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Four-Manifold Invariants and Physics

C Nash, National University of Ireland, Maynooth,

Ireland

Introduction

Manifolds of dimension 4 play a distinguished role

in physics and have done so ever since special and

general relativity ushered in the celebrated four-

dimensional spacetime. It is also the case that

manifolds of dimension 4 play a distinguished role

in mathematics: many generalities about manifolds

of a general dimension do not apply in dimensi on 4;

there are also phenomena in dimension 4 with no

counterpart in other dimensions.

This article descr ibes some of the more important

physical and mathematical properties of dimension 4.

We begin with an account of some topological and

geometric properties for manifolds in general, but

avoiding dimension 4, and then embark on the

dimension 4 discussion. The references at the end

will serve to take the reader further into the subject.

Topological, Piecewise-Linear, and

Differentiable Structures for Manifolds

In dealing with topological spaces which are mani-

folds, one distinguishes three types of manifolds M:

topological, piecewise-linear, and differentiable (also

called smooth). It is possible to describe the more

important differences between these three types

using topological techniques.

Consider then a manifold M of dimension n; M will

always be assumed to be compact, connected and

closed unless we indicate the contrary. The type of M is

determined by examining whether the transition

functions g



are homeomorphisms, (invertible) piece-

wise-linear maps, or diffeomorphisms. Now, since the

transition functions are maps from one subset of R

another, we introduce the groups TOP

, PL

,and

DIFF

which are all the homeomorphisms, piecewise-

linear maps, and diffeomorphisms of R

,respectively.

We are naturally led to the three sets of inclusions:

TOP

 TOP

TOP



 PL

 PL



DIFF

 DIFF

DIFF



½1

For each of the three sets of inclusions we pass to

the direct limit and construct the three limiting

groups

TOP; PL; DIFF ½2

With these three groups are associated the classifying

spaces BTOP, BPL an d BDIFF. The transition

functions g



are those of the tangent bundle to M;

and there are three possible tangent bundles depending

on the type of M and we denote these tangent bundles

by TM

TOP

, TM

,andTM

DIFF

in an obvious nota-

tion. Then to determine the tangent bundles TM

TOP

,andTM

DIFF

one simply selects an element of the

homotopy classes

½M; BTOP; ½M; BPL; and ½M; BDIFF½3

respectively.

Given this threefold hierarchy of manifold struc-

tures one wishes to know when one can straighten

out a topological manifold to make it piecew ise

linear; and also, when can one smooth a piece-

wise-linear manifold to make it differentiable?

386 Four-Manifold Invariants and Physics

If dim M  5ofM these two questions can be

formulated as lifting problems.

TOP versus PL for dim M 6¼ 4

Taking the first of them, so that we are comparing

piecewise-linear an d topological structures on M,

one can check BPL fibers over BTOP with fiber

TOP/PL yielding

TOP=PL ! BPL

# 

BTOP

½4

A method for straightening out a PL manifold is now

apparent: now a topological manifold is a choice of

map  : M !BTOP,andafactorization of  through

BPL will give M a PL structure. We show this below

BPL



%#¼   



!

BTOP

½5

The existence of the map  : M !BPL satisfying

 =    provides M with a PL structure and is a

lifting of the map  from the base BTOP to the total

space BPL.

This lifting method, for passing from TOP

structures to PL structures, does work, provided

dim M  5, since we have the stability result that

TOP

’

TOP

; n  5 ½6

For the map  to exist the obstructions to the lifting

which are cohomology classes of the form

kþ1

ðM; 

ðTOP=PLÞÞ ½7

must vanish. However, Kirby and Siebenmann have

shown that

TOP=PL ’ KðZ

; 3Þ½8

where K(Z

,3) is Eilenberg–Mac Lane space so that

its sole nonvanishing homotopy group is in dimen-

sion 3 giving us



TOP=PLðÞ¼

if n ¼ 3

0 otherwise



½9

Any obstruction to ’s existence is a class e (M),

say, in

ðM; Z

Þ dim M  5 ½10

When e(M) vanishes, the map  exists and furnishes

M with a PL structure; if e(M) = 0 it is natural to go

on to ask how many (homotopy classes of) such ’s

exist? Standard obstruction theory says the relevant

homotopy classes are just the whole cohomology

group

ðM; 

ðTOP=PLÞÞ ½11

which, since k = 3, is just

ðM; Z

Þ½12

So, for dim M  5, we see that when a closed

topological manifold M acquires a PL structure by

the lifting process just described, then the possible

distinct PL structures are isomorphic to

ðM; Z

Þ½13

which is not zero in general.

Finally, if dim M  3, then the notions PL and

TOP coincide, so we are left with the case dim M = 4

which we shall come to below. Now we wish to

describe the next step in the sequence TOP, PL,

DIFF which is the smoothing problem.

