+,w"]) E Uzi,c+ and l 2 4do
and do 2 lo there is a unique solution u = u(w+,w') to the equation

F+(iz(w+,w') + u) = 0
This solution u satisfies the estimate
—§6I
"‘4ng S 006 3

where 6 is the minimum of the lowest eigenvalues of Laplacians Am, m E x(L). More-
over the assignment ([w+,w']) —) :yz(w+,w‘) + u defines a unique smooth map 71" :
Ugh -+ M£++k_ (X;) that is a difleomorphism onto an open subset of M2++k_(X1).

This difi'eomorphism factors through to 7: : Ufl.k— = Ufl,k_/SO(3) -—> Mgr”? (X1).

We can use the above theorem also for the purpose of reversing gluing. Let X;
contain a fixed cylinder Cd = (-do, do) X L for some d 2 do. Define A111,”)C+ (X1, C4)
to be the set of those (.0 E M1,_+k+ (X1), for which the following holds:

/ “Poi”: < 50/2
Cd

/X_ ||Fwi||j —87r2k- < 50/2

I
j + “mug—8791c, <eo/2
XH-

Let ML.“ (X1, C40) denote the corresponding set of connections based at some point
n E Coo. The following claim describes the domain of the “ungluing”:

 

 

Lemma 2.16 Assume that l and do satisfy the assumptions of the gluing theorem.
Then there is a smooth map

Mitt—.19. (XI, C(10); u 012.,“

the union taken over all m E x(L) such that CS(m) E k+ E —k_ mod Z.

Proof: According to Lemma 2.12 for every connection w from ML,“ (X1, Coo) we
can find a unique flat connection I‘,, s 6 {0,1, . . . , [123] + 1} such that I‘, is the closest
to wlcdo. Define two approximately ASD connections (12* = :1, (ml x1 , 6’) on the bundle
Ei over the spaces with infinite cylindrical ends X i. The assumptions on I and do
guarantee that we can use the gluing theorem (2.15) for perturbing these approximate
connections to a pair of ASD connections [w',w+] 6 (7,31,, +. Cl

Now we are interested in describing the image of the above “ungluing” map. Our
main statement is the following:

10

Lemma 2.17 For every 6 satisfying 0 < e < so there is a length lo such that for
every I _>_ lo the set

It... (x+i [1:50 _ ellggMIc_(X-? [1’50 _ 6l)

is in the image of ungluing map p defined on ML,“ (X;,Cdo).
Proof: Let us begin with a simple topological lemma:

Lemma 2.18 Let 8, denote a ball in R" centered at the origin and with a radius 5.
Let f : Bc —) R" be a continuous map such that for every :1: 6 Be, dist(f(:c),a:) < 6.
Then the center of Be is in the image off.

Proof: Let S; = 38‘. Suppose that f maps B, to R" \ {0}. Then the restriction
f : S, -—> R“ \ {0} is homotopically trivial. On the other hand the map

F(:c,t)=ta:+(1—t)f(:c)

provides a homotopy between f and identity map (the interval ta: + (1 — t) f (1:) never
passes through 0 as dist( f (:r),:z:) < 6). This contradicts the triviality of f. D

Definition 2.19 Let U be an open set in the Riemannian manifold M. For a positive
number 6 define the sets

U+¢ = {m E MIdist(m,U) < e}
U_c = {m E M|B(m,e) C U}

Corollary 2.20 Let M be a manifold with a radius of injectivity bounded from below
( for example if M is precompact) and let U be an open subset in M. Then there exists
an 61 such that for every 6 < 61 and every map f : U —> M such that dist( f (11:), 2:) < e
the set U.c is in the image off.

Let us come back to the proof of Lemma 2.17. Consider the map
p 0 7 = M2,.(X+, [’+,€o]) goMiJX', [l—,€ol) -* MAO“) 53M;_(X-)

Both components 7 and p are defined by the gluing theorem. If we replaced these
maps by corresponding approximate gluing, then after restricting the connections
over X f: and X; we would get an identity. The theorem (2.15) tells us that while
stretching the neck we can make the actual gluing as close to the approximate one as
we want. Thus it remains to show that the L2 norm of the curvature can be estimated
by an L3 norm of the connection, which is due to the Sobolev embedings theorems

(see [P]):

IIFA - FA+al|L2 = Ildaa + a A (III S ”0|ng + Clllallu S Czllalng

11

3 The description of instantons on the cylindrical
end.

Certain S' U (2) bundles over R X L can be obtained from S U (2) bundles E over a
sphere S4 by dividing by a Z, action. It turns out that the condition for bundles
to arise in this way is closely related to the existence of ASD connections on these
bundles. Following [FL], we can describe the Zp-invariant bundles over .S'4 as

(01 x SU(2),T2)q(Di x SU(2),T1) (3.1)

where f = e277” acts on D; C C2 via {(21,22) = (216,225), and for g E SU(2),

T.(.)=..(expgm o ),..dT.(.)=..(expgm' 0 )

exp {- m exp {- m’

We will refer to the numbers m and m’ as the weights of the Z, action. The transition
function F : S3 x SU(2) —-> S3 x SU(2) of (3.1) can be written as F(a:,g) = (:r, f(a:)g)
for f : 53 -—) 3U (2) The Z, equivariance of F means that f(:c£) = T;1 f (x)T1. No—
tice that the degree I: of the map f is equal to the second Chern class of the bundle E.
We denote the bundle obtained from the above construction divided by a Zp action
by E(k, m, m’). [FL] adapted to the case of S'U (2) bundles over R X L(p,1), gives a
construction of Zp equivariant bundles with weights (m, m’) and degree k = m’2 —m2.
Let us now denote by M k /p(R x L, [m, m’]) the moduli space of ASD connections on
a bundle E (k, m, m’). Before quoting the results of [Au] describing moduli spaces of
ASD connections on E (k, m, m’), we need to comment on orientations. The construc-
tion of ASD connections is based on the identification of S"1 with quaternions H; thus
when R X L is the boundary of a complex manifold M, the orientations in [Au] agree
with the convention that the weight m is from “the manifold side” and m’ is from the
other side. In our particular application we are studying the spaces R X L as the ends
of the space N = W, thus we have to reverse orientations coming from the complex
structures. We can do it in two ways: either by changing the convention about the
order of weights or by changing L(p, q) to L(p, —q). In this paper we have chosen the
second way, thus even though the boundary of N is L(p, —1), while quoting Austin’s
paper we fix q = +1. The main construction in [Au] (Lemma 5.1) gives a description
of ASD connections on the bundles E (In, m, m’) for which there exists a solution (a, b)
to a system:

a E m + m' (mod p)
b E m' — m (mod p)
a - b = k (3.2)

These bundles are called 31 equivariant. (The reason for this name is that the Z,
action on these bundles extends to an 5'1 action). To state the general result we need
one more definition:

12

Definition 3.3 A Z, equivariant bundle E (k, m, m’) is a composite of Z, equivariant
bundles {E(k,-,m,-, m2) 2;, if and only iffor all i = 1, 2, . . . , n we have:

ki > 0; k = Z; ks; m 5 m1 (mod P); m:- E m,“ (mod 1’)? m1, 5 m’ (mod P)

Lemma 3.4 (Compare Lemma 5.2 in [Au].) A Z, equivariant bundle supports an
invariant ASD connection if and only if it is a composite of S'1 equivariant bundles.

The bundles over R X L may have noninteger second Chern numbers (which we will
often refer to as energies). In fact the minimal amount of energy on a nontrivial moduli
space on R X L is 1. This ostensibly leads to large number of possible distributions of

energy on the cylinder. We would like to use (3.4) to limit the number of possibilities.

Lemma 3.5 For fixed m and m’ let k be the minimal amount of energy for which
E(lc,m,m’) supports an ASD instanton. Then

k— m’2—m2 ifm’>m
— m’z—m2+p(m—m’) ifm’<m

Proof: As m,m’ S p/2, then (a,b) = (m + m’, m’ — m) is the smallest nonegative
solution to (3.2) in the case m’ > m, and (a, b) = (m — m’,p— m —m’) is the smallest
solution in the case m’ < m. The product of these numbers gives the required
result for .S'1 equivariant bundles. Now assume that E(k,m,m’) is a composite of
{E(k;,m,,m$)}?=1. The in the case of m’ > m we have:

k=zk.2m3—m2+zp(m.~—mz>2m2-m:
iEI
Here I is the set of those indexes i for which m,- > m2. In the case m’ < m we have:
’8 Z mf-mi+2:ezp(mi—mi) Z mE-mi‘l’Zi‘flMme-mi) = mf-mi+P(m1-m$.)
As an example we can make the following list representing the minimal energies
required for “tunnelling” between flat connections m and m’ on R x L(4,1).

m\m’ 0 1 2
0 0 1/4 1
1 3/4 0 3/4 (3.6)
2 1 1/4 0

 

 

The dimension of the moduli space M = Mk/p(R x L(p,1), [m, m’]) is

, 8k ”“1 ' ' ’ '
dimM = $ — 3 + n + 3 Zeofl?) - (MR?) — mflfln— )
P

i=1

where n E {0,1,2} is the number of m,m’ different from 0,p. By using Fourier
expansions, this formula can be simplified to:

dim Mmz/p(R X L(p,1),[0,m]) = 4m — 3 for m S p/2 (3.7)

13

Using this result and the gluing theorem we can complete our calculations of the
dimensions of moduli spaces M,‘(Y,m), where K. is in fiZ. A similar result has been

observed in [MMR] in Lemma 13.4.1:
Lemma 3.8 The dimension of MK(Y,m) is equal to

1 8 2
8,. — g-(0(Y)+x(Y)) - E,- + —’p1+1—4m

when m is nontrivial, and

[\DIC—O

8.. — gem + w» —

if m is trivial.

