Marathe K. Topics in Physical Mathematics

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68 2 Topology

inequality χ(M) ≥

|τ(M)|,whereχ(M) is the Euler characteristic and τ(M)

is the signature of M . A number of new obstructions are now known. Some of

these indicate that their existence may depend on the smooth structure of M

as opposed to just the topological structure. These obstructions can be inter-

preted as implying a coupling of matter ﬁelds to gravity (see [269,270, 94]).

The basic problem is to understand the existence and moduli spaces of these

metrics on a given manifold and perhaps to ﬁnd a geometric decomposition

of M with respect to a special functional. One of the most important tools for

developing such a theory is the Chern–Gauss–Bonnet theorem which states

that

χ(M)=

8π



(|Rm|

−|K|

)dv =

8π





|W |

−

|K|



dv.

This result allows one to control the full Riemann curvature in terms of the

Ricci curvature Ric. It is interesting to note that in [242], Lanczos had arrived

at the same result while searching for Lagrangians to generalize Einstein’s

gravitational ﬁeld equations. He noted the curious property of the Euler class

that it contains no dynamics (or is an invariant). He had thus obtained the

ﬁrst topological gravity invariant (without realizing it). Chern’s fundamental

paper [74] appeared in the same journal seven years later. Chern–Weil the-

ory and Hirzebruch’s signature theorem give the following expression for the

signature τ(M ):

τ(M)=

12π



(|W

−|W

−

)dv.

The Hitchin–Thorpe inequality follows from this result and the Chern–

Gauss–Bonnet theorem. In dimension 4, all the classical functionals are con-

formally (or scale) invariant, so it is customary to work with the space of

unit volume metrics on M.

2.7 The Hopf Invariant

As we remarked in the preface, mathematicians and physicists have often

developed the same ideas from diﬀerent perspectives. The Hopf ﬁbration and

Dirac’s monopole construction provide an example of this. Each is based on

the observation that S

(the base of the Hopf ﬁbration) is a deformation

retract of R

\{0} (the base of the Dirac monopole ﬁeld). Thus non-triviality

of π

) can be interpreted as Dirac’s monopole quantization condition.

Other Hopf ﬁbrations and Hopf invariants also arise in physical theories, as

we indicate later in this section.

Let f : S

2n−1

→ S

,n > 1, be a continuous map. Let X denote the

quotient space of the disjoint union D



under the identiﬁcation of

2.7 The Hopf Invariant 69

x ∈ S

2n−1

⊂ D

with f (x) ∈ S

.Themapf is called the attaching

map.ThespaceX is called the adjunction space obtained by attaching

the 2n-cell e

:= (D

2n−1

)toS

by f and is denoted by S

∪

.The

cohomology H

∗

(X; Z)ofX is easy to calculate and is given by

(X; Z)=



Z if i = n or 2n,

0otherwise.

Let α ∈ H

(X; Z)andβ ∈ H

(X; Z) be the generators of the respective

cohomology groups. We note that in de Rham cohomology α, β can be identi-

ﬁed with closed diﬀerential forms. It follows that α

is an integral multiple of

β. The multiplier h(f) is completely determined by the map f and is called

the Hopf invariant of f.Thuswehave

= h(f)β.

It can be shown that h(f) is, in fact, a homotopy invariant and hence deﬁnes

amap(alsodenotedbyh)

[f] → h(f )ofπ

2n−1

) → Z.

To include the case n = 1 we note that the double covering c

: S

→ S

has

adjunction space RP

and Hopf invariant 1. The following theorem is due to

H. Hopf:

Theorem 2.14 Let F

:= C(S

2n−1

),n>0, denote the space of contin-

uous functions from S

2n−1

to S

. Then we have the following:

1. If n>1 is odd, then h(f)=0, ∀f ∈F

2. If n is even, then for each k ∈ Z there exists a map f

∈F

such that

h(f

)=2k.

3. If there exists g ∈F

such that h(g) is odd, then n =2

, where m is a

nonnegative integer.

4. Let π ∈F

(resp., F

) be the real (resp., complex) Hopf ﬁbration. Then

h(π)=1.

We note that the real Hopf ﬁbration π : S

→ RP

∼

occurs in the

geometric quantization of the harmonic oscillator [271,272] while the complex

Hopf ﬁbration π : S

→ CP

∼

occurs in the geometric construction

of the Dirac monopole. It can be shown that the last result in the above

theorem can be extended to include the quaternionic and octonionic Hopf

ﬁbrations (which arise in the solution of Yang–Mills equations on S

and

, respectively) and that this extended list exhausts all F

that contain a

map with Hopf invariant 1. This result is part of the following extraordinary

theorem, which links several speciﬁc structures from algebra, topology, and

geometry.