PL versus DIFF for dim M 6¼ 4

Similar ideas are used to address the question of

smoothing a piecewise-linear mani fold – however,

the results are different. Let us assume that M is a

closed PL manifold with dim M  5. This time the

fibration is

PL=DIFF ! BDIFF

# 

BPL

½14

The smoothing of a piecewise-linear M can also be

handled with obstruction theory and leads us immedi-

ately to the consideration of the homotopy groups



(PL=DIFF). This time the nontrivial homotopy

groups of the fiber are much more numerous than in

the piecewise-linear case. In fact one has



PL=DIFFðÞ¼

0ifn  6

if n ¼ 7

if n ¼ 8

992

if n ¼ 11

½15

The obstructions to passing from a PL to a DIFF

structure on M now lie in

kþ1

ðM; 

ðPL=DIFFÞÞ ½16

and the number of distinct liftings comprises the

cohomology group

ðM; 

ðPL=DIFFÞÞ ½17

Four-Manifold Invariants and Physics 387

As an illustration of all this, consider the case

M = S

; then the first nontriviality occurs when n = 7

and so the obstruction to smoothing S

lies in

ðS

; 

ðPL=DIFFÞÞ ½18

which is of course zero – this means that S

can be

smoothed, a fact which we know from first

principles. However, by the obstruction theory

introduced above, the resulting smooth structures

are isomorphic to

ðS

; 

ðPL=DIFFÞÞ¼H

ðS

; Z

Þ¼Z

½19

Hence, we have the celebrated result of Milnor and

Kervaire and Milnor that S

has 28 distinct

differentiable structures, 27 of which correspond to

what are known as exotic spheres.

Lastly, if dim M  3, then PL and DIFF coincide –

this leaves us with the case of greatest interest

namely dim M = 4.

The Strange Case of Four Dimensions

In four dimensions there are phenomena which have

no counterpart in any other dimension. First of all,

there are topological 4-manifolds which have no

smooth structure, though if they have a PL structure,

then they possess a unique smooth structure. Second,

the impediment to the existence of a smooth structure

is of a completely different type to that met in the

standard obstruction theory – it is not the pullback of

an element in the cohomology of a classifying space,

that is, it is not a characteristic class. Also the four-

dimensional story is far from completely known.

Nevertheless, there are some very striking results

dating from the early 1980s onwards.

We begin by disposing of the difference between

PL and DIFF structures: our earlier results together

with the vanishing statement



PL=DIFFðÞ¼0; n  6 ½20

mean that every PL 4-manifold possesses a unique

DIFF structure. Thus, we can take the crucial

difference to be between DIFF and TOP.

In Freedman (1982) all, simply connected, topo-

logical 4-manifolds were classified by their intersec-

tion form q.

We recall that q is a quadratic form constructed

from the cohomology of M as follows: take two

elements  and  of H

(M; Z ) and form their cup

product  [  2 H

(M; Z); then we define q(, )by

qð; Þ¼ð [ Þ½M½21

where ( [ )[M] denotes the integer obtained by

evaluating  [  on the generating cycle [M] of the

top homology group H

(M; Z )ofM. Poincare´

duality ensures that such a form is always non-

degenerate over Z and so has det q = 1; q is then

called unim odular. Also we refer to q, as ‘‘even’’ if

all its diagonal entries are even, and as ‘‘odd’’

otherwise.

Freedman’s work yields the following:

Theorem (Freedman). A simply connected

4-manifold M with even intersection form q belongs

to a unique homeomorphism class, while if q is

odd there are precisely two nonhomeomorphic

manifolds M with q as their intersection form.

This is a very powerful result – the intersection

form q very nearly determines the homeomorphism

class of a simply connected M, and actually only

fails to do so in the odd case where there are still

just two possibilities. Further, every unimodular

quadratic form occurs as the intersection form of

some manifold.

As an illustration of the impressive nature of

Freedman’s work, choose M to be the sphere S

since H

; Z) is trivial, then q is the zero quadratic

form and is of course even; we write this as q = ;.

Now recollect that the Poincare´ conjecture in four

dimensions is the statement that any homotopy

4-sphere, S

say, is actually homeomorphic to S

Well, since H

; Z) is also trivial then any S

also

has intersection form q = ;. Applying Freedman’s

theorem to S

immediately asserts that S

belongs to

a unique homeomorphism class which must be that

of S

thereby establishing the Poincare´ conjecture.

Freedman’s result combined with a much earlier

result of Rohlin (1952) also gives us an example of a

nonsmoothable 4-manifold: Rohlin’s theorem asserts

that given a smooth, simply connected, 4-manifold

with even intersection form q, then the signature –

the signature of q being defined to be the difference

between the number of positive and negative eigen-

values of q – (q)ofq is divisible by 16.

Now write

q ¼

2 1000000

12100000

0 1210000

00121000

00012101

00001210

00000120

00001002

¼E

½22

is actually the Cartan matrix for the exceptional

Lie algebra e

), then, by inspection, q is even, and

by calculation, it has signature 8. By Freedman’s

theorem there is a single, simply connected, 4-mani-

fold with intersection form q=E

. However, by

388 Four-Manifold Invariants and Physics

Rohlin’s theorem, it cannot be smoothed since its

signature is 8.