Proof: By Lemma 3.5, there is a moduli space of ASD connections on N with
boundary value m and with energy at. What is more, this is the smallest energy
giving a nontrivial moduli space on N with this boundary value. Gluing together
the bundles over Y and N we obtain a bundle over X with energy it + ”I‘D—2. From
the gluing theorem (2.15) we have that dim MK(X, m) = dim M,‘(Y, m) + hm+ dim
Mm2/,(N, m), which for m nontrivial, by using (2.8), gives the following equation:

3 1 m 2 3
81¢ — §'(U(X)+1+X(X)—2) — 5 + a—+1+4m—3 = 8(rc+—T:—)— 5~(o(X)+x(X))
here we used 0(X) = a'(Y) — 1 and x(X) = x(Y) + 2. Solving this equation for pm
and plugging the result into (2.8) gives the first formula. The second formula can be
obtained similarly after gluing in the trivial connection on N. C]
Similar calculations give:

dimMK(N,m) = {8n — 3 + 4m — 8—2:: if m is nontrivial (39)
SK. — 3 . if m is trivial.
One can notice that the first formula gives 8n — 3 for m = 0 and ‘23, so it is valid in
both cases. '
In order to formulate the result which allows us to quantize moduli spaces on the
cylinders, we need to notice a simple fact connected with gluing. This fact has also
been mentioned in [F31]:

Lemma 3.10 Let M° = Mfi(X,m) for a nontrivial element m of the character
variety. Then the base point fibration M° —> M reduces, i.e. there exists an 51-

bundle Q such that M" = Q X 50(3).
51

Proof: Consider the map 0" : M° —» 52 C R(L). This map restricts over each
fiber as 0°|50(3) : 50(3) —> 52 that sends g —> gég’l. Thus for fixed point pt E 52
(0°)’1(pt) is an 51 bundle Q over 52 for which M" = Q X 50(3). Cl

51

The moduli spaces on the cylinders R X L have two boundary value maps

63: : Mf,(R X L(p,1),[m,m']) —; ’R(L)

14

with n E Z,,. Let us recall that according to our convention on the cylinders we
consider as positive the direction from N to Y. Thus a: points toward N, and
61 points toward Y. With this understood, define MEP(R x L(p,1),[m,m’]) to be
(afl)'1(pt) for any pt E Im (82). Thus when m is a trivial element of x(L), then
MEP(R X L(p,1), [m, m’]) is the based moduli space M:(R X L(p,1), [m, m’]). When
m is nontrivial, then according to Lemma 3.10 the space MEP(R X L(p,1), [m,m’])
is an 51 reduction of the base—point fibration. This space should be thought of as
the gluing parameter bundle over an unbased moduli space. This gluing parameter
is attached at the end of the cylinder which is closer to N. The next lemma shows
that these moduli spaces provide good quantization of the whole moduli space:

Lemma 3.11 dimM§P(R x L(p,1),[m,m’]) Z 4- [m — m’].

Proof: This is a direct consequence of Lemma 3.5 describing the lowest energy on
the moduli space M §P(R X L(p, 1), [m, m’]) and the dimension formula (3.7).

Case 1: m’ > m. The moduli space Mme/p(R X L(p, 1), [0,m’]) has a boundary
corresponding to a splitting

M70712/P(R X L(p,1),[0, m]) g: Mfmn—m2)/p(R X L(p,1), [m, m’])

where all of the moduli spaces involved have the minimal energy required for each
tunneling. From (3.7) we have: 4 - m’ — 3 = 4 - m — 3 + dim M§P(R x L(p,1), [m, m’]),
from which the lemma follows.

Case 2: m’ < m. From the index formula we get that dim Mm(RXL(p, 1), [0, 0])

8m — 3. This space allows a splitting Ming/AR X L(p,1), [0,m]);an(p_m)/p(R X
L( p, 1), [m, 0]). Analyzing the dimensions of both spaces we get:
- Z
8m — 3 = 4m — 3 + dlmMm?p_m)/p(R x L(p,1), [m, 0])
Thus dim M:?p_m)/p(R X L(p, l), [m, 0]) = 4m.
Now consider the space M(pm_m2)/,,(R x L(p,1),[m, 0]) and its splitting
M°m__~;m2 ...-..(R x L(p,1).{m,m'1);g Maya (R x L(p.1).{m'.01)

Again the energies have been chosen according to Lemma 3.5 so as to be the smallest
ones. Analyzing the dimensions of the involved spaces we get the equation:

4m = dim MEP(R X L(p,1), [m,m']) + 4m’,

which concludes the proof of the lemma. D

4 Dividing the moduli space into open sets.

Now we can come back to the computation of the Donaldson invariant D(zo"). We
assume here that the numbers p and n are fixed. The goal of this section is to list all

15

open sets defined in (2.14) that cover the intersection
M;,,(X)nVz (1:11 V... (4.1)
To fix notation let us assume that the class z E A(Y) has the degree d, and
dikao(X) = 2 (d + n)

Since we are using a generic metric on X, the intersection of divisors on Y-side is
transverse, so the intersection Ugh n V, n '61 V0,. (see (2.14) for the definition of
Ugh”) is empty if dim MkY(Y, m) S 2d. This implies that dim Mk” (N,m) must
not exceed 2n — 1 if m is trivial nor 2n — 3 if m is trivial. In order to choose the sets
Ufi,” in the most efficient way, let us define the set

J = {(172.10 Im e x(L), k e ZI;1,M,.(N,m)¢@,
and 0 < dika(N,m) + hm S 2n}

(hm is defined in (2.9)). We set also j = ,7 U {(0,0)}. Note that the bound on the
dimension of the spaces M k(N , m) implies that the set ,7 is finite. In (I we define a
partial order by saying that (m1, k1) S (mg, to) if there is a nonempty moduli space
Mk2-k1 (R X L, [m], 7712])-

Definition 4.2 To every element (m,lc) E j assign its degree deg (m,lc) to be the

length of the longest chain in ,7 from (0,0) to (m, k) linearly ordered by S. Define
deg(J) = maxxej deg(x).

The next lemma shows how to compute the above degrees.

Lemma 4.3 For every (m, k) E J we have the following relation:
4 ' (deg (m, k) — 1) + 1 = dim Mk(N,m)

Proof: Set a = deg (m,k). The definition of 0 implies that the moduli space
M k(N , m) has a boundary component corresponding to a fibered product of (a — 1)
moduli spaces Mk,(R X L, [m,,mS-D and the last a‘h space MkN(N, mN). Thus from
(3.11) we have: dim Mk(N,m) _>_ 4(a — 1) + 1. I

We prove the inequality in the opposite direction by induction on the degree of
elements of J. Lemma 3.5 gives us the following inequalities in J:

(ch):(m+1,k+2m+1

 

)S(m,k+1)

Thus deg (m, k + 1) _>_ deg (m, k) + 2. Using the dimension formula (3.9) we get:

dika+1(N,k+1) = dika(N,k)+8=4(deg(m,k)-1)+1+8
S 4(deg(m,k+1)—3)+1+8=4(deg(m,lc+1)—1)+l

El

16

Corollary 4.4 Let F be a graph representing the set .7 with the relation S. Then
(m,lc) S (m’,lc’) are immediate neighbors in the graph I‘ if and only if m’ = m :i: 1

and
k'-k={

Proof: The fact that (m,lc) S (m’,lc’) for m’ and 19’ defined above follows from
Lemma 3.5. The Lemma 4.3 gives us:

~[2m+1] ifm’=m+1
-[-2m+1]+1 ifm’=m—1

'U IH'U IH

deg(m’, lC’) - deg(m, k) = ildika:(N,m') — dika(N, m)]
idimemR x L, [m,mv = 1 (4.5)

thus there are no other points of J between (m, k) and (m’, k’). D

For each element of J we wish to assign an open set in the moduli space M k(X )
Let us fix 3 = deg(J). Notice that by changing the metric on X we can identify in
X a subset isometric to a cylinder [0,1] X L for an arbitrary large I. We can adjust its
length I so as to have a collection of disjoint subcylinders Cf, . . . , Cf), R1, C11, . . . , 0},
R2, C12, . . . C3, . . . R,” Cf, . . . C,’ satisfying the following:

1. For each C,- and B..- there is some r E [0,1] such that these cylinders are equal
to [r,r+ 1] X L;

2. For every connection w with energy I: + n on X there are 3 + 1 numbers
io,i1,i2, . . . ,i, (1 S i, S t) such that

./C" [le2 < 50

'r

where so is the number defined in Theorem (2.11).

Each cylinder R,- splits the manifold X into two manifolds with boundary, which we
denote as Y,- and N,. We also denote by ng the cylinder that is cut from [0,1] x L by
the cylinders R,- and Rj.

Definition 4.6 For every element (m, k) E ..7 we define an open set U)? C Mk0 (X)
as the set of all w that satisfy:

1. f [Fol2 < so, where i = deg(m,lc).
R.-

2. f IF,,,|2 = k + e for some I5] < 60
Ni

3. If (r,n) is a point in R,- then the character m is closest to w|{,}xL in the sense

of Lemma 2.12.

Having this assignment in mind we shall understand that the collection of open
sets {Ufl‘} has also defined a partial order S and deg(UL") makes sense as well. Each
set UL" has uniquely assigned region R,- with i = deg(m,k) and let us choose one

17

point x,- E 12,-. According to Lemma 2.17 each U)? can be described as the fibered

product:
“11-10%, m) 3,; M11111. m)/50(3) (4.7)

Thus every connection in UL" can be written as w = [wy,wN]. In general, if a given
connection w allows the representation (4.7) for some i, then we say that w allows
splitting at the region 12,-. In the next lemma we shall prove that the sets UL” suffice

for covering the intersection of divisors V n 01, V. As the proof of that lemma

requires splittings of moduli space at multiple regions R1, 122,” .,12, on the cylinder
[—n, n] X L, we need to be more precise about the notation of cylindrical—end manifolds
Y and N. Thus by Y and N we shall mean finite cylinder manifolds ending at the
first splitting regions (including these regions). For a given region 12 we shall mean by
the terms Y—side or N -side the corresponding connected components of the manifold
X cut by 12. (See the picture below):

 

 

 

 

 

 

 

 

N sidcof R Y sidcof R
Figure 1. The splitting of the manifold X.