70 2 Topology

Theorem 2.15 Let S

n−1

⊂ R

,n>0 denote the standard (n − 1)-sphere

in the real Euclidean n-space with the convention that S

:= {−1, 1}⊂R

and R

:= {0}. Then the following statements are equivalent:

1. The integer n ∈{1, 2, 4, 8}.

2. R

has the structure of a normed algebra.

3. R

has the structure of a division algebra.

4. R

n−1

admits a cross product (or a vector product).

5. S

n−1

is an H-space.

6. S

n−1

is parallelizable (i.e., its tangent bundle is trivializable).

7. There exists a map f : S

2n−1

→ S

with Hopf invariant h(f)=1.

The relation of conditions (1) and (3) with condition (4) and its general-

izations have been considered in [126]. It is well known that complex numbers

have applications to 2-dimensional geometry and its ring of Gaussian in-

tegers (i.e., numbers of the form m + ni,wherem, n ∈ Z)isusedinmany

classical questions in arithmetic. Similarly, the quaternions are related to 3-

dimensional and 4-dimensional geometry and they contain rings of integers

with many properties similar to those of Gaussian integers. The octonions

have applications to 7- and 8-dimensional geometry. In elementary algebra

one encounters the construction of the complex numbers in terms of certain

real matrices of order 2. This doubling procedure of constructing C from

R was generalized by Dixon to construct H from C and the octonions O from

H. This procedure leads to identities expressing the product of two sums of

squares as another such sum. The familiar identity from high school

+ b

)(c

+ d

)=(ac + bd)

+(ad − bc)

is the special case n = 1. Many interesting further developments of these

ideas can be found in the book [83] by Conway and Smith.

2.7.1 Kervaire invariant

In 1960 Kervaire deﬁned a geometric topological invariant of a framed diﬀer-

ential manifold M of dimension m =4n + 2 generalizing the Arf invariant for

surfaces. Ten years earlier Pontryagin had used the Arf invariant of surfaces

embedded in S

k+2

with trivialized normal bundle to compute the homotopy

groups π

k+2

)fork>1. This group can be identiﬁed with the cobordism

group of such surfaces. The cobordism of manifolds and the corresponding

cobordism groups were deﬁned by Thom in 1952. Algebraically the Arf invari-

ant is deﬁned for any quadratic form over Z

. Kervaire deﬁned a quadratic

form q on the homology group H

2n+1

(M; Z

) by using the framing and the

Steenrod squares. The Kervaire invariant is the Arf invariant of q.Agen-

eral reference for this section is Snaith [356]. Kervaire used his invariant to

2.7 The Hopf Invariant 71

obtain the ﬁrst example of a non-smoothable 10-dimensional PL-manifold.

In the smooth category the ﬁrst three examples of manifolds with Kervaire

invariant 1 are S

×S

, S

×S

,andS

×S

. In these three cases the Ker-

vaire invariant is related to the Hopf invariant of certain maps of spheres. No

further examples of manifolds with Kervaire invariant 1 were known for many

years. The problem of ﬁnding the dimensions of framed manifolds for which

the Kervaire invariant is 1 came to be know as the Kervaire invariant 1

problem.

In 1969 Bill Browder proved that the Kervaire invariant is 0 for a mani-

fold M if its dimension is diﬀerent from 2

j+1

− 2,j∈ N. By 1984, it was

known that there exist manifolds of dimensions 30 and 62 with the Kervaire

invariant 1. Then on April 21, 2009, during the Atiyah 80 conference at Ed-

inburgh, Mike Hopkins announced that he, Mike Hill, and Doug Ravenel had

proved that there are no framed manifolds of dimension greater than 126 with

Kervaire invariant 1. The case n = 126 was open as of January 2010. Mike

Hopkins gave a very nice review of the problem and indicated key steps in the

proof at the Strings, Fields and Topology workshop at Oberwolfach (June,

2009). This section is based in part on that review. The proof makes essential

use of ideas from a generalized cohomology theory, called topological mod-

ular forms, or tmf theory. It was developed by Hopkins and collaborators.

Witten has introduced a homomorphism from the string bordism ring to the

ring of modular forms, called the Witten genus. This can be interpreted in

terms of the theory of topological modular forms or tmf.Wehaveal-

ready seen in Theorem 2.15, how the spheres S

enter from various

perspectives in it. We have also discussed their relation to the Hopf ﬁbration

and to diﬀerent physical theories. The relation of the other manifolds with

other parts of mathematics and with physics is unclear at this time.

The discussion of homotopy and cohomology given in this chapter forms a

small part of an area of mathematics called algebraic topology, where these

and other related concepts are developed for general topological spaces. Stan-

dard references for this material and other topics in algebraic topology are

BottandTu[55], Massey [279], and Spanier [357]. A very readable introduc-

tion is given in Croom [90].