The next breakthrough was due to Donaldson

(1983). Donaldson’s theorem is applicable to defi-

nite forms q, which by appropriate choice of

orientation on M we can take to be positive definite.

One has:

Theorem (Donaldson). Asimplyconnected, smooth

4-manifold, with positive-definite intersection form

q is always diagonalizable over the integers to

q = diag(1, ..., 1).

One can immediately deduce that no, simply

connected, 4-manifold for which q is even and

positive definite can be smoothed!

For example, the manifold with q = E

 E

has

signature 16 (by Rohlin’s theorem). But since E

even, then so is E

 E

and so Donaldson’s

theorem forbids such a manifold from existing

smoothly.

In fact, in contrast to Freedman’s theorem, which

allows all unimodular quadratic forms to occur as

the inte rsection form of some topological manifold,

Donaldson’s theorem says that in the positive-

definite, smooth, case only one quadratic form is

allowed, namely I.

Donaldson’s work makes contact with physics

because it uses the Yang–Mills equations as we now

outline.

Let A be a connection on a principal SU(2) bundle

over a simply connected 4-ma nifold M with posi-

tive-definite intersection form. If the curvature

2-form of A is F, then F has an L

norm which is

the Euclidean Yang–Mills action S. One has

S ¼kFk

¼

trðF ^FÞ½23

where F is the usual dual 2-form to F. The minima

of the action S are given by those A, called

instantons, which satisfy the famous self-duality

equations

F ¼F ½24

Given one instanton A which minimizes S one can

perturb about A in an attempt to find more

instantons. This process is successful and the space

of all instantons can be fitted together to form a

global moduli space of finite dimension. For the

instanton which provides the absolute minimu m of

S, the moduli space M is a nonc ompact space of

dimension 5.

We can now summarize the logic that is used to

prove Donaldson’s theorem: there are very strong

relationships between M and the moduli space M;

for example, let q be regarded as an n  n matrix

with precisely p unit eigenvalues (clearly p  n and

Donaldson’s theorem is just the statement that

p = n), then M has precisely p singularities which

look like cones on the space CP

. These combine to

produce the result that the 4-manifold M has the

same topological signature Sign(M)asp copies of

; and so they have signature a – b, where a of

the CP

’s are oriented as usual and b have the

opposite orientation. Thus,

Sign ðMÞ¼a  b ½25

Now by definition, Sign (M) is the signature (q)of

the intersection form q of M.But,byassumption,q is

positive definite n  n so (q) = n = Sign (M). Hence,

n ¼ a  b ½26

However, a þ b = p and p  n so we can say that

n ¼ a  b; p ¼ a þ b  n ½27

but one always has a þ b  a  b so we have

n  p  n ) p ¼ n ½28

which is Donaldson’s theorem.

Donaldson’s Polynomial Invariants

Donaldson extended his work by introducing poly-

nomial invariants also derived from Yang–Mills

theory and to discuss them we must introduce

some notation.

Let M be a smooth, simply connected, orientable

Riemannian 4-man ifold without boundary and A be

an SU(2) connection which is anti-self-dual so that

F ¼F ½29

Then the space of all gauge-inequivalent solutions to

this anti-self-duality equation – the moduli space

– has a dimension given by the integer

dim M

¼8k  3ð 1 þ b

Þ½30

Here k is the instanton number which gives the

topological type of the solution A. The instanton

number is minus the second Chern class c

(F) 2

(M; Z) of the bundle on which the A is defined.

This means that we have

k ¼c

ðFÞ½M¼

8

trðF ^ FÞ2Z ½31

The number b

is defined to be the rank of the

positive part of the intersectio n form q of M.

Four-Manifold Invariants and Physics 389

A Donaldson invariant q

d, r

is a symmetric integer

polynomial of degree d in the 2-homology H

(M; Z)

of M

d;r

: H

ðMÞH

ðMÞ

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

d factors

! Z ½32

Given a certain map m

: H

ðMÞ!H

4i

ðM

Þ½33

if  2 H

(M)and represents a point in M,we

define q

d, r

() by writing

d;r

ðÞ¼m

ðÞm

ðÞ½M

½34

The evaluation of [M

] on the RHS of the above

equation means that

2d þ 4r ¼ dim M

½35

so that M

is even dimensional; this is achieved by

requiring b

to be odd.

Now the Donaldson invariants q

d, r

are differential

topological invariants rather than topological invari-

ants but they are difficult to calculate as they require

detailed knowledge of the instanton moduli space

. However they are nontrivial and their values

are known for a number of 4-manifolds M. For

example, if M is a complex algebraic surface, a

positivity argument shows that they are nonzero

when d is large enough. Conversely, if M can be

written as the connected sum

M ¼ M

where both M

and M

have b

> 0, then they all

vanish.

Topological Quantum Field Theories

Turning now to physics, it is time to point out that

the q

d, r

can also be obtained, Witten (1988), as the

correlation functions of twisted N = 2 supersym-

metric topological quantum field theory.

