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288 9 4-Manifold Invarian t s

We now consider the special case of E =ad(P )=P ×

g, the Lie algebra

bundle associated to P with ﬁber g, and deﬁne

: Λ

(M,E) → Λ

(M,E)byd

:= p

◦ d

Now the curvature 2-form F

splits into its self-dual part (F

)

∈ Λ

(M,E)

and its anti-dual part (F

)

−

∈ Λ

−

(M,E). Thus, for a self-dual (resp., anti-

dual) gauge connection ω

−

◦ d

=(F

)

−

=0 (resp.,d

◦ d

=(F

)

=0). (9.8)

Using equation (9.8) and a suitable part of the above diagram we obtain the

fundamental instanton deformation complex (or simply the instanton

complex)

0 −→ Λ

(M,E)

−→ Λ

(M,E)

−

−→ Λ

−

(M,E) −→ 0

Similarly, one has the anti-instanton deformation complex (or simply

the anti-instanton complex). It can be shown that the instanton (resp.,

anti-instanton) complex is an elliptic complex. The formal dimension of the

instanton moduli space is determined by the cohomology of this complex. We

deﬁne the twisted Dirac operator

: Λ

(M,E) → Λ

(M,E) ⊕ Λ

−

(M,E)

(α):=δ

α ⊕ d

−

α, α ∈ Λ

(M,E),

where δ

is the formal adjoint of d

.ItcanbeshownthatthetwistedDirac

operator D

is elliptic. We now compute its analytic index by studying the

cohomology of the original instanton complex. There are three Hodge–de

Rham Laplacians

1. Δ

:= δ

on Λ

(M,E),

2. Δ

:= d

+ δ

−

on Λ

(M,E),

3. Δ

:= d

−

and their associated harmonic spaces, which determine the cohomology

spaces H

, 0 ≤ i ≤ 2, of the instanton complex. We denote by h

the dimen-

sion of H

for all i.LetM

⊂A/G denote the moduli space of irreducible

instantons of instanton number k. For a self-dual connection ω the tangent

space T

can be identiﬁed with ker d

−

/ Im d

. Thus, at least formally

dim M

= h

. Under suitable conditions on M and the gauge group, one

can show that h

=0=h

and that M

is a manifold. Thus, the index

Ind(D

)=−h

+ h

− h

= h

equals the formal dimension of the moduli

space M

(M,G). This index can be computed by using the Atiyah–Singer

index theorem and leads to the following result:

9.2 Moduli Spaces of Instantons 289

dim M

(M,G)=2ch(P ×

)[M] −

dim G(χ(M)+σ(M )), (9.9)

where ch is the Chern character, g

= g⊗C is the complexiﬁcation of the Lie

algebra g, χ(M) is the Euler characteristic of M and σ(M) is the Hirzebruch

signature of M .IfM is 1-connected then b

=1=b

and b

=0=b

.Thus,

χ(M)=2+b

=2+b

+ b

−

and σ(M )=b

− b

−

.Thisgives

dim M

= c(P ) −

dim G(2 + 2b

)=c(P ) − dim G(1 + b

where c(P ) is a constant depending on P . For example, if P is an SU(N)-

bundle over M with instanton number k,then

dim M

=4kN − (N

− 1)(1 + b

In the particular case of N =2,weget

dim M

(M,SU(2)) = 8k − 3(1 + b

We shall use this result in our discussion of the Donaldson polynomial invari-

ants of M later in this chapter.

If M = S

then χ(M)=2andσ(M) = 0; thus formula (9.9) reduces to

dim M

,G)=2ch(P ×

)[S

] −dim G. (9.10)

We now use this formula to obtain dimensions of moduli spaces of instantons

over S

for some standard non-Abelian gauge groups.

Proposition 9.8 Let k denote the instanton number of the principal bundle

P (S

,G). Then we have the following results.

1. Let G = SU(n),n≥ 2.Then

dim G = n

− 1 and ch(P ×

)[S

]=2nk.

Thus, equation (9.10) becomes

dim M

,SU(n)) = 4nk − n

+1. (9.11)

Applying formula (9.11) to the particular case of G = SU(2) we obtain

dim M

,SU(2)) = 8k − 3.

For instanton number k =1, we get

dim M

,SU(2)) = 5,

corresponding to the BPST family of solutions. In the last section we have

given an explicit geometric construction of the (8k − 3)-parameter family

of instantons with instanton number k.

290 9 4-Manifold Invarian t s

2. Let G = Spin(n),n >3,whereSpin(n) is the universal covering group of

SO(n).Then

dim G =

(n − 1) and ch(P ×

)[S

]=2(n − 2)k.