Lemma 4. 8 Every connection that belongs to the intersection M ko( ) O V, D '61 V,
is contained in one of the setsU m, (m, i) E J. Moreover deg (J)_ < —+—

Proof: First we shall prove that s = dng is a sufficient number of the splitting
regions. Assume that w is a connection that does not allow splitting at any of the
regions 121, . . . ,R,. According to the definition of the length I, there are cylinders
03;, r = 0, 1, . . . , s such that w has energy less than so in these regions, thus it allows
the splitting at C5. Let m, denote the gauge equivalence class of the limiting flat

connection on C; , and k, the energy on the N —side of 03;. As (.0 E M1,,(X )flVz '61 V,,

then for every r the dimension bound dim Mk(N, m,) S 2n, defining the set J, is
satisfied, thus the pair (m,, la.) E J for every r. By Lemma 2.17, no can be represented
as a fibered product of connections [wN,w1, . . . ,w,,wy]. The first connection wy lies
in a moduli space of nonnegative dimension. If one of connections w,- defined on
[l,-,l,+1] x L D 12,- was flat, then it would allow the obvious splitting along 12,- as
w; = [w,-,_,w,,+]. In here and are the restrictions of w,- onto different connected
components of [l,-, 1,4,1] X L \ 12,-. As each w,- is nonflat, then we have a sequence of
strict inequalities in J

(mm/m) < (mhkl) < < (m,,k,)

18

from which we get that the last pair (m,., 111,.) has the degree greater than or equal to
s + 1, thus contradicting the definition of 3.

Next assume that w allows splitting at 12,- with 11:,- being its energy on the N — side
and m,- being the nearest flat connection on 12,- in the sense of Definition 4.6. We
still need to prove that for some i there is the equality deg(m,, k,) = i. 30 assume
there is an to which satisfies: for every i = 1, 2, . . . , 3 either (.0 does not split at 12,- or
deg(m,, k,) 75 i for (m,, k.) defined by a splitting at 12,-. Consider first the case when
w splits at 12,- with deg(m,, k,) < i. We claim that there exists another region 12,- for
which the equality deg(m,, k,-) = j holds. This is proved by induction with respect
to i. For i = 1 the statement is obviously true, since each element of J has degree at
least one. Thus we may assume that:

For every r < i and every connection on that allows a splitting at the region 12, with
deg(m,, k,) < r there existj < r such that w splits at R,- and deg(m,, k,) = j.
Consider now a connection (.0 that splits at 12,- with deg(m,, 16;) < i. Then we can write
to as [wy,wN]. The connection wN must allow splitting at 12,-1 for some i > jl _>_ 1, as
otherwise wN could be written as aw = [wjv'v w}! , . . . , a251,], showing that deg(m,, k,) 2
i in contradiction to our assumption. Splitting am at 12,-l we get: am 2 [wrnwm]
and the numbers(m,-,, k,,) corresponding to that splitting. If deg(m,,, k,-,) < jl, then
we are done by an inductive assumption. If deg(m,,,k,-,) = jl, then no splits at
12,-1 with the right equality deg(m,,,lc,-,) = jl. We claim that the last possibility:
deg(m,-1 , k,,) > j], implies that the connection can splits at some other region 12,-,2 for
jl < jz < i. If that claim is not true, then le can be written as a composition of
i — j] nontrivial connections on the cylinder between the regions 12,-1 and 12,-. Thus
we have:

deg(m,, k,) 2 deg(mjn kj1))+(i-J'1) > i

again contradicting our assumption. Thus we can write le = [wT2+,wT2—]. Define

wN, = [awl ,wT—]. Same argument as above aplied for (42)»), instead of any, shows that
we either find the required splitting for w, or we will find another splitting of wT2+ at
the region 12,-, for jg < j3 < i. As there can be only i such splittings, this process
must terminate by finding the 12,- with the required properties.

The proof in the case when deg(m,,k,~) > i is the same as in the the previous
case, exept that one has to reverse the “orientation” of the whole proof, i.e. start the
induction from the point that is closest to the Y—side and replace all N ’s with Y’s,
and “+” with “—”.

Let us now consider an element (m,k) E J with maximal degree. There is
a corresponding connection in M),(N,m) with the splitting [wN,w1, . . . ,w,_1] that
defines its degree. According to Corollary 3.11 the smallest dimension of a moduli
space Mk(R x L, [i, 3]) including the gluing parameter from the N -side is no less than
4 - (i —j). Thus

e—l
2(d + n) = dikaY(Y) + hm, + ZdimeflR x L, [m,-,mg])+
i=1

+ dikaN(N,mN) 22d+hx+4(8-1)+l 2261-1-48—2

19

from which the last statement of the lemma follows. D _
For example, when evaluating D(zo4) for o - o = —4 we get the following set J
encoding the covering of the moduli space:

(0,0)

(11%)

(2,1 (0,1)

In general on the graph representing J we can denote only the values of the
equivalence classes of the flat connections m E x(L). To each edge of the graph J we
can assign the minimal energy for tunneling between the boundary values denoted at
the ends of the edge that has been established in Lemma 3.5. Then the energy of an
arbitrary vertex can be obtained by summing the energies of the edges while going
from the top of J to the given vertex.

Example 4.9 When computing D(zo“) for 11-17 = —6 we obtain the following graph,
that stabilizes:

x1 ........... 1

.. ........... l\o
i\l

$3 ........... \2[\0

4 ...................... :£\[

Figure 2. The graph J for o - a = ——6.

On this graph the points x1, x2, . . . denote the chosen points from the regions 121 , R2, ..
All points on the graph lying on the level indicated by x,- correspond to open sets U1”
that split at the region 12,-. The above graph, according to Lemma 4.8, terminates at
the level [11,1]. The meaning of the columns of this graph will be explained later.

Lemma 4.10 The intersection of the divisors V, :11 V,,. is empty in U;? unless the

moduli space M k:(N ,m) contains a reducible connection.

Before proceeding with the proof we want to comment on the terminology. The
set UL" for a generic metric does not contain reducible connections, as b+(X) Z 3.

20

Nonetheless we shall informally say that A is a reducible connection in Uf‘ if the
restriction of A to N is a reducible connection in M k(N , m).

Proof: The open set U ,f,‘ has dimension 2(d + n). On the other hand, this set viewed
as fibered product has the dimension

dikao_kr(Y,m) + hm + dikar(N,m)

As b+(X) = b+(Y) Z 3, then the intersection Mk,_k:(Y,m) O V,, is transverse since
it contains only irreducible connection. Thus dika,(Y,m) 2 2d. If the divisors
V, intersected along irreducible connection as well, then we would have: 2(d + n) 2
2d + hm + 2n, in contradiction to the fact that hm = 1 or 3. Cl

5 Computing intersection numbers in open sets
U12”

5.1 Description of the neighborhoods of the reducible con-
nections

Let us recall that our main purpose is to compute the intersection number
M,(X) n V, n fl V,,. (5.1)
3:

or dually, to evaluate the cohomology class p(2)p"(o) on the moduli space M k(X )
As we saw in the previous paragraph, this intersection is included in the set covered
by the sum of open sets UL" and takes place near connections whose restriction to
N is reducible. Thus our first task is to describe the neighborhoods of reducible
connections in the moduli spaces M I," (N, m,), and then we will describe a procedure
for keeping the intersection (5.1) inside these neighborhoods. Before proceeding,
we want to notice that there is one kind of noncompactness of the sets UK that
can be taken care of immediately. This noncompactness comes from the completely
concentrated connections (so—called bubbles). Recall that the sets UL" are obtained
via fibered products iY(Y, m) a; M}: (N, m), where the point x belongs to the region

12, on which the energy estimate excludes bubbling. Thus both basepoint fibrations
extend to the 50(3) bundles over the Uhlenbeck compactifications of these moduli
spaces, and from this point on we will assume that the sets UL" contain completely
concentrated connections.

When a reducible connection occurs in the top stratum, its neighborhood is mod-
elled on C" / 51. For the sake of later generalizations we set the following notation:

Definition 5.2 e Every neighborhood of a reducible connection is modeled on
some stratified space divided by as 51 action. Let K denote a compact neighbor-
hood of the total space of this 51 action. The space K / 51 describes a compact
neighborhood of the reducible connection. In this case K = {z E C"|||z|| S 1};
the general description of K is given below.

21

e Let M denote the link of reducible connection. Thus M = UK.

e Let r(K/51) be the image of K/Sl under the deformation retraction shrinking
the cone parameter to zero. ( Thus in the case of top stratum connection r(K/51)
is a point.) There is a corresponding deformation retraction of the set K, which
we will denote by the same symbol. As 51 acts trivially on r(K), then we have

Tar/51) -_—v. r(K).

The set M/51 is often referred to as the link of the reducible connection. In this case
M /51 = CPn'l. In the proof of Proposition (5.1.21) in [DK] it is shown that the
Pontrjagin class of the restriction of the adjoint bundle to the link of the reducible
connection on the bundle L EB L is equal to

p1(P‘d) = h2 X1+4-h x c1(L)+4-1 X c§(L)
where h is the positive generator of H2(CP"‘1). Thus we have the following:

Lemma 5.3 On the link of a reducible connection on the top stratum in Mk(N,m)
we have the following identities in H2(M/51):

M0) = -<01(L),0)'h
Ma?) = -i-h2

The second equation holds for points x E N.