The action S for this theory is given by

S ¼

ﬃﬃﬃ









D



 þ iD









 iD





½



;





½



;







½; 

½; 



½36

where F



is the curvature of a connection A



and

(, , ,



, 



) are a collection of fields introduced

in order to construct the right supersymmetric

theory;  and  are both spinless while the multiplet

(



, 



) contains the components of a 0-form, a

1-form, and a self-dual 2-form, respectively.

The significance of this choice of multiplet is that

the instanton deformation complex used to calculate

dim M

contains precisely these fields.

Even though S contains a metric, its correlation

functions are independent of the metric g so that S

can still be regarded as a topological quantum field

theory. This is because both S and its associated

energy momentum tensor T ( S=g) can be written

as BRST commutators S = {Q, V}, T = {Q, V

} for

suitable V and V

With this theory, it is possible to show that the

correlation functions are independent of the gauge

coupling and hence we can evaluate them in a small

coupling limit. In this limit, the functional integrals

are dominated by the classical minima of S, which

for A



are just the instantons



¼F





½37

We also need  and  to vanish for irreducible

connections. If we expand all the fields around the

minima up to quadratic terms and do the resulting

Gaussian integrals, the correlation functions may be

formally evaluated.

A general correlation function of this theory is

given by

<P >¼

DF exp½SPðFÞ ½38

where F denotes the collection of fields present in

S and P(F) is some polynomial in the fields.

S has been constructed so that the zero modes in

the expansion about the minima are the tangents to

the moduli space M

. This suggests doing the DF

integration as follows: express the integral as an

integral over modes, then integrate out all the

nonzero modes first leaving a finite-dimensional

integration over the compactified moduli space

. The Gaussian integration over the nonzero

modes is a boson–fermion ratio of determinants,

which supersymmetry constrains to be 1, bosonic

and fermionic eigenvalues being equal in pairs.

This amounts to writing

<P >¼

½39

where P

denotes some n-form over M

and

n = dim

. If the original polynomial P(F)is

judiciously chosen, then calculation of < P> repro-

duces evaluation of the Donaldson polynomials q

d, r

390 Four-Manifold Invariants and Physics

The Seiberg–Witten Equations

The Seiberg–Witten equations constitute another

breakthrough in the work on the topology of

4-manifolds, since they greatly simplify the calcula-

tion of the data supplied by the Donaldson

polynomial invariants. We shall discuss this later

below but turn now to the equations themselves.

If we choose an oriented, compact, closed,

Riemannian manifold M , then the data we need for

the Seiberg–Witten equations are a connection A on

a line bundle L over M and a ‘‘local spinor’’ field .

The Seiberg–Witten equations are then

¼ 0; F

¼



 ½40

where @=

is the Dirac operator and  is made from

the gamma matrices 

according to =(1=2)

[

, 

]dx

^ dx

We call a local spinor because global spinors

may not exist on M; however, in dimension 4,

orientability guarantees that a spin

structure exists

on M (a choice of spin

structure on M is an extra

piece of data in the Seiberg–Witten case); is then

the appropriate section for the spin

bundle and

behaves locally like a spinor coupled to the U(1)

connection A. Le t Spin

(M) denote the set of

isomorphism classes of spin

structures on M then,

for the case b

> 1 – the case b

= 1 has some

technicalities – the Seiberg–Witten invariants deter-

mine a map SW of the form

SW : Spin

ðMÞ!Z ½41

We emphasize that A is just a U(1) abelian

connection and so F = dA, with F

denoting the

self-dual part of F.

We shall now have a look at an example of a new

result obtained directly from the Seiberg–Witten

equations. The equations clearly provide the

absolute minima for the action

S ¼

j@=



 j

½42

If we use a Weitzenbo¨ ck formula to relate the

Laplacian r



to @=



plus curvature terms, we

find that S satisfies

j@=



 j

j j

Rj j

½43

jFj

j j

Rj j

þ 

ðLÞ½44

where R is the scalar curvature of M and c

(L) is the

Chern class of L.

We notice that the action now looks like one for

monopoles. But now suppose that R is positive and

that the pair (A, ) is a solution to the Seiberg–

Witten equations, then the left-hand side (LHS) of

this last expression is zero and all the integrands on

the RHS are positive so the solution must obey = 0

and F

= 0. A technical point is that if M has b

> 1,

then a perturbation of the metric can preserve the

positivity of R but perturb F

= 0tobesimplyF = 0

rendering the connection A flat. Hence, in these

circumstances, the solution (A, ) is the trivial one.

This means that we have a new kind of vanishing

theorem in four dimensions.

Theorem (Witten 1994). No 4-manifold with b

1 and nontrivial solution to the Seiberg–Witten

equations admits a metric of positive scalar curvature.