Thus, equation (9.10) becomes

dim M

,Spin(n)) = 4(n −2)k −

(n − 1),k≥ n/2. (9.12)

For small values of n some of the gauge groups are locally isomorphic. For

example, Spin(6) is locally isomorphic to SU(4).Thus,(9.11)and(9.12)

lead to the same dimension for the corresponding moduli spaces in this

case.

3. Let G = Sp(n).Then

dim G = n(2n +1), and ch(P ×

)[S

]=2(n +1)k,

and we obtain

dim M

,Sp(n)) = 4(n +1)k − n(2n +1),k≥ n/4. (9.13)

We note that the groups SU(2), Spin(3),andSp(1) are isomorphic and hence

the corresponding moduli spaces are homeomorphic. Each of these groups have

been used to study the topology and geometry of the fundamental moduli space

of BPST instantons from diﬀerent perspectives (see, for example, [103, 172,

173, 181]).

9.3 Topology and Geometry of Moduli Spaces

Study of the diﬀerential geometric and topological aspects of the moduli

space of Yang–Mills instantons on a 4-dimensional manifold was initiated by

Donaldson [104]. Building on the analytical work of Taubes [364, 363]and

Uhlenbeck [385, 386, 387], Donaldson studied the space M

(M), where M

is a compact, simply connected, diﬀerential 4-manifold with positive deﬁ-

nite intersection form. He showed that for such M the intersection form is

equivalent to the unit matrix. Freedman had proved a classiﬁcation theorem

for topological 4-manifolds [136], which shows that every positive deﬁnite

form occurs as the intersection form of a topological manifold. A spectacu-

lar application of this classiﬁcation theorem is his proof of the 4-dimensional

Poincar´e conjecture. Donaldson’s result showed the profound diﬀerence in

the diﬀerentiable and topological cases in dimension 4 to the known results

in dimensions greater than 4. In particular, these results imply the exis-

tence of exotic 4-spaces, which are homeomorphic but not diﬀeomorphic to

the standard Euclidean 4-space R

.SoonmanyexamplesofsuchexoticR

9.3 Topology and Geometry of Moduli Spaces 291

were found ([157]). A result of Taubes gives an uncountable family of ex-

otic R

and yet this list of examples is not exhaustive. The question of the

existence of exotic diﬀerentiable structures on compact 4-manifolds has also

been answered in the aﬃrmative (see, for example [343]). Using instantons

as a powerful new tool Donaldson has opened up a new area of what may

be called gauge-theoretic mathematics (see the book by Donaldson and

Kronheimer [112], Donaldson’s papers [105,107, 109]and[136]).

We now give a brief account of Donaldson’s theorem and its implications

for the classiﬁcation of smooth 4-manifolds. In Chapter 2 we discussed the

classiﬁcation of a class of topological 4-manifolds. In addition to these there

are two other categories of manifolds that topologists are interested in study-

ing. These are the category PLof piecewise linear manifolds and the category

DIFF of smooth manifolds. We have the inclusions

DIFF ⊂ PL ⊂ TOP .

In general, these are strict inclusions, however, it is well known that every

piecewise linear 4-manifold carries a unique smooth structure compatible with

its piecewise linear structure. Freedman’s classiﬁcation of closed, 1-connected,

oriented 4-manifolds does not extend to smooth manifolds. In the smooth cat-

egory the situation is much more complicated. In fact, we have the following

theorem:

Theorem 9.9 (Rochlin) Let M be a smooth, closed, 1-connected, oriented,

spin manifold of dimension 4.Then16 divides σ(M), the signature of M .

Now as we observed earlier, 8 always divides the signature of an even form

but 16 need divide it. Thus, we can deﬁne the Rochlin invariant ρ(μ)of

an even form μ by

ρ(μ):=

σ(μ)(mod2).

We note that the Rochlin invariant and the Kirby–Siebenmann invariant

are equal in this case, but for non-spin manifolds the Kirby–Siebenmann

invariant is not related to the intersection form and thus provides a further

obstruction to smoothability. From Freedman’s classiﬁcation and Rochlin’s

theorem it follows that a topological manifold with non-zero Rochlin invariant

is not smoothable. For example, the topological manifold |E

| := ι

−1

)

corresponds to the equivalence class of the form E

and has signature 8

(Rochlin invariant 1) and hence is not smoothable.