Chapter 3

Manifolds

3.1 Introduction

The mathematical background required for the study of modern physical

theories and, in particular, gauge theories, is rather extensive and may be di-

vided roughly into the following parts: elements of diﬀerential geometry, ﬁber

bundles and connections, and algebraic topology of a manifold. The ﬁrst two

of these parts are nowadays fairly standard background for research workers

in mathematical physics. In any case, physicists are familiar with classical

diﬀerential geometry, which forms the cornerstone of Einstein’s theory of

gravitation (the general theory of relativity). Therefore, in this chapter we

give only a summary of some results from diﬀerential geometry to establish

notation and make the monograph essentially self-contained. Fiber bundles

and connections are discussed in Chapter 4. Characteristic classes, which are

fundamental in the algebraic topology of a manifold, are discussed in detail

in Chapter 5. The study of various classical ﬁeld theories is taken up in Chap-

ter 6. The reader familiar with the mathematical material may want to start

with Chapter 6 and refer back to the earlier chapters as needed.

There are several standard references available for further study of the

material on diﬀerential geometry that are discussed here in Chapters 3

and in Chapters 4 and 5; see, for example, Greub, Halperin, and Vanstone

[162,163,164], Kobayashi and Nomizu [225,226], Lang [245], and Spivak [358].

Some basic references for topology and related geometry and analysis are

Palais [310], Porteous [319], and Booss and Bleecker [47], For references that

also discuss some physical applications see Abraham and Marsden [1], Abra-

ham, Marsden, and Ratiu [2], Choquet-Bruhat, DeWitt-Morette, and Dillard-

Bleick [79], Choquet-Bruhat and DeWitt-Morette [78], Curtis and Miller [93],

Felsager [124], Jost [214], Marsden [278], Nakahara [295], Sachs and Wu [331],

Scorpan [343],andTrautman[378].

K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 3, 73

 Springer-Verlag London Limited 2010

74 3 Manifolds

3.2 Diﬀerential Manifolds

The basic objects of study in diﬀerential geometry are manifolds and maps

between manifolds. Roughly speaking a manifold is a topological space ob-

tained by patching together open sets in a Banach space. For many physi-

cal applications this space may be taken to be ﬁnite-dimensional. However,

inﬁnite-dimensional manifolds arise naturally in ﬁeld theories and, hence, in

this chapter our discussion of manifolds applies to arbitrary manifolds unless

otherwise stated.

Deﬁnition 3.1 Let M be a topological space and F a Banach space. A chart

(U, φ) is a pair consisting of an open set U ⊂ M and a homeomorphism

φ : U → φ(U) ⊂ F,

where φ(U) is an open subset of F . M is called a topological manifold or

simply a manifold modeled on the Banach space F if M admits a family

A = {(U

,φ

)}

i∈I

of charts such that {U

}

i∈I

covers M. This family A of

charts is said to be an atlas for M.If(U

,φ

), (U

,φ

) are two charts and

:= U

∩ U

= ∅, then

:= φ

◦ φ

−1

: φ

) → φ

)

is a homeomorphism. The maps φ

are called transition functions of the

atlas A. Various smoothness requirements on A are obtained by using the

transition functions. For example, if the φ

are C

-diﬀeomorphisms (i.e.,

and φ

−1

are of class C

), 0 <p≤ +∞ ,thenA is called a diﬀerential

(or a diﬀerentiable) atlas of class C

. Two diﬀerentiable atlases are said to

be compatible if their union is a diﬀerentiable atlas. Let A be a diﬀerentiable

atlas of class C

on M.The maximal diﬀerentiable atlas of class C

containing

A is called the diﬀerential structure on M of class C

determined by A.

A diﬀerential structure of class C

∞

is also called a smooth structure.The

corresponding atlas is called a smooth atlas. A manifold M together with a

diﬀerential structure is called a diﬀerential (or a diﬀerentiable) manifold.

By deﬁnition the dimension of M , dim M, is the dimension of F .IfF is

then M is called a real manifold of dimension m.IfF is C

and the

transition functions are holomorphic (complex analytic), then M is called a

complex manifold of complex dimension m.

One often considers a class of manifolds subjected to additional topologi-

cal restrictions such as compactness, paracompactness etc.. For example, the

result that locally compact Hausdorﬀ topological vector spaces are ﬁnite-

dimensional, implies that locally compact Hausdorﬀ manifolds are ﬁnite-

dimensional. We remark that a ﬁnite atlas (i.e., an atlas that consists of

a ﬁnite family of charts) always exists on a compact manifold.

The charts allow us to give an intrinsic formulation of various structures

associated with manifolds.

3.2 Diﬀerential Manifolds 75

Deﬁnition 3.2 Let M,N be real diﬀerential manifolds and f : M → N.