Now, for technical reasons , we assume that the

d, r

have the property that

d;rþ2

¼ 4q

d;r

½45

A simply connected M with this property is called of

simple type. We also define

by writing

d;0

; if d ¼ðb

þ 1Þ mod 2

d;1

; if d ¼b

mod 2

(

½46

The generating function G

() is now given by

ðÞ¼

d¼0

ðÞ½47

According to Kronheimer and Mrowka (1994),

() can be expressed in terms of a finite number

of classes (known as basic classes) 

(

2 H

(M))

with rational coefficients a

(the Seiberg–Witten

invariants) resulting in the formu la

ðÞ¼exp½  =2

exp½

 ½48

Hence, for M of simple type, the polynom ial

invariants are determined by a (finit e) number of

basic classes and the Seiberg–Witten invariants.

Returning now to the physics we find that the

quantum field theory approach to the polynomial

invariants relates them to properties of the moduli

space for the Seiberg–Witten equations rather than

to properties of the instanton modul i space M

The moduli space for the Seiberg–Witten equa-

tions, unlike the insta nton case, is compact and

generically has dimension

ðLÞ2ðMÞ3ðMÞ

½49

Four-Manifold Invariants and Physics 391

(M) and (M) being the Euler characteristic and

signature of M, respectively. When

ðLÞ¼2ðMÞþ3ðMÞ½50

we get a zero-dimensional moduli space consisting

of a finite collection of points

; ...; P

g½51

Now each point P

has a sign 

= 1 associated

with it coming from the sign of the determinant of

elliptic operator whose index gave the dimension of

the moduli space. The sum of these signs is an

integer topological invariant denoted by n

, that is,

j¼1



½52

Returning now to our formu la for G

(), one

finds that

ðÞ¼2

pðMÞ

exp½ =2

exp½c

ðLÞ½53

pðMÞ¼1 þ

ð7ðMÞþ11ðMÞÞ ½54

and the sum over L on the RHS of the formula is

over line bundles L that satisfy

ðLÞ¼2ðMÞþ3ðMÞ½55

that is, it is a sum over L with zero-dimensional

Seiberg–Witten moduli spaces.

Comparison of the two formulas for G

() – the

first mathematical in origin and the second physi-

cal – allows one to identify the Seiberg–Witten

invariants a

and the Kronheimer–Mrowka basic

classes 

as the c

(L)’s.

The results described thus far are for simply

connected 4-manifolds but this condition is not

obligatory for and there is also a theory in the non-

simply-connected case (Marin˜ o and Moore 1999).

The physics underlying these topological results is

of great importance since many of the ideas

originate there. It is known that the computation

of the Donaldson invariants there uses the fact that

the N = 2 gauge theory is asymptotically free. This

means that the ultraviolet limit being one of weak

coupling is tractable. However, the less tractable

infrared or strong-coupling limit would do just as

well to calculate the Donaldson invariants since

these latter are metric independent.

In Seiberg and Witten’s work, this infrared

behavior is actually determined and it is found

that, in the strong-coupling infrared limit, the theory

is equivalent to a weakly coupled theory of abelian

fields and monopoles. There is also a duality

between the original theory and the theory with

monopoles which is expressed by the fact that the

(abelian) gauge group of the monopole theory is the

dual of the maximal torus of the group of the

nonabelian theory.

We recall that the Yang–Mills gauge group in this

discussion is SU(2). Seiberg and Witten’s results

mean the replacement of SU(2) instantons used to

compute the Donaldson invariants by the counting

of U(1) monopoles. This calculation of the non-

abelian Donaldson data by abelian Seiberg–Witten

data theory is much like the representation theory of

a nonabelian Lie group G where everything is

determined by an abelian object: the maximal torus.

The theory considered by Seiberg and Witten

possesses a collection of quantum vacua labeled by a

complex parameter u which turns out to parametrize a

family of elliptic curves. A central part is played by a

function (u) on which there is a modular action of

SL(2, Z). The successful determination of the infrared

limit involves an electric–magnetic duality and the

whole matter is of very considerable independent

interest for quantum field theory, quark confinement,

and string theory in general.

Seiberg–Witten Theory and Exotic

Structures on 4-Manifolds

We saw earlier that, when dim M 6¼ 4, a manifold

may posses s a finite number of differentiable

structures, S

having 28 distinct smooth structures.

However, in dimension 4, Seiberg–Witten theory has

been used to show that there are many 4-manifolds

with a countable infinity of smooth structures. We

just mention two: the K3 surface has infinitely

many smooth structure s as does the manifold

#5CP

: This is another instance of how dimen-

sion 4 differs from all other dimensions. This infinite

variety of exotic smooth structures in four dimen-

sions is also of great interest to physics.

An outstanding four-dimensional matter still is the

smooth Poincare´ conjecture which asks whether a

smooth 4-manifold M homotopic to S

is diffeo-

morphic to S

?SuchanM is certainly homeomorphic

to S

because this is the standard Poincare´conjecture

proved by Freedman and, if the answer to this question

is yes then S

would be an example of a 4-manifold

with no exotic smooth structures. There is at present

no consensus on the answer to this question.