The description of reducible connections in the lower strata of Mk(N, m) has been
worked out in [KoM]. For every sequence of integer numbers n1 _>_ n2 2 - - - _>_ n,
there is a stratum 5t corresponding to s bubbles with multiplicities described by the
numbers n,-. Let us denote by k = k,- — 2:2, n,- and by P = L 63L the 5U(2) bundle
with c2(P) = k. Let F N be a principal 50(4) bundle associated to a tangent bundle
TN, and let us define Fr = P X F N to be a fibered product of the bundles P and

N
F N over N. Then Fr is a principal 5 0(3) X 50(4) bundle over N. Following the
notation from [KoM] and [FM 2], for each n > 0 define Z, to be the space of gauge
equivalence classes of ideal ASD connections on 54 with standard metric that satisfy:

1. they are based at the south pole
2. they are centered at the north pole
3. they are concentrated in the 6 ball around the north pole

The notions of “concentration of energy” and “6 ball” in the last statement should be
understood in the following way: for a fixed annulus A contained in the complement
of the 6 ball, every ASD connection (.0 with the “energy concentrated in the 6 ball” can
be glued along A to another ASD conneCtion on the manifold N. The spaces Zn have
a natural 50(3) action that changes the framing at the south pole, and the 50(4)
action that rotates the sphere. In addition to that there is an R+ action on each Zn

22

corresponding to conformal contraction toward north pole. Thus there exists a space

Zn such that Z, = cZn. For everyj=1,2,. . . ,3 define 01,-: Fr X can. As
S0(3)xSO(4)

is proven in [FM 2] the neighborhood K of the reducible connection in the strata 5 t
is

(on x 1'1 G1,) /51 (5.4)

j=l

divided by the symmetry group permuting the Cl,- factors. The top stratum described
above fits into the above description for s = 0. In Theorem 4.4.2 of [KoM] it has
been shown that given a reducible connection w, for r = n1 + no + - - - + n,, different
strata of the form (5.4) can be glued together to form a smoothly stratified space
GP(w,r). This space allows the projection 1r : GP(w,r) —> E’(N) induced by an 51
equivariant deformation retraction onto the r—fold symmetric product E”(N). The
notation from Definition 5.2 extends to the case of a reducible connection in the lower
strata. Lemma 4.7.4 in [KoM] gives a description of the class ”(0) on the link of given
reducible connection:

Theorem 5.5 Let M be a link of reducible connection (.0 E GP(w, r). Then the class
p(0‘) E H2(M/51) is equal to

11(0) = WSW?) - (01(L),0') ' CM
where

1. CM is the first Chern class of the 51 bundle M —) M/51.
2. 2"(0) E H2(2"(N) ) is the class induced from the Poincare dual to 0’ in H2(N)

by symmetrization.

Similarly we conclude that
—4p(x)|M/51 = -47r"2‘(x) + Ci,

Next we want to apply the gluing theorem to “attach” connections on the Y— side
to the neighborhoods of reducible connections on N — side. Recall that the 5 0(3)
fibration Mio_k(Y,m) reduces to an 51 fibration when m is nontrivial. In that case
let Q denote this Sl—fibration. For the sake of uniform notation let Q be the whole
total space of Mio_k(Y,m) when m is trivial. Notice that even in the later case we
can still think of Q as of the total space of an 51 fibration with the base Q/ 51.

When the boundary value m is nontrivial, the basepoint fibration on the Y—side
is Q :1 5 0(3) and the basepoint fibration Up”: —1 UL" near a “reducible” connection

can be described through the fibered product of:

Q :1 30(3) 30(3) x K

\ /s.

S2

23

which gives the basepoint fibration Q :1 (5 0(3) X K) -—1 Q :1 K. This 5 0(3) fibration
reduces to an 51 bundle with a total space Q; (51 X K) = Q x K.

The case when m is trivial is even easier to describe. The 5 0(3) fibration over
U)? is:
QX (SO(3);K) anK

which again admits a reduction whose total space is Q x (51 :1 K ) = Q X K. i

5.2 Desigularization of Q SXI K

To make the future computations easier we want to “enlarge” the set Q :51 r(K) =
Q/51 x r(K), onto which Q X K deformation retracts. For any 51 bundle Q let us
denote by Q0: Q X D2, the disc bundle associated to a circle bundle Q. Consider a
projection 71': Q :5 M C —-> Q X K that 18 identity for every nonzero radius parameter

of the disc D2, and for the radius zero 7r( [q] [m])- — ( [q] r(m)). Here [.] denotes the
equivalence class with the respect to the 51 action. This map is well defined because
of the 51 equivariance of the retraction r. If a is any top dimensional cohomology

class in H“”"(Q;<l K, QSX1 M) then
<a, [<2 5x, K, 61> = <a, mac :1 1140,61» = (1‘0:ng 140,31) (5.6)

Thus we can evaluate the pullback 1r"‘a in the enlarged space Q X M C, getting the
1
same answer. 5

5.3 Closing neighborhoods of reducible connections

Next we wish to describe a procedure that will keep the intersection of divisors in (5.1)

inside neighborhoods of reducible connections. Below, we describe compact spaces

C containing neighborhoods Q x M 0 together with a technique for translating the
l

intersection problem from U ,2" into the sum of these compact sets.

In case m is trivial, let N denote the based moduli space MiN(N , m) and G =
5 0(3), and in the case when m is nontrivial, let N be the 51 reduction of the above
basepoint fibration and G = 51. Then the set UL" = Q EN.

Let M),(N,m) = N/G contain 51 reducible connections A1,A2,. . . ,A,, each
contained in the compact nieghborhood K,/51. In the set UL" each A,- is contained
in a neighborhood of the form Q X K,. Let q,- : Q X M,- -—> Q/51 X M,-/51 denote

$1 $1

projections of the boundary of the above neighborhoods. The cohomology classes 11(0)
and p(x) can be obtained from pullback via q,- of the first Chern class of the bundle
M,- —+ M,/51. Since the dimension of Q/5l X M,-/51 is 2 less than the dimension

24

of Uf‘, the top dimensional cohomology class will evaluate trivially on it. Thus it is
reasonable to compactify Q :1 M? as:

C,- = Q x Mf’ u Q/s1 x M,/Sl (5.7)
S] QilQ X Mi
51

The space that we added to Q X M? is homotopy equivalent to either of the two
51

spaces: QC X M,- or Q X Mic. Thus the projection may be replaced by an inclusion
SI SI

giving:
. ,.. _c c , .
C, _ Q; M, Q >L<1JM.-Q :51 M, (5.8)
8
~ .0 .0
_ Q 5).: M, Q ’EJM‘ Qg M, (5.9)
5'

To define the class p(a) we have to pull back the bundle M,- —» M,-/51 over the
space Q X M,, thus it is more convenient to use the construction (5.8). Then [1(0) is
31

a multiple of the first Chern class of the 51 bundle:

c C , C C .
QXM‘ gym-Q XM’TQsfiM' cyan-Q 51M
51
The total space of this bundle is the boundary of Q0 X Mic, which can be considered
as the 53 bundle 5(Qc X M?) over Q/S1 X M,/51. Thus C,- is P (Q0 X MC),
the projectivization of the bundle Q0 X Mic. The first Chern class of the bundle
5 (QC X Mic) —> P (QC X M?) is the class h that restricts as a generator of each

fiber CP1 .

The next two lemmas justify our choice of the above compactification:

Lemma 5.10 Let w be a fixed divisor in Mk(N,m). Let M“ be a complement of
the sum of the neighborhoods LI,- K,/51. Then there is a one-to-one correspondence
between divisors V on UL” that pull back from the divisor w, and collections of divisors

V,- on C,- such that V,fl(Q/51 X M,/51) = [Q/51]X [13,-], and (131, . . . ,i},) = 6[w]. Here
6 is the boundary map a from the Mayer— Vietoris sequence:

-- —> H.1M‘) ea 3114K.) -> H.(M1(N.m)) 3» <19 Ham/5‘) -—»
Proof: Given the divisor V C UL" that pulls back from w C Mk(N,m) define

v,- = w] M, )51 . Using the deformation retractions r,- : C,- \ Q X M? —» Q/5l X M,-/ 51,
51

we can construct the divisors V,- = VIQ x Mr: Q UM r‘1([Q/51] x v,).
51 ' x i

51

25

Going in the other direction: given a collection of divisors (V,, 13,-) define

V: {V U -110
a IQ,M,C,,Mip~( I)

51
where pN : Q X N —> N/G is the projection, and w] is a restriction of the class w
G

onto the set N/G \ (LI,- pN(Q X M,-0 )) . It can be directly seen that the composition
51

of these two assignments gives the identity. [3

Lemma 5.11 Let w1,. . .,wr be fixed divisors on Mk(N,m) and V1,. . . , V, denote
pull-backs of w,- ’s to UL". Assume also that the sum of codimensions of V,- ’s is equal
to the dimension of UL”. Then the following signed intersections are equal:

8
r r e
:91 V; _ 2.21 V12

i=1
where V,,- is the divisor on 0,- corresponding to V,- as in Lemma 5.10.

Proof: Let V},- be a perturbation of V,,,- into transversal position. Since Q/5l X
M,-/5l has codimension 2 in C,-, the one dimensional set {61 V},- | t E [0,1]} misses

Q/51 X M,-/ 51, the set added in the compactification. As 6[w,~] = (13,-,1,.. . ,v,,,),
and this is a homological condition, we can use Lemma 5.10 for all V3,, to get a
corresponding perturbation of the divisors V3,. Because the dimension of Mk(N, m)

is lower than the dimension of U L" the perturbed divisors cannot intersect outside the
set Q x (N \ (LI,- K,)). Thus the intersection number of V,- is equal to the intersection
51

number of V,,. D

The top stratum of the moduli space M kN(N , m) has a reducible connection on
the bundle L if and only if c§(L) = kN. Thus if the reducible connection exists, it is
unique. Similarly on the stratum in which the background connections have energy
kN — 3 there exists a unique reducible connection if and only if c§(L’) = kN — 3.
Thus every reducible connection A,- E M kN(N ,m) has uniquely assigned 1,, the first
Chern class of the reduction of the bundle L,- and s,- = kN — c§(L,-). We can perform
the closing—up construction for every link M,- of A,- separately getting the following
equivalent of Theorem 5.5 for the spaces 0,:

Corollary 5.12 Under the above correspondence of cohomology classes we have the
following equalities in H‘(C,-):

11(0) = 71‘3” (0) - 1,- ° hi

where the class h,- restricts as the generator of each fiber CP2 H P(QC X MJ-C).
Similarly

—41r‘2’i(x,) + h? ifr S i

c2, ifr > i

-4#($r) = {

26

We will need some properties of the characteristic classes of the bundle P(QC x
M C) which we now evaluate. The Kiinneth formula provides an embedding

H*(M/S'1)—> H"'(Q/Sl x M/Sl)

that sends x E H‘(M/ 51) to 1 (8 x. In the description that follows by cohomology
classes x E H‘(M/51) or y E H‘(Q/51) we shall mean their images 1 (8) x or y (8)1 in
H‘(Q/ 51 x M/51). From the definition of characteristic classes we get the relation
h'2 = c1(QC X MC)h — co(QC X MC). Notice that c1(QC X M0) = cM + cQ and

. 1
c2 = chM, which in vector notation (a, b) = a-h+b can be wrltten as h2 = A 0 for
A = ( cM + cQ 1
—chM 0
Lemma 5.13 We have

). By elementary computations we obtain the following lemma:

n+2 n+2 n+1 n+1
C - C C '— C
hn+2z L—i—Hh—L—LCMCQ
CM — 60 CM - CQ

Proof: Using the expression for h2 we get:

h-(ah+b)=ah’+bh=[a(cM+co)+b]-h — a-cMcQ=A(:)

n+1_ n 1

h —A (0

1 1
S=—(. —c.)