We say that f is diﬀerentiable (or smooth) if, for each pair of charts,

(U, φ), (V,ψ) of M and N, respectively, such that f (U) ⊂ V ,therepresen-

tative ψ ◦f ◦ φ

−1

of f in these charts is diﬀerentiable (or smooth). The set

of all smooth functions from M to N is denoted by F(M,N).WhenN = R

we write F(M ) instead of F(M, R). A bijective diﬀerentiable f ∈F(M,N)

is called a diﬀeomorphism if f

−1

is diﬀerentiable. The set of all diﬀeomor-

phisms of M with itself under composition is a group denoted by Diﬀ(M).

Diﬀeomorphism is an equivalence relation.

The class of diﬀerential manifolds and diﬀerentiable maps forms a category

(see Appendix C) that we denote by DIFF . The class of complex manifolds

and complex analytic maps forms a subcategory of DIFF. We discuss some

important complex manifolds but their physical applications are not empha-

sized in this book. An excellent introduction to this area may be found in

Manin [257] and Wells [399,398]. The class of topological manifolds and con-

tinuous maps forms a category that we denote by TOP. An important prob-

lem in the topology of manifolds is that of smoothability, i.e., to ﬁnd when

a given topological manifold admits a compatible diﬀerential structure.

We note that a diﬀerential structure on a topological manifold M is said to

be compatible if it is contained in the maximal atlas of M .Itiswellknown

that a connected topological manifold of dim < 4 admits a unique compatible

smooth structure. We observe that a topological manifold of dim > 3may

admit inequivalent compatible diﬀerential structures or none at all.

Deﬁnition 3.3 Let M be a diﬀerential manifold with diﬀerential structure

A and let U ⊂ M be open. The collection of all charts of A whose domain is

asubsetofU is an atlas for U, which makes U into a diﬀerential manifold.

This manifold is called an open submanifold of M . More generally, a subset

S ⊂ M is said to be a submanifold of M if, ∀x ∈ S, there exists a chart

(U, φ) with x ∈ U such that

1. φ(U) ⊂ G ⊕ H,whereG,

H ar

e Banach spaces;

2. φ(U ∩S)=φ(U) ∩ (G ×{b}),forsomeb ∈ H.

Then, denoting by π

the projection onto the ﬁrst factor, (U ∩ S, π

◦ φ)

is a chart at x. The collection of all these charts is an atlas on S,which

determines a diﬀerential structure on S. With this diﬀerential structure, S

itself (with the relative topology) is a diﬀerential manifold. We observe that

an open submanifold is a special case of a submanifold.

Let R

:= {x =(x

,...,x

) ∈ R

| x

≥ 0}.ThesetR

:= {x ∈

| x

=0} is called the boundary of R

.LetU be an open subset of R

in the relative topology. We denote by bd(U)thesetbd(U):=U ∩R

and call

bd(U) the boundary of U .WedenotebyInt(U)thesetInt(U ):=U \bd(U )

and call it the interior of U.IfU and V are subsets of R

and f : U → V ,

we say that f is of class C

, 1 ≤ p ≤ +∞ (smooth if p =+∞), if there

exist open neighborhoods U

of U and V

of V ,andamapf

: U

→ V

76 3 Manifolds

class C

such that f coincides with f

on U. The derivative Df

(x),x∈ U ,

is independent of the choice of U

. Thus we may deﬁne Df(x):=

(x),x∈ U. If f is a smooth isomorphism (i.e., diﬀeomorphism), then f

induces a diﬀeomorphism of Int U onto Int V and of δU onto δV .IfM is a

topological space, a chart with boundary for M is a pair (U, φ)whereU

is an open set of M and φ : U → φ(U) ⊂ R

is a homeomorphism onto the

open subset φ(U)ofR

. With obvious changes with respect to the deﬁnition

of diﬀerentiable manifold, we have the notions of an atlas with boundary

and of an n-manifold with boundary.Thus,ann-manifold with boundary

is obtained by piecing together open subsets of the upper half-space R

.This

construction can be extended to the inﬁnite-dimensional case. In this case in

place of R

we have a Banach space F andinplaceofR

we have the

half-space

:= {x ∈ F | λ(x) ≥ 0},

where λ is a continuous, linear functional on F (for details see Lang [245]).

The boundary of an n-manifold M with boundary denoted by ∂M,isthe

subset of the points x ∈ M such that there exists a chart with boundary (U, φ)

with x ∈ U and φ(x) ∈ R

. The interior of M is the set Int M := M \∂M.A

diﬀerentiable manifold with empty boundary is called a diﬀerentiable man-

ifold without boundary and, in this case, we recover our previous deﬁnition

of a diﬀerentiable manifold. The diﬀerentiable structure of an n-manifold

with boundary M induces, in a natural way, a diﬀerentiable structure on

∂M and Int M with which ∂M and Int M are manifolds without bound-

ary of dimensions n − 1andn, respectively. Thus, the boundary ∂(