Exotic Structures on Open 4-Manifolds

If M is an open manifold, that is, a noncompact

manifold without boundary, and M = R

then, for

392 Four-Manifold Invariants and Physics

n 6¼ 4, there is only one smooth structure; but for

n = 4, there are exotic differentiable structures on

. In fact, Gompf showed that there is a continuum

of exotic differentiable structures that can be placed

on R

Symplectic and Ka¨ hler 4-Manifolds

Many 4-manifolds are symplectic, and symplectic

manifolds are central in physics; there are many results

obtained using Seiberg–Witten theory concerning the

topology and geometry of symplectic manifolds. The

exotic K3 structures referred to above are all symplec-

tic and so there is no shortage of symplectic structures

even within one homeomorphism class. Taubes

obtained far-reaching new results for symplectic

4-manifolds including establishing an equivalence

between the Seiberg–Witten invariants in the symplec-

tic case and the Gromov invariants.

Ka¨ hler manifolds possess, simultaneously, com-

patible, Riem annian, symplectic and complex struc-

tures and, beginning with Witten’s work, there are

many results to be found for Ka¨ hler 4-manifolds

using Seiberg–Witten techniques.

4-Manifolds with Boundary

There is a very important extension of the Donaldson–

Seiberg–Witten theory to 4-manifolds M with bound-

ary @M = N.When@M 6¼ , the Donaldson invariants

are not numerical invariants but take values in HF(N)

where HF(N) denotes what is called the Floer

homology of the 3-manifold N. Topological quantum

field theory is the ideal setting for this theory since it

naturally treats manifolds with boundaries. The Floer

homology groups HF(N) act as Hilbert spaces for the

quantum fields defined on the boundary. There is now

a full interplay of 4-manifold theory and 3-manifold

theory as well as Yang–Mills theory in three and four

dimensions. This interplay is often realized by taking

two 4-manifolds M

and M

with the same boundary

N and joining them along N to obtain a closed

4-manifold M so that

M ¼ M

[

½56

Given a 3-manifold N, and an SU(2) connection

A, Floer studied the critical points of the Chern–

Simons function f(A) defined by

f ðAÞ¼

8

tr A ^ dA þ

A ^ A ^ A



½57

where f(A) is regarded as a function on the infinite-

dimensional space A of connections. The function f(A)

changes by an integer under a gauge transformation

and so descends to a single-valued gauge-invariant

function on the space of gauge orbits A=G if one

considers exp (2kif ( A)) where k 2 Z (G being the

group of gauge transformations). Morse theory

applied to this infinite-dimensional setting gives an

infinite Morse index to each critical point, a pathology

which is avoided by only defining the difference of the

index between two critical points using spectral flow.

The critical points correspond, via gradient flow and a

consideration of the instanton equations

F ¼F ½58

on the 4-manifold N  R, to the flat connections on

the 3-manifold N. The latter are identifiable as the

set of (equivalence classes) of representations of the

fundamental group 

(M) in the gauge group SU(2),

that is, with

Homð

ðNÞ; SUð2ÞÞ=Ad SUð2Þ½59

For the Seiberg–Witten formulation, let

A denote a

connection on the 3-manifold N with curvature

A). Then the Chern–Simons function f(A)is

replaced by the abelian Chern–Simons function

together with a quadratic fermi on term resulting in

the function f

(

A), defined by

AÞ¼

D=

 þ

A ^ Fð

AÞ

½60

where D=

denotes the self-adjoint Dirac operator in

three dimensions acting on a spinor  on N; because

of the presence of the Chern–Simons function

(

A) is only defined up to a multiple of 8

in a

manner similar to the case for f (A). Gradient flow

together with the Seiberg–Witten equations on the

4-manifold N  R result in critical points corre-

sponding to the solutions to

 ¼ 0; Fð

AÞ¼

 ½61

which is a three-dimensional version of the Seiberg–

Witten equations.

The critical point theory of these two functions f (A)

and f

(

A) permit the construction of the instanton

Floer homology groups HF

inst

(N)andHF

(N),

respectively. In fact, there are several kinds of Floer

homology: Lagrangian Floer homology, instanton

Floer homology, Heegard–Floer homology, Seiberg–

Witten–Floer homology and conjectures concerning

their relations to one another.

There are still many unanswered questions of

joint interest to mathematicians and physicists in the

entire area of 4-manifold theory .

See also: Electric–Magnetic Duality; Gauge Theoretic

Invariants of 4-Manifolds; Floer Homology; Topological

Quantum Field Theory: Overview.

Four-Manifold Invariants and Physics 393

Further Reading

Atiyah MF (1989) Topological quantum field theories. Institut

des Hautes Etudes Scientifiques Publications Mathematiques

68: 175–186.

Atiyah MF (1995) Floer homology. In: Hofer H, Weinstein A,

Taubes C, and Zehnder E (eds.) Floer Memorial Volume.

Boston: Birkha¨user.

Donaldson SK (1984) An application of gauge theory to four

dimensional topology. Journal of Differential Geometry 96:

387–407.