A direct check verifies the relation:
A = s( ”Q 0 ) s-1
0 CM

A"=S(Cg (.).)S-1
0 cM

Thus we have

Set

thus

from which the result follows. CJ

The situation is different when the reducible background connection is trivial. In
that case there is no direct relation between p(o)2 and p(x). We need to supply a
further discussion with some more notation. Notice first that whenever the moduli
space M k(N , m) admits a trivial background connection, then k must be an integer
and m = 0. Thus on the Y side we always have the 50(3) fibraton Q = MfiY(Y,0).
Let M be a restriction of the basepoint fibration over the link of the trivial connection
M/50(3) C M),(N,m) with the projection 1r : M/50(3) —» 2"(N).

We have the following analog of Theorem 5.5:

27

Lemma 5.14 We have the following equalities of classes in H‘(M/50(3)):
11(0) = W'E’(0)
#(931') = WE’RE) - 2PM
where PM is the first Pontrjagin class of the fibration M —> M / 5 0(3).

For any 5 0(3) fibration Q define QG = Qsdl )c50(3) and let pQ denote its first
3
Pontrjagin class. A similar construction to (5.8) produces an 5 0(3) fibration:

6(QG x W) _. 6(QG x PG) (5.15)
50(3)
which we shall'denote by 5g(QG x 170) —1 Pa(QG x ’PG), or even 50 —1 PG when
the two other 5 0(3) bundles will be obvious from the context. Let H denote the
Pontrjagin class of this fibration. As in the case of 51 fibrations, there is a further
projection P0(QG x MG) —1 Q/G X P/G’, whose fiber is 250(3), the suspension of
5 0(3). The fibration 50 —> PG restricts over each fiber of PG as the fibration:

50(3) * 50(3) —. 250(3)

and the first Pontrjagin class is 2. As before, the computations of the intersections of
the dual classes in the neighborhood of the trivial connection can be translated into
the intersection of corresponding divisors in Pg(QG X M G). As in the case of 51 re-
ducible connections, there is a correspondence between the classes that pullback from
the moduli space containing M / 5 0(3) and the classes in the compactification (5.15).
Using the above correspondence and Theorem 5.5 we get the following corollary:

Corollary 5.16 Under the above correspondence of cohomology classes we have the
following equalities:

11(0) = n’E’(0) (5.17)

where the class H restricts as twice generator of each fiber Pa(QG X M G). Similarly

. _ —47r‘2’(x,-) + H ifj Si
"4"” ‘ {p5 ifj > 2'
It follows from the corollary that in this neighborhood, the top power of the classes
p(0) evaluates trivially, so it seems that these neighborhoods are redundant. But our
method of computing 11(0)“? on the whole moduli space M1,, (X) requires evaluating
in PG(QG X MG) terms of the form 11(0)“ -p(x1)"’1 - - - p(x,)°“ for various 0, 01, . . . , 0,,
which may give a nontrivial contribution in Pg. 30 let us state the following analog
of Lemma 5.13:

Lemma 5.18 With the notation above we have:

+2 +2 +1 +1

'PPP
PM - [’0 PP - PQ Q

28

Proof: As 51 C 5 0(3), then there is an 51 fibration Q -—> Q/51 with the first Chern
class cQ E H2(Q/51). Let p : Q/S'1 -—+ Q/50(3) denote the projection of the fibration

with the fiber 52. Then since p‘ (Q X C3) = Q X C3 we have:
30(3) 51

'_ ="‘ 3 = " C3 = (33=—2 5.19
p( m) p (02(QSO><(3)C )) c2 (1» (0,3,3) )) c219,, ) c5 ( )
Observe also that for every fiber 52 C Q/51, c2[52] = 2. Thus from Leray—Hirsh
theorem we get that the map (H°(52)€BH2(52)) ®H*(Q/50(3)) —+ H‘(Q/51) sending
i‘(cQ)®a —> p“(a)UcQ is monomorphism, and in particular also p" is monomorphism.
After these general remarks about 5 0(3) fibrations let us come back to the con-

struction described in the Lemma. Let h denote the first Chern class of the bundle
8(QG X 790) —> 0(QG X ’1’"). Then using Lemma 5.13 and (5.19) we have:
31

 

 

 

 

 

 

p'(H) = 112 = (h(CQ + 613) — CQcp) (5.20)
Thus: 4 4 c3 c3
p“(H") =h“ = CP'CQ -h — P- Qcpcq =
cP—cQ cp—cQ
c1 -.4, c1: —c5
- h c + c — - c c
C?) _czq ( P Q) CP __ CQ P Q
which by (5.20) can be written as:
at — c5 . c1» — c5
(fig—C22 .[p H+cPcQ] — Cp—cQ .cch_

= p‘(H) - (c?) + cg) + (CPCQ) - l-CPCQl = Pi (HIP? + Pol — PPPQ)

As p‘ is a monomorphism, the above proves the same relation that we had in Lemma
5.13 with Chern classes replaced by Pontrjagin classes. Thus the result follows. Cl

Thus in this paragraph we have established the relations between the class 11(0)
and Pontrjagin classes of basepoint fibration in each of the sets U)? separately. The
next chapter deals with the problem of matching these descriptions together.

5.4 Compactifications of the U1"

The results of the previous paragraph are based on the assumption that the spaces
M/51 and Q / 5 1 are compact. However this assumption is not valid in general. Even
after taking the Uhlenbeck compactification, the sets

U1“ = 1.0) 3,; MW)

29

still remain open. This is because there are connections whose energy may escape
through the cylindrical end. We want to describe the boundary of the sets U ,2" con-
taining such connections. As we shall see in the next section, we need to understand
the characteristic classes of the bundles over UL“ that over the boundary 8U)? pull
back from certain set r(U) that has the dimension at least 2 less than dim U ,5". Let
us denote by UL" the compact space U1," yr(U), where p : 3U)? —) r(U). Since there

is an isomorphism of cohomology groups H‘°”(U,T) 9: Ht°P(U,’,",8), we shall refer to
our construction as either compactifying the spaces UL" or as to splitting the global
characteristic classes into the sum of relative ones.

We shall begin the description of the connections in the boundary of U ,2" with the
generalization of the definition of based connections. Let us recall that if MAX)
denotes the space of ASD connections in A, then MflX) = MAX) 0X E,. Thus we

E

can similarly define a moduli space of connections based at two points as

Mi'”(X) = MAX) 39w, x 13,) (5.21)

Note that MZ’”(X) can be also described as a fibered product MflX) M)?” MZ(X).
k
The projection Mi’y(X) -> MflX) is in fact the quotiening Mfi’“(X) by 50(3),,

acting on By. We shall use this definition for the moduli spaces of connections on the
cylinders R x L based at the different ends of R. We indicate this by ”MflR X L).
Similarly, for a finite interval [a, b] C R we can define the space z./\/1Z([a, b] X L) to be
a space of connections over the cylinder [a — 1, b + 1] X L having small energy on the
subcylinders [a — 1, a] x L and [b, b+ 1] x L containing the points x and y.

The noncompactness of U ,2" is a direct consequence of noncompactness of the
spaces MkY(Y) and M),(N), so it is sufficient to describe the compactification of
these factor spaces. Recall that in order to glue back these factor spaces we need to
work with spaces of connections having small energy on the fixed cylinder 12. We shall
assume this throughout this paragraph. Let us focus first on the case of compactifying
the space M kY(Y). The boundary component corresponding to the splitting at one
region can be described as the fibered product:

MiY(Y,m,-) )g ”Mada — 1, b +1] X L, [m,-,m])/ (50(3),, X 50(3),) (5.22)

The first Pontrjagin class of the 50(3), action defines the point class —4p(x). The
total space of this 5 0(3)x-fibration needs to be glued to the space M£N(N, m) to get
the compactified set UL". The time translation parameter of the space

Mina“ - 1,1» +1] X L, [m,m'])

is the collar parameter of the fibered space M k(Y, m). Denote by M3,,([a — 1, b+ 1] X
L) = Mk,([a — 1,b+1] X L)/R and let

p = M1_(Y.m.-) g; M21(T,1m.-,m1)/SO(3). —»

30

—* M),_(Y,m,-) X M],,([a —1,b+1] X L, [m,-,m])

be the projection. We define the compactification of M kY(Y, m) as
Mammy [Mum-n.) x MS..([a — 1, b +11 x L,1m.-,m1)]

Notice that the space that we have added in this compactification has codimension
at least 2 in Mia/(Km). (One dimension is “lost” while dividing by time trans-
lations, and we also “lose” the gluing parameter between Y and the cylinder T).
Thus we are in the situation described at the begining of this section, which allows
us to define the compactified space MkY(Y,m), or equivalently the relative class
H‘°”(M;,,,(Y,m), 6(MkY(Y, m)) ). We shall use the same symbol Mky(Y, m) for the
compactified space. We define the classes p(x,~) on this compactification as pull-
backs of the corresponding classes on the product M 1,_(Y, m,-) x M5,,([a — 1, b + 1] X
L, [m,, m]). Notice that with the class p(y) for y in the splitting region, pulling back
from Mi_(Y,m,-) or M2,”([a — 1, b + 1] X L, [m,-,m]) gives the same result.