For nearly two decades very little progress was made beyond the result

of the above theorem in the smooth category. Then, in 1982, through his

study of the topology and geometry of the moduli space of instantons on

4-manifolds Donaldson discovered the following unexpected result, which has

led to a number of important results, including the existence of uncountably

many exotic diﬀerentiable structures on the standard Euclidean topological

space R

292 9 4-Manifold Invarian t s

Theorem 9.10 (Donaldson) Let M be a smooth, closed, 1-connected, ori-

ented manifold of dimension 4 with positive deﬁnite intersection form ι

Then ι

∼

(1), the diagonal form of rank b

, the second Betti number of

Proof : We given only a sketch of the main ideas involved in the proof. The

theorem is proved by consideration of the solution space of the instanton ﬁeld

equations

∗F

= F

where ω is the gauge potential on a principal SU(2)-bundle P over M with

instanton number (Euler class) 1 and F

is the corresponding SU(2)-gauge

ﬁeld on M with values in the bundle ad(P). The instanton equations are a

semi-elliptic system of nonlinear partial diﬀerential equations, which are a

nonlinear generalization of the well-known Dirac equation on M. The group

G of gauge transformations acts as a symmetry group on the solution space

of the instanton equations. The quotient space under this group action is the

moduli space M

(M)(orsimplyM) of instantons with instanton number

1. Recall that the action of G is not, in general, free due to the existence of

reducible connections. The space M depends on the choice of the metric on

the base M . By perturbing the metric or the instanton equations we obtain

a nearby moduli space (also denoted by M),whichcanbeshowntobean

oriented 5-dimensional manifold with boundary and with a ﬁnite number k

of singular points. In fact, moduli space M has the following properties:

1. Let k be half the number of solutions to the equation

(α, α)=1,α∈ H

(M,Z).

Then for almost all metrics on M, there exists a set of k points

,...,p

in M such that M\{p

,...,p

} is a smooth 5-manifold.

The points p

are in one-to-one correspondence with equivalence classes of

reducible connections.

2. Each point p

has a neighborhood U

⊂M, which is homeomorphic to a

cone on CP

3. The moduli space M is orientable.

4. There exists a collar (0, 1] ×M ⊂M. Attaching M to the open end of the

collar at 0 we obtain the space

M := M∪M. This space is a compact

manifold with the manifold M appearing as part of the boundary.

The proofs of the properties listed are quite involved and we refer the reader

to the books by Freed and Uhlenbeck [135], Lawson [247], and Donaldson

and Kronheimer [112] for details. Using the above properties, one can show

that the moduli space

M provides a cobordism between the manifold M

and a disjoint union X of k copies of CP

. Since the intersection form is a

cobordism invariant, M and X have isomorphic intersection forms. Thus, the

intersection form of M is equivalent to the identity form. 

9.3 Topology and Geometry of Moduli Spaces 293

The above theorem, combined with Freedman’s classiﬁcation of topological

4-manifolds, provides many examples of non-smoothable 4-manifolds with

zero Kirby–Siebenmann invariant. In fact, we have the following classiﬁcation

of smooth 4-manifolds up to homeomorphism.

Theorem 9.11 Let M be a smooth, closed, 1-connected, oriented 4-

manifold. Let

∼

denote the relation of homeomorphism and let ι

denote

the intersection form of M .Ifι

= ∅ then M

∼

and if ι

= ∅ then we

have the following cases:

1. M is non-spin with odd intersection form

∼

j(1) ⊕ k(−1),j,k≥ 0,

and

∼

j(CP

)#k(CP

);

i.e., M is homeomorphic to the connected sum of j copies of CP

and

k copies of

, that is, CP

with the opposite complex structure and

orientation.

2. M is spin with even intersection form

∼

mσ

⊕ pE

,m,p≥ 0,

where





is a Pauli spin matrix and E

is the matrix associated to the exceptional

Lie group E

in Cartan’s classiﬁcation of simple Lie groups. i.e.,

⎛

⎜

⎝

2 −1000000

−12−100000

0 −12−10000

00−12−1000

000−12−10−1

0000−12−10

00000−12 0

0000−1002

⎞

⎟

⎠

and

∼

m(S

× S

)#p(|E

|).

That is, M is homeomorphic to the connected sum of m copies of S

×S

and p copies of |E

|, the unique topological manifold corresponding to the

intersection form E

Gauge-theoretic methods have become increasingly important in the study

of the geometry and topology of low-dimensional manifolds. New invariants

294 9 4-Manifold Invarian t s

are being discovered and new links with various physical theories are being

established. In this chapter we are studying their application to 4-manifolds.

We shall consider applications to invariants of 3-manifolds and links in them

in later chapters.