Donaldson SK (1996) The Seiberg–Witten equations and

4-manifold topology. Bulletin of the American Mathematical

Society 33: 45–70.

Donaldson SK (2002) Floer Homology Groups in Yang–Mills

Theory. Cambridge: Cambridge University Press.

Donaldson SK and Kronheimer PB (1990) The Geometry of Four

Manifolds. Oxford: Oxford University Press.

Fintushel R and Stern RJ (1998) Knots, links and 4-manifolds.

Inventiones Mathematicae 134: 363–400.

Fintushel R and Stern RJ (2004) Double node neighborhoods and

families of simply connected 4-manifolds with b

= 1;

arXiv: math.GT/0412126.

Freedman MH (1982) The topology of 4-dimensional manifolds.

Journal of Differential Geometry 17: 357–453.

Gadgil S (2004) Open manifolds, Ozsvath–Szabo invariants and

exotic R

s, arXiv:math.GT/0408379.

Gompf R (1985) An infinite set of exotic R

’s. Journal of

Differential Geometry 21: 283–300.

Gromov M (1985) Pseudo-holomorphic curves in symplectic

manifolds. Inventiones Math. 82: 307–347.

Kronheimer PB and Mrowka TS (1994) Recurrence relations and

asymptotics for four manifold invariants. Bulletin of the

American Mathematical Society 30: 215–221.

Mandelbaum R (1980) Four-dimensional topology: an introduc-

tion. Bulletin of the American Mathematical Society 2: 1–159.

Marin˜ o M and Moore G (1999) Donaldson invariants for non-

simply connected manifolds. Communications in Mathemati-

cal Physics 203: 249–267.

Morgan J and Mrowka T (1992) A note on Donaldson’s

polynomial invariants. International Mathematics Research

Notices 10: 223–230.

Nash C (1992) Differential Topology and Quantum Field Theory.

London–San Diego: Academic Press.

Park J, Stipsicz AI, and Szabo Z (2004) Exotic smooth structures

on CP

#5CP

, arXiv:math.GT/041221.

Rohlin VA (1952) New results in the theory of 4 dimensional

manifolds. Doklady Akademii Nauk SSSR 84: 221–224.

Seiberg N and Witten E (1994a) Electric–magnetic duality,

monopole condensation, and confinement in N = 2 super-

symmetric Yang–Mills theory. Nuclear Physics B 426: 19–52.

Seiberg N and Witten E (1994b) Electric-magnetic duality,

monopole condensation, and confinement in N = 2 super-

symmetric Yang–Mills theory – erratum. Nuclear Physics B

430: 485–486.

Seiberg N and Witten E (1994c) Monopoles, duality and chiral

symmetry breaking in N = 2 supersymmetric QCD. Nuclear

Physics B 431: 484–550.

Taubes CH (1995a) More const raints on symplectic forms from

Seiberg–Witten invariants. Mathematical Research Letters

2: 9–13.

Taubes CH (1995b) The Seiberg–Witten and Gromov invariants.

Mathematical Research Letters 2: 221–238.

Witten E (1988) Topological quantum field theory. Communica-

tions in Mathematical Physics 117: 353–386.

Witten E (1994) Monopoles and four manifolds. Mathematical

Research Letters 1: 769–796.

Fractal Dimensions in Dynamics

upanovic

and D Z

ubrinic

, University of Zagreb,

Zagreb, Croatia

Introduction

Since the 1970s, dimension theory for dynamics has

evolved into an independent field of mathematics.

Its main goal is to measure complexity of invariant

sets and measures using fractal dimensions. The

history of fractal dimensions is closely related to

the names of H Minkowski (Minkowski content,

1903), H Hausdorff (Hausdorff dimension,

1919), G Bouligand (Bouligand dimension, 1928),

L S Pontryagin and L G Schnirelmann (metric order,

1932), P Moran (Moran geometric constructions,

1946), A S Besicovitch and S J Taylor (Besi covitch–

Taylor index, 1954), A Re´nyi (Re´nyi spectrum

for dimensions, 1957), A N Kolmogorov and

V M Tihomi rov (metric dimension, Kolmogorov

complexity, 1959), Ya G Sinai, D Ruelle, R Bowen

(thermodynamic formalism, Bowen’s equation,

1972, 1973, 1979), B Mandelbrot (fractals and

multifractals, 1974), J L Kaplan and J A Yorke

(Lyapunov dimension, 1979), J E Hutchinson (frac-

tals and self-similarity, 1981), C Tricot, D Sulliva n

(packing dimension, 1982, 1984), H G E Hentschel

and I Procaccia (Hentschel–Procaccia spectrum for

dimensions, 1983), Ya Pesin (Carathe´odory–Pesin

dimension, 1988), M Lapidus and M van Franken-

huysen (complex dimensions for fractal strings,

2000), etc. Fractal dimensions enable us to have a

better insight into the dynamics appearing in various

problems in physics, engineering, chemistry, medi-

cine, geolog y, meteorology, ecology, economics,

computer science, image processing, and, of course,

in many branches of mathematics. Concentrating on

box and Hausdorff dimensions only, we describe

basic methods of fractal analysis in dynamics, sketch

their app lications, and indicate some trends in this

rapidly growing field.