While applying the same construction to the moduli space M),(N,m) we get a
similar splitting of that moduli space into factor spaces MzN(N, m’) and the mod-
uli space on the nearest cylinder M%T([a — 1, b + 1] X L, [m’,m]). In this situation

ZN(N, m’) is not an 50(3) fibration because of the presence of reducible connec-
tions. Thus we shall apply the construction similar to (5.8), relating the classes p(0)
and p(x,-) to the Pontrjagin class of the basepoint fibration based at x, which is at the
end of N. This basepoint fibration based at x allows us to define the extension of the
boundary value map 8 : M£Y(Y, m) —+ 72(L), and thus the gluing of the compactified
spaces §Y(Y) and M}: (N) As a result we get a compactification of the set UL". In
general it may happen that the fibered product defining the boundary of M 1,(Y) has
a bigger number of factors. In this case we apply inductively the above procedure for
constructing the compactification of an unbased moduli space. This construction is
similar to the Floer compactification of the moduli spaces on the cylinder.

Now we want to ask the question if the relations between 11(0) and the point
classes proven in Corollaries 5.12 and 5.17 hold on the sets C,- after compactifying the
moduli spaces M),(N,m) and MkY(Y, m). With the definition of the compactified
11(0) as above, the answer is positive only when there is only one reducible connection
A,- is in the the set M1,, (N, m1). (We used the relation between this connection and
the Pontrjagin class of the basepoint fibration based at x to define the compactified 2
class 11(0).) Otherwise the cohomology class

11(0) — n’E’(0) — (c1 (L) , 0) - ha (5.23)

whose evaluation inside the set C,- is zero, has nontrivial support in the neighborhoods
of the other reducible connections A,-, j at i.

31

5.5 The relative Mayer—Vietoris sequence

So far we have been dealing only with local computations in a single open set U)?
coming from covering the part of the moduli space Mk(X) that contains the inter-

section of V,. 0161 V,. N ow we want to show how to combine these local computations
into a global one.

Assume first for simplicity that the whole moduli space M may be covered by
just two open sets U1 and U2. As the moduli spaces of connections on bundles
over R x L allow time translations, then from (5.22) we see that U1 (1 U; can be
written as (—2, 2) X V’, where V’ is codimension 1 submanifold of M. Let us put
V1 = U1 \ (—1,2) X V’ and V,» = U; \ (—2, 1) x V’ (thus V}’s are deformation retracts
of Ug’s, but they do not cover M). Then the cohomology group H*(M) is a part of
the relative Mayer—Vietoris sequence:

—’ H*(M1V1UV1’) TH‘(M1V1)€BHI(M1 V2) “*H'(M50) —’

II II II (524)
H-(EV') H“(U2,0U2) H*(U1,8U1)

It follows from this sequence, that any top dimensional class in H‘°P(M) is equal to the
sum of two relative classes in H‘(U1,3U1) and in H*(U2,8U2) modulo a “correction
term” from H‘(EV’ ).

In general, when the set M is covered by open sets U1, U2, . . . , Ur, let us denote the
complements of these sets by V,- = M\U,-. For any subset {i1, i2, . . . ,ik} C {1, 2, . . . ,r}
let V,,,,-, ,,,,, ,-,, = V,, U- - -UV;-,,, and Vil1i”°""* = V,, n- - ~flV,-,,. Then we have the following
generalization of the relative Mayer-Vietoris sequence:

Theorem 5.25 Let M be a manifold containing the submanifolds V1,...,V,., not
necessarily covering M. Then for every class 11 E H‘°P(M, K,2,,,,,) there exist classes
11,-, ,,,,, ,-,, E H’OP(M,V‘1"""*) for {i1,i2, . . .ik} C {1,2,.. . ,r}, such that:

(1‘ 1 [M1 V1.2 ----- fl) = :(-1)k+l Z (”imam-i1: 1 [M1 vi1,i2,...ik])
k {1,,;,,...i,,}

Proof: We shall prove this theorem by induction on the number of sets V,-. For
r = 2 the theorem follows from the relative Mayer—Vietcris Theorem described above.
Assume thus that this theorem is valid for all n < r. The sequence (5.24) for M1 = V1
and M2 2’ W 3

’ 00000

Ht°p(M, V1 U V2,3,...,r) —* Ht0p(Ma V1) EB Ht°p(M, V2,3 ..... r) —* Ht°p(M, V1,2,...,r) —* 0
(5.26)
Hence, using the inductive assumption, there exist classes 11,-, ,,,,, 5, E Ht°P(M,V1 U
V2,3,,,,,,.), where {i1,...,i,} C {2,3,...,r}, a class #1 E H(M, V1), and a class a E
H’°P(M, V1 U V23 ,) such that

II = #1 + XX-Ukl'l Z 1111,12,...“ — a
k

{”1'i2""”k}c
C{2,3,...,r}

,...

32

eeeee

eeeee

a E H"'(M, V1 U mer) under the excision isomorphism. Using again the inductive
assumption for the set U1 containing sets V,- we get

a = z:(-—l)""'1 Z 6111,12,...1',‘

k {‘1112»---‘k}C
C{2,3,..

where 01,-,,,-,,,,,,-, E H’°P(U1,V). By excision H‘°P(U1,V) c: H‘°P(M, V131"? """" f“), thus
a,-,,,-,,,,,,-,, correspond to p1,,,,,-,,_,,,-,,, which concludes the proof. Cl

Corollary 5.27 When the sets U1,U2,...,U,. cover the space M, then eyery class
[1 E H‘°P(M) is equal to a combination of the classes 11,-,,,-,,,,,,,-,, E H‘°”(U,-,,,-,,,,,,,-,,),
where U,,,,-,,,,,,,-, is a compactification of the intersection U,-,,,-,,,,,,,-,, described in previous
paragraph.

Proof: When the sets {U,}f=, cover M, then Vlgmr = 0. Thus from Theorem
5.25 it is sufficient to prove that H‘°”(M,V‘1""""‘*) 2 H’°p(U,-,,,-,,,,,,,-,). The exci-
sion isomorphism gives us H*(M, Vi1 ""°""*) 5: H*(U,-,,,-,,,,,,,-,,, 6). Recall that the com-
pactified sets U are of the form U UK, where K is a set of codimension at least

2. Let us denote CK = EU UK. Then we have another excision isomorphism:

H‘(U,-,,,-,,,_,,,-,,,0) 2 H’(U,CK). As the set CK deformation retracts onto K, which
is of codimension at least 2, then we have the last isomophism H‘°”(U, C K) 9.: H‘°P(U)
from the exact sequence of the pair (U, C K)- D

Let (I) denote the set of these classes 0 E H‘°P(Mk(X)) that are equal to a linear
combination of terms of the form a,,,,,,,_,,,p(0)°‘ - p(x1)"'1 - - - p(x,)°". The specific
compactification of the classes [1(0) and p(x,-) that we defined before allows us for
every (15 E (I) to define a “restriction map” r,-,,,-,,_,_,,-,, : Hi°P(Mk(X)) -+ H‘°P(U:,,;,,...,i,,)
such that d) is a combination of the classes r,-,,,-,,_,,,,-,,(¢) in the sense of Lemma 5.27.

The following lemma will allow us to avoid dealing with the classes r,-,,,-2 ,,,,, ,-,,(45) for
k > 1:

Lemma 5.28 Let M and V1,...,V,. be as in Theorem 5.25. Let p E H‘°”(M) be
a cohomology class such that p,- = 0 E H‘°”(M,V,-) for every i = 1,2,.. . ,r. Then
1“ = 0 E Htop(M, W,2,...,r)-

Proof: This is proved by an induction on the number of sets V;- as in (5.25). For
r = 2 the statement follows from the relative Mayer—Vietoris sequence. The inductive
assumption implies that 11 E Hi°P(M, V2,3,,,,,,) is zero. Thus the sequence (5.26) gives
us the lemma. Cl '

33

6 The proof of Theorem 1.3

The proof of Theorem 1.3 is based on the idea of replacing the classes 11(0) with some
appropriate point classes p(x,) by using relations described in Lemmas 5.12 and 5.16.
In other words we would like to find a linear combination of the form:

11(0)“ + Z “0.01 ..... 0511(010 ' 11(331)crl ' ' ' ”(33:1)“a (6°11

a<n

that would evaluate as zero on every set U,. In the above formula, as well as in later
notation, we shall often skip the factor 11(2) that appears in every term. Then from
Corollary 5.27 it follows that (6.1) is zero on the whole of M p(X ), thus giving the
main result.

We shall begin by defining a form that will evaluate trivially on a part of the
diagram J defined in (4.6). Let {U,-}f=, be the open sets of the cover corresponding
to the first column of J.

Theorem 6.2 Let 0 - 0 = —p, and let' 3 be the number of elements in x(L(p,1) ).

Let n > 23, and set r = "—333- when n is even, and r = L225}- when n is odd. There

is an a = (01,02, . . . ,a,) such that 0(a) defined by:

1(a) = 1(0)" — as#(0)2’u(x,)" + ,,_,,(,)2s—2,(,,_,).+1 + . . . + ,,,(,)2,(,,).+.-1
when n is even, and
(13(0) = 11(0)" - as#(0)2"‘#(x.)’ + a.-1#(0)2"3#(:c.._1)’“ + ~ - - + alu(a)‘#(x1)’+"‘

when n is odd, evaluates trivially on the set 1<l,J< U,.
_1_e

Proof: We shall give the proof in the case when n is even. The proof in the other
case is analogous. For convenience let us set p(x,) = —4;1(x,). Using Corollary 5.12
for the set U, we get the relation: 11(0)2 2 s2 - p(x,) for i S 8. Thus, in U, we get:

¢(a) = (32" + a, + gab, + . . . + —(32;3_1 (11) ”(0)28p($8)r

While considering the other sets U,- we have to use the following lemma, which is a
direct consequence of Lemma 5.13:
Lemma 6.3 Leti > k, and let the numbers a, ,3, and 7 be such that 7 + a Z
dimME(N, k). Then in the set U), we have:

P

#(0)7u($k)°’#($s)” = MOP/100°”

34

From this Lemma and the relation 11(0)2 = k2p(x,-) for i S k we get in Uk:

(2(a) = #(0)2k ((k2)r+s—kp(xk)r+s—k + (k2)‘""a, , P($k)"kp($s)' + . . .