9.3.1 Geometry of Moduli Spaces

In the Feynman path integral approach to quantum ﬁeld theory, one is in-

terested in integrating a suitable function of the classical action over the

space of all gauge-inequivalent ﬁelds. In addition, one assumes that an “ana-

lytic continuation” can be made from the Lorentz manifold to a Riemannian

manifold, the integration carried out and then the results transferred back

to the physically relevant space-time manifold. Although the mathematical

aspects of this program are far from clear, it has served as a motivation for

the study of Euclidean Yang–Mills ﬁelds, i.e., ﬁelds over a Riemannian base

manifold. Thus, for the quantization of Yang–Mills ﬁeld, the space over which

the Feynman integral is to be evaluated turns out to be the Yang–Mills mod-

uli space. Evaluation of such integrals requires a detailed knowledge of the

geometry of the moduli space. We have very little information on the ge-

ometry of the general Yang–Mills moduli space. However, we know that the

dominant contribution to the Feynman integral comes from solutions that

absolutely minimize the Yang–Mills action, i.e., from the instanton solutions.

If Y denotes the Yang–Mills moduli space, then Y = ∪Y

,whereY

is the

moduli space of ﬁelds with instanton number k. The moduli space M

self-dual Yang–Mills ﬁelds or instantons of instanton number k is a subspace

of Y

. Thus, one hopes to obtain some information by integrating over the

space M

. Several mathematicians [103, 172, 173,255, 281] have studied the

geometry of the space M

and we now have detailed results about the Rie-

mannian metric, volume, form and curvature of the most basic moduli space

. We give below a brief discussion of these results.

Let (M,g) be a compact, oriented, Riemannian 4-manifold. Let P (M,G)

be a principal G-bundle, where G has a bi-invariant metric h.Themetricsg

and h induce the inner products  , 

(g,h)

on the spaces Λ

(M,ad P )ofk-

forms with values in the vector bundle ad P . We can use these inner products

to deﬁne a Riemannian metric on the space of gauge connections A(P )as

follows. Recall that the space A(P ) is an aﬃne space, so that for each ω ∈

A(P ) we have the canonical identiﬁcation between the tangent space T

A(P )

and Λ

(M,ad P ). Using this identiﬁcation we can transfer the inner product

 , 

(g,h)

on Λ

(M,ad P )toT

A(P ). Integrating this pointwise inner product

against the Riemannian volume form we obtain an L

inner product on the

space A(P). This inner product is invariant under the action of the group G

on A and hence we get an inner product on the moduli space A/G. Recall that

G does not act freely on A, but G/Z (G) acts freely on the open dense subset

9.3 Topology and Geometry of Moduli Spaces 295

of irreducible connections. After suitable Sobolev completions of all the

relevant spaces, the moduli space O

= A

/G of irreducible connections can

be given the structure of a Hilbert manifold. The L

metric on A restricted

to A

induces a weak Riemannian metric on O

by requiring the canonical

projection A

→O

to be a Riemannian submersion. The space I

irreducible instantons of instanton number k = −c

(P ), is deﬁned by

= M

∩O

where k = −c

(P ). The space I

is a ﬁnite dimensional manifold with sin-

gularities and the weak Riemannian metric on O

restricts to a Riemannian

metric on I

The space I

was studied by Donaldson ([106]) in the case where M is a

simply connected manifold with positive deﬁnite intersection form and where

the gauge group G = SU(2). He proved that there exists a compact set

K ∈I

such that I

−K is a union of a ﬁnite number of components, one of

which is diﬀeomorphic to M × (0, 1) and all the others are diﬀeomorphic to

×(0, 1). Using the Riemannian geometry of I

discussed above we can

put this result in its geometric perspective as follows:

Theorem 9.12 Let M be a simply connected manifold with positive deﬁnite

intersection form and let the gauge group G = SU(2).Then

1. I

is an incomplete manifold with ﬁnite diameter and volume.

2. Let

be the metric completion of I

.Then

−I

is the disjoint union

of a ﬁnite set of points {p

} and a set X diﬀeomorphic to M.Thereexists

an >0 such that M × [0,) is diﬀeomorphic to a neighborhood of X in

with the pull-back metric asymptotic to a product metric and for each

there is a neighborhood diﬀeomorphic to CP

×(0,) with the pull-back

metric asymptotic to a cone metric with base CP

See [173,172] for a proof and further details.

In general, the L

metric on a moduli space cannot be calculated explicitly,

since it depends on global analytic data about M. However, for the funda-

mental BPST instanton moduli space I

, the metric and the curvature have

been computed by several people (see, for example, [103, 172, 181, 203]). We

have the following theorem.