394 Fractal Dimensions in Dynamics

Fractal Dimensions

Box Dimensions

Let A be a bounded set in R

, and let d(x, A)be

Euclidean distance from x to A. The Minkowski

sausage of radius " around A (a term coined by

B Mandelbrot) is defined as "-neighborhood of A,

that is, A

:= {y 2R

: d(y, A) <"}. By the upper

s-dimensional Minkowski content of A, s  0,

we mean

s

ðAÞ:¼

lim

"!0

Ns

2½0; 1

Here jj denotes N-dimensional Lebesgue measure.

The corresponding upper box dimension is defined by

dim

A :¼ inffs  0: M

s

ðAÞ¼0g

The lower s-dimensional Minkowski content M



(A)

and the corresponding lower box dimension

dim

are defined analogously. The name of box dimen-

sion stems from the following: if we have an "-grid

in R

composed of closed N-dimensional boxes

with side ", and if N(A, ") is the number of boxes of

the grid intersecting A, then

dim

A ¼

lim

"!0

log NðA;"Þ

logð1="Þ

and analogously for

dim

A. It suffices to take any

geometric subsequence "

= b

k

in the limit, where

b > 1 (H Furstenberg, 1970). There are many other

names for the upper box dimension appearing in the

literature, like the Cantor–Minkowski order, Min-

kowski dimension, Bouligand dimension, Borel

logarithmic rarefaction, Besicovitch–Taylor index,

entropy dimension, Kolmogorov dimension, fractal

dimension, capacity dimension, and limit capacity.

If A is such that

dim

A = dim

A, the common value

is denoted by d := dim

A, and we call it the box

dimension of A. If, in addition to this, both M



(A)

and M

d

(A) are in (0, 1), we say that A is

Minkowski nondegenerate. If, moreover, M



(A) =

d

(A) =: M

(A) 2(0, 1), then A is said to be

Minkowski measurable.

Assume that A is such that d := dim

A and

(A) exist. Then the value of M

(A)

1

is called

the lacunarity of A (B Mandelbrot, 1982). A

bounded set A R

is said to be porous (A Denjoy,

1920) if there exist >0 and >0 such that

for every x 2A and r 2(0, ) there is y 2R

such

that the open ball B

r

(y) is contained in B

(x) nA.

If A is porous then it is easy to see that

dim

A < N

(O Martio and M Vuorinen, 1987, A Salli, 1991).

We proceed with two examples. Let

A := C

(a)

, a 2(0, 1=2), be the Cantor set obtained

from [0, 1] by consecutive deletion of 2

middle

open intervals of length a

(1  2a) in step k 2N [

{0}. Then dim

A = ( log 2)=( log (1=a)) (G Bouligand,

1928), and A is nondegenerate, but not Minkowski

measurable (Lapidus and Pomerance, 1993). For

the spiral  of focus type defined by r = m’



polar coordinates, where  2(0, 1) and m > 0 are

fixed, ’  ’

> 0, we have dim

=2=(1 þ )

(Y Dupain, M Mende´s-France, C Tricot, 1983). It

is Minkowski measurable (Z

ubrinic

and Z

upanovic

2005), and the larger m, the smaller the lacunarity;

see Figure 1.

Hausdorff Dimension

For a given subset A of R

(not necessarily

bounded) and s  0 we define H

(A):= lim

" !0

inf {

i = 1

} 2[0, 1], where the infimum is taken

over all finite or countable coverings of A by open

balls of radii r

 ". The value of H

(A) is called

s-dimensional Hausdorff outer measure of A. The

Hausdorff dimension of A, sometimes called the

Hausdorff–Besicovitch dimension, is defined by

dim

A :¼ inffs  0: H

ðAÞ¼0g

If A is bounded then dim

A  dim

A  N.

We say that A is Hausdorff nondegenerate (or d-set)

if H

(A) 2(0, 1) for some d  0. Cantor sets share

this property, and dim

(a)

= (log2)=( log (1=a)),

where a 2(0, 1=2) (Hausdorff, 1919).

Gauge Functions

The notions of Minkowski contents and Hausdorff

measure can be generalized using gauge functions

h : [0, "

) !R that are assumed to be continuous,

increasing, and h(0) = 0. For example,

h

ðAÞ:¼

lim

"!0

hð"Þ

and similarly for M



(A) (M Lapidus and C He,

1997), while for H

(A) it suffices to change r

with

h(r

) in the above definition of the Hausdorff outer

measure (Besicovitch, 1934). Gauge functions are

used for sets that are Minkowski or Hausdorff

Figure 1 Spirals of equal box dimensions (4/3) and different

lacunarities (0.43 and 0.05).

Fractal Dimensions in Dynamics 395