+ Egg—{I’Wklrfl-kai)
= ”(0)2kp($k)r+s—k ((k2)r+e-k + (162)8-ka, + . , , + (k2;k_1a1)

Thus we get a system of 3 equations with 3 variables, whose matrix is the Vander-
monde matrix:

(32)“1 . .. (32) 1
1 1 1

whose determinant is 1119-9-54 j2 — i2). In particular, this determinant is nonzero,
thus we can always find the numbers 01,. . . , 0,. D

In order to remove the assumption n > 23 from the above theorem we need to
define the relative invariants, that have been introduced in [F32].

Definition 6.5 Let Y is a 4—manifold with a cylindrical end and MkY(I/, m) a com-
pactified moduli space. Assume that m is nontrivial, and that c denotes the first
Chern class of the 51 fibration 3‘1(pt). Then define the relative Donaldson invariant
Dy[m] : A(Y) —+ R by:

Dr[ml(z - C“) = (11(2) - 6”, Min/(Y1 m))

When m = 123 we set Dy[m](z) = Dx,,(z) to be a twisted 50(3) invariant. When
m = 0 we set Dy [m](z) = Dx(z), the usual invariant.

The class wo of that twisted invariant must evaluate as zero on Y and on N, thus
it is a unique Z2 class that comes from H1(L) in the cohomology sequence:

_; H1(L) —) H2(X) —+ H2(Y) ea H2(N) _,
Since the Z2 reduction of the class 0 satisfies these conditions, we have w; = 0.

Lemma 6.6 Let n = 2k S 23 (where s is the number of elements in x(L)). Then
there exist coefiicients a1, . . . ,ak._1 such that the form

431: = ”(0)21: - ak—lfl(a)2k-2P($k—1) - - -- — a1p(0)2p(x1)""1

evaluates on the sum of the sets Uf=,U,- as (2k)!Dy[k](z) in the case when k is a
trivial element of x(L), and evaluates as £1’—;°13D}z[k](1'z - c) otherwise.

35

Proof: Let W(k — 1) denote the k X (k —— 1) matrix

and let W(k — 1)j denote the (k — 1) X (k — 1) matrix obtained from W(k — 1) by
deleting j ‘1‘ column from the right. Then according to Theorem 6.2 for a ,- = W
the form 45), evaluates trivially on the sets U,- for 1 S i < k. 30 it remains to evaluate
43;, on the set Uk:

¢klUk = _(k2)k + ak_1(k2)k-l + . . . + alkz 2
k2
= det Wk 1 [_(k2)k_l ' det Wk-l + elk—.1(l€2)k"2 det W(k — 1)"’1 + . ..
+ a, det W(k _1)1] =
W 1
= "2 det k ‘ H (k2 - i2) = L2,?) (6.7)

det Wk“ — OSiSk—l

When k is a trivial element of x(L) and U), corresponds to the set with the biggest
amount of energy on the N -side that covers the intersection of divisors V,, then the
bundle Q apearing in the Lemma 5.13 is 50(3) —> 52 = 50(3)/51. As the first
Chern class of this bundle is equal to 2, then we have an additional factor of 2 in the
formula for ¢klUk in this case. [I]

To avoid dealing with separate statements for two different cases 11 > 2s and
n S 23 we shall introduce the following notation:

Definition 6.8 Define the Donaldson invariants needed for evaluation D(z0"):

2m k—2d—
Di(z0 xm+2d

D[m](c-z) ifk—2d—m=0.

m) {D(z02mx:,'f§’d'm ifk — 2d — m > 0.
0 ifk—2d-m<0.

when n = 2k, and

2 1 k 23 0(202m-1xin-4-22dd-m) ifk - 2d -— m > 0.
01(20 m_ aim—+2117”): D[m](z) if k _ 2d _ m = 0.
0 ifk—Zd—m < 0.

in the case n = 2k +1.

The particular subscript of xm+2d in the above formulas is taken from the fact that in
the (d + l)“t column of the diagram J the element with boundary value m lies in the
level xm+2d. Now we are ready to prove the theorem, from which (1.3) will follow:

Theorem 6.9 Let X be a smooth manifold containing an embedded sphere 52 with
self-intersection 02 = -—p. Let s be the number of elements in x(L(p,1)), and let

36

z E A(X) be a cohomology class such that z - 0 = 0. Then there exist constants a;
such that:

D(02"z) = Z ZaiDL(02mxk"2“‘m2) (6.10)
m=1 d=0
when n = 2k, and:
D(a”‘+‘z)= 2 2d am D.L( 0.2m—lxk—2d—mz) (6.11)
m=l d=0

when n=2k+1.

Proof: We first concentrate on the case when n = 2k. Let U1,U2,. . . ,U, be the
open sets that correspond to the first column of the diagram J (if there are fewer
than 3 sets in the first column, thanks to the definition of D _L we can assume that
we completed that list by empty sets). Theorem 6.2 and Lemma 6.6 imply that there
are constants 0?, a3, . . . , 02 such that

8
= Z: a9,01( 202mx" m)
m=1
evaluates trivially on the first column of J. Let J (1) be a diagram obtained from J
by erasing its first column. For every vertex of J (1), like in (4.6) we can assign an
open set in Mk_1(X) (the moduli space of connections on X, with one charge less
than the space that we started with.) Repeating the argument from Theorem 6.2 we
can show that there exists a polynomial

¢1 = 0.11/40 )Z’Il(xs+2)r‘1 + 01.11107 )28-2#($s+1)' + ° ' ' + “111(0 )2#(33)r+8-2
+ aifl($2)'+’-l

such that 0° — 01 evaluates trivially on the first column of J (1). The difference 0° — 01
should be understood in the following way: as the diagram J (I) has been obtained
from J by erasing some of its elements, then we can say that there is an embedding
J (1) H J. Then 01 we evaluate on the open set corresponding to a vertex of J (1),
whereas the 0° we evaluate on the corresponding set of J. Repeating Theorem 6.2
for consecutive diagrams J (i) we prove the theorem. Cl

Proof of Theorem 1.3: Lemma 6.6 with Theorem 6.9 implies that the relative
Donaldson invariants Dy[k] can be expressed in terms of D(02") and D(02-lx"‘j) for
j < k. Thus all relative invariants in 6.10 can be replaced with invariants defined on
X, which completes the proof. Cl

7 Examples of computations.

The theory that we have developed gives more than just Theorem 1.3. By using the
results of this thesis we can compute the actual coefficients in some of the formulas
given by the (1.3). Below we give some examples of such computations.

37

Example 7 .1 Let 0 be the cohomology class of an embedded 2—sphere with self—in-
tersection -4. Although Theorem 1.3 does not cover the case of D(z0“), the only
relative Donaldson invariant that appears in Theorem 6.9 is a twisted 5 0(3) invariant
with 1.02 = 0, that is why we get the formula of the form:

D(z04) = A - D(z) + B - D(z02x) + C - D,(z)
We have already seen that the diagram J in this situation is
‘ 1

l\o

and let us denote the corresponding open sets by U1“, U12 and U9. Then from
Lemma 6.6 we have that 0° = ;1(0)4 — p(0)2p(x1) evaluates trivially on U11), Since
the coefficient B is obtained from the term p(0)2p(x1) = —4/1(0)211(x1), then we have
B = —4. The coefficient C we obtain by evaluating 0° on U3. The dim M 1(N , 2) = 5,
thus the neighborhood of the reducible connection is modelled on C3/51 = CPz.
From Lemma 5.12 it follows that U,2 = P(50(3)C X 720), where 7,, —» CP" is a line
bundle with c1(7,,) = [CPI]. If h is the generator of the fiber CPl of P, then we

have:
11(0) = 2h; p(wl) = h”

Thus in U,2
11(0)4 - #(0)2P($1) = ic([50(3)/Sll) ' (24 — 22) ' 117(2) = i24 ' Da(2)
Here 0 = 2[52] is the first Chern class of the fibration 50(3) -+ 50(3)/51.

In order to determine the sign in the above equation we have to refer to the orien-
tations of moduli spaces defined in [D2]. By using a complex structure on the elliptic
complex

0°(TX (g) L) _. 01(TX 55 L) _. 111(TX 55 L)

where L —-> X is a complex line bundle, one can assign an orientation 0(L) to a
reducible connection respecting the splitting L 63 L in the case of 5 U (2) connections,
or respecting the splitting L GB R in the case of 5 0(3) connection. The Example 4.3
from [D2] worked out for an arbitrary reducible bundle L EB L gives:

Claim 7.2 The orientation 0(L) of a moduli space in the neighborhood of reducible
connection on a bundle E = L EB L is —1'z/\ (standard orientation on CP"), where
it is the normal pointing away from the reduction. This orientation compares to the
standard one with a sign (-1)CI(L)2.

For the connections over the space N we have to change the sign in the above
Claim, due to the fact that N has the orientation opposite to the complex one.
The decomposition X = Y#N yields a splitting of arbitrary line bundle L into

38

L ’5 Ly #LN, where c1(L) = c1(LN) on N. The neighborhood of reducible connec-
tion on LN EB LN is described as 5 0(3) :5 C“ has the orientation 0(L). An arbitrary

connection may be related to a reducible one through additions and subtractions of
instantons. The set U,2 is a neighborhood of the reducible connection on LN EB LN
with c1(LN) = 2[52] E H2(N). The associated 50(3) connection is defined on
Lg, 69 R. Thus as a bundle L on the whole X we can take the line bundle with
c1(L) = —PD[52]- — —0, since —0[52 ]- — c1(LN )- — 4. Thus the complex orientation
on U,2 compares with the standard one with the sign (- -1)"— - (— —1),4 giving the “”+
sign at D,(z).