Theorem 9.13 There exists a diﬀeomorphism φ : R

→I

for which the

pull-back metric has the form

∗

= ψ

(r)g,

where g

is the L

metric on I

and g is the standard Euclidean metric on

,andψ is a smooth function of the distance r.

The explicit formula for the function ψ is quite complicated and is given

in [172]. The geometry of the moduli space of instanton solutions in the par-

ticular case of CP

has been studied in detail in [67,171]. The moduli spaces

296 9 4-Manifold Invarian t s

of Yang–Mills connections on various base manifolds have also been studied

in [201, 202, 203]andin[171]. We now state the result on the curvature of

the moduli space M

of irreducible instantons on a compact, Riemannian

manifold with compact, semisimple gauge group. The calculation of this cur-

vature is based on a generalization of O’Neill’s formula for the curvature of

a Riemannian submersion.

Theorem 9.14 Let X, Y ∈ T

A be the horizontal lifts of tangent vectors

∈ T

[α]

,where[α] is the equivalence class of gauge connections

that are gauge-equivalent to α. Then the sectional curvature R of M

at [α]

is given by

R(X

 =3b

∗

(Y ),G

∗

(Y ))

+ b

−

(X),G

−

(Y ))

−b

−

(Y ),G

−

(Y )),

where b is bracketing on bundle-valued forms, b

∗

its adjoint, and b

−

is b

followed by orthogonal projection onto Λ

−

(M,ad P ) and G

are the Green

operators of the corresponding Laplacians on bundle-valued forms.

9.4 Donaldson Polynomials

Donaldson’s theorem on the topology of smooth, closed, 1-connected 4-

manifolds provides a new obstruction to smoothability of these topological

manifolds. A surprising ingredient in his proof of this theorem was the mod-

uli space I

of SU(2)-instantons on a manifold M. This theorem has been

applied to obtain a number of new results in topology and geometry and has

been extended to other manifolds. The space I

is a subspace of the moduli

space M

of Yang–Mills instantons with instanton number 1. The space M

in turn is a subspace of the moduli space Y

of all Yang–Mills ﬁelds on M .

In fact, we have



⊂



⊂A/G .

Donaldson has used the homology of these spaces M

, for suﬃciently large

k, to obtain a family of new invariants of a smooth 4-manifold M , satisfying

a certain condition on its intersection form. We now describe these invariants

known as Donaldson’s polynomial invariantsi, or simply as the Don-

aldson polynomials.

The Donaldson polynomials are deﬁned by polarization of a family q

symmetric, multilinear maps

: H

(M) ×···×H

(M)



 

d(k)times

→ Q,

9.4 Donaldson Polynomials 297

where k is the instanton number of P (M, SU(2)) and the function d(k)is

given by

d(k):=4k −

(χ(M)+σ(M)) = 4k −

(1 + b

), (9.14)

where χ(M) is the Euler characteristic of M, σ(M) is the signature of M ,

and b

is the dimension of the space of self-dual harmonic 2-forms on M .We

shall also refer to the maps q

as Donaldson polynomials.

A basic tool for the construction of Donaldson polynomials is a map that

transfers the homology of M to the cohomology of the orbit space O

irreducible connections on the SU(2)-bundle P . To describe this map we

begin by recalling the K¨unneth formula for cohomology of a product of two

manifolds

× M

)

∼



k=i+j

) ×H

If α ∈ H

×M

)wedenotebyα

(i,j)

=(β, γ), its component in H

)×

). For a ﬁxed i ≤ k we deﬁne a map

f : H

) ×H

× M

) → H

)

f(a, α)=



(i,j)







γ.

Given a ﬁxed element α ∈ H

× M

), the above map induces the map

(μ

)

: H

) → H

k−i

deﬁned by (μ

)

(a)=f(a, α).

We now apply this formula to the case when M

is a smooth, closed, 1-

connected 4-manifold M, M

is the moduli space O

of irreducible instantons

on M, k =4,i =2,andα = c

(P), the second Chern class of the Poincar´e

bundle P over M ×O

with structure group SU(2), to obtain Donaldson’s

map

μ := (μ

(P)

)

: H

(M) → H

). (9.15)

We note that the rational cohomology ring of O

is generated by cohomology

classes in dimensions 2 and 4 and that the map μ and the map (μ

(P)

)

can

be used to express these generators in terms of the homology of M.Using

the map μ we deﬁne maps μ

,d≥ 1,

: H

(M) ×···×H

(M)



 

d times

→ H

),d≥ 1,

,...,a

)=μ(a

) ∧μ(a

) ∧···∧μ(a

). (9.16)