There are no 51 reducible connections inside the set U? but there is a trivial
one. Following the discussion in [Y], the restriction of the basepoint fibration to
the link of the trivial connection in M1(N,0) is F r+(N ) —» N, where F r+(N ) is the
associated 50(3)-principal bundle of AiTN. Then U? = Pa(50(3)G x Fr+(N)G),

and following Corollary 5.16 we have in this set:
#(0) = 1r"'(0); P($1) = -4W’($1)+ H

where H is a generator of the fiber 250(3) of Pa(50(3)G X Fr+(N)G). Thus 0° =
—H[250(3)] -02[N] = —2- (—4) = 8. The additional “—” sign has to be added again

due to the anti—complex orientation of N. Putting all three terms together we get

Theorem 7.3 When 0 is a class of 52 with self-intersection —4, then the following
formula holds:

D(204) = -—8 - 0(2) + 24 - D,(z) — 4 - D(z02x) (7.4)

7 .1 Second stratum connections over CP2

The next example requires the description of the reducible connections in the second
stratum of the moduli space. The following presentation is based on [Y, Oz]. Let
(.0 be a reducible connection on the bundle L EB L over X with cL = c1(L). Assume
that n is such that the neighborhood of w E Mcz (X) 18 modeled on C" / 51 Then
an open neighborhood of {w} X X in the compactified moduli space Mc2+1(X) is
homeomorphic to the bundle:

Cu 3:, (PL x (Fr+X SOX(3)CSO(3) )/ U(1)) (7.5)

X

By the symbol P if Q we have denoted the fibered product of two fibrations P and Q

over X. (the pullback of Q over P.) Let PL denote the principal 51 bundle associated
to L, on which U ( 1) acts in a natural way from the right. The U ( 1) action on c5 0(3)
is a square of the right action of 51 H 5 0(3) via the map

1 0 0
exp(i9)—> 0 c080 sin0

0 —sin9 c030

39

For any complex line bundle L we have L X 5 0(3) 2 L2 :5 5 0(3), where the 51 action
SI 1

on 50(3) is the square of the 51 action. Thus in what will follow we shall replace
the fibration PL with Pp getting the standard U (1) = 51 actions on all bundles
appearing in the formula. The stabilizer 5§T acts in the standard way on C" and on

the left on the space PL 3: (F r+X SOX(3)c50(3) )/ U (1) Thus in the neighborhood
of {10} X X the bundle M —1 M/51 used in Theorem 5.5 is:

M = S (0" x (P12 § (Fr+X 55,3)c50(3) )/S‘)) -

_. p (on x (P, 3, (Fr+X $513050“) )/51)) (7.6)

where the projection is the quotient by the 551‘ action. In the above formula we have
used the symbol “5” even though the fiber of the boundary of

n 1
C x (PL: if (Fr+XSOX(3)cSO(3) )/S )

is not a sphere, but 52’“1 at 50(3), a space that is double covered by 52"”. For
later computations we need to evaluate the powers of the first Chern class CM of the
fibration M —+ M/Ssi'r on M/Sl.

When the 5 0(3) fibration F r+X lifts to an 5 U (2) fibration Er+X (i.e. when 1.02
of the manifold X vanishes), then, following [Oz], we have an 51 equivariant double
cover map:

” 1 1
PL 3(((Fr+X 513((2)05U(2) )/5 —> Pp 3(((Fr+XSOX(3)c50(3) )/5

where 51 acts in the standard way on 5 U (2) The fibration

~ 1
PL 3; (Fr-,.X Slfiflcsmz) )/3

can be orientation reversing identified with the sphere bundle of the complex vector
bundle L‘1 (X) Aic with the complex orientation. For the whole space M/S1 we have
the following:

Lemma 7.7 (Compare Proposition 2 in [02]) If X is a four manifold with w; = 0,

then there is a fiberwise covering map p of degree 2"

p: P (0" 613(12 o 1110)) .. p (on x (PL. 3, (Fr-,.X ails-1650(3) )/51))

Moreover p*(cM) = —2hp, where hp is the positive generator of the fiber CP"+l of
P (on o (L s 211(3)) —. X.

40

Proof: In order to make the lift an 51 equivariant we need to lift the 51 action on
PL: and on each fator of C" = c(§'l * - - - =1: 5:). This gives us a 2"+1 cover of the

 

n times
fibrations before projectivization. Dividing by the 51 action “cancels” one factor of

2.
On the fibers the map p is

S2n+3 A 5271-1 * 50(3)

7

GP“ P (521-1 ... 30(3) )/51

A
7

By using technics similar to the desingularization of the moduli space (5.6) and
Theorem 5.13 we get:

(6111, (52“1 * 50(3) )/51) = (CnM+19P(SZC X 30(3)C)) = 65013152] = 2
Hence
(1)1634“), CPW) = Twill”, (13'2“l * 50(3) )/31) = 2"+I(7"+1,CP"+1)

giving the second conclusion modulo the sign. The minus in this formula is due to the
fact that the identification of 2" cover space with P(C" EB (L ® A1,C)) is orientation
reversing. Cl

Now we need to adjust the above results for the case of the space N E CPn/Zp.

7 .2 Second stratum connections over the orbifold N

Let T : CP1 —1 N denote the quotient by Z, action defined by 7'[zo : 21 : zo] =
[exp(27"‘)zo : 21 : 22]. This action has a fixed sphere {zo = 0} and one fixed point
{21 = z; = 0}. The quotient space N is not a manifold, having singularities at the
fixed points of 1. Near the fixed point sphere N is diffeomorphic to —p degree disc
bundle over 52, thus the singularity there can be resolved. The neighborhood of the
fixed point is modelled on the open set in R" divided by Z, action with a single
fixed point at the origin. Such spaces are called orbifolds. We will need the following
properties of N:

Lemma 7.8 Let 0 E H2(N) be a cohomology class Poincare dual to the sphere with
self—intersection —p. Also let x E H4(N) denote the generator of this group. Then
we have

T'(0) = m; 7"(3) = m2
where 7 E H2(CP2) is a generator of this group.

41

Proof: For the generator :1: E H4(N) we have r*(x) = p72, thus proving second
equality. Also we have:

(7100,5162) = P' (USN) = ~10? = ((imYfifi)

Since PD(0) is a fixed by the action of T, then we have the “+” sign in the first
equality. El

Over the orbifold N the line bundles L and the bundle A1(N ) are defined as
Z, quotients of the corresponding bundles over W. Thus if hN denotes the class
obtained by quotiening the generator h of the fiber CP2 of P(C€B(L®A1’C)) —+ C133,

then we have:

(112070), P(C EB (13 (8 A10)» = (11117110), N) = £010,512?) = (hm—(7P7) (7-9)

and

(hire), P(C EB (13 69 A13)» = (7"(2), N) = (72.33?) (7-10)

In particular this answer does not depend on the intersection number p. Now we are
ready to come back to the computations of next examples.

Example 7.11 For the 0 with the self—intersection -4, same as in the previous
example, we want to increase the number of the classes 0 appearing in the Donaldson
invariant. According to Theorem 1.3 we expect the formula of the form:

D(20°) = a1 D(z02x2) + aoD(z04x) + a3D(zx) + a4D(202) (7.12)

Each term in the above formula corresponds to different open set represented on the
diagram:

i\.
\1

Solving the system of equations described in Theorem 6.2 gives us that the form

050 = #(0)6 - 5#(0)4P($2) - ("4)Il(‘7)217(-7-‘1)2

evaluates trivially on the sets U11), and U12. This shows that the coefficient 01 =
—4 - (—4)2 = —64, and a2 = 5 - (-—4) = —20. Again it remains to evaluate 0° on U?
and U51), Similarly like in the previous example, in U? we have

3° = 4 . 52w] . H(250(3)] -p($1)[Mk(Y, 0)] = 4 . (—4) . 2 . (-4)D(zx) -_- 1280(25)

42

This gives us the coefficient at D(zx) with the sign determined like in the previous
example. Since the graph J terminates on one level below the set U,°, then we also
get 01 = —32p(x2). In U51), we have:
11(0) = “‘10) - 11; p(wi) = -47r‘($i) + h”
Thus:
3" = («‘(a) - hr — 5 («*(a) — hr- (-451...) + 1.2) + 4 («‘(a) — h)?-
(—47r"(x,-) + h“)2
= h6 — 6h5n‘(0) + 15h“(1r"‘(0))2 — 5(h4 — 4h31r"(0) + 6h2(7r"(0))2) -
(—47r"'(x,-) + h?) + 4 ((7r’(0))2 — 2515);. + 52) . (54 — 852515;,»
= 155515) (-—6 + 20 — 8) + h47r"(x,-) (1552 + 20 - 3052 + 452 — 32) (7.13)
= 6h51r‘(0) + 32h41r"(x,-)
The set U51), is P(Q$ x M C), where M is the neighborhood of the connection (11 X N
described in (7.6). Thus

(1157110111554) = (5,, 2y) - <h3r(a),M> = 0(02) . __;_ - (—2)3 . «mew

Here CF, = c1(L (8) A10) = —27 and the extra minus in the front of the whole formula
is due to the fact that the identification of the 2" cover of M and L <8) A10 was
orientation reversing. Thus the contribution of the term 6h57r“(0) in U51), is 48D(02).

Similar computations give (h41r*(x), U51“) = -2D(02). The contribution of 01 in U1),
is equal to D(02), thus adding these terms together we get:

D(z0°) = —64D(z02x2) — 20D(z04x) — 128D(zx) — 48D(z02)

Example 7.14 As the other way of generalizing the above considerations we shall
prove that for 0 - 0 = —6 we have

D(z0°) = —192D(zx) — 64D(202x2) — 20D(z04x) - 108D(202) + 720D,(z)
The graph J for this situation is

i\.
l\l

and as before the form 0° = 11(0)6 —5p(0)4p(x2) — (—4)p(0)2p(x1)'2 evaluates trivially
on the sets Ui/s and UZ/G, giving the coefficients at D(z02x2) and at D(20“x). In U3),
we have:

¢° = 1452] . (36 —5-3“ +133)= 2(729 -405+36) = 720

43

which gives the constant at D,(z) and 01 = —48p(x2). Repeating the reasoning from
the previous example we get that 0° = 4 - (—6) - 2 - (—4)D(zx) = 192D(zx) in U9.
From (7.13) in U71“, we have:

0° = 6h57r*(0) + 54h41r'(x,-)

As the pull—back over CP2 of the link of the reducible connection in Ui/s does not
depend on the self—intersection p of 0, we can repeat the computations from the
previous example, obtaining:

l¢01U7I/S> + (<19, Ui/e) = (6 ' 8 + 54 ' (-2) - 48) ‘D(02) = 4080072)

BIBLIOGRAPHY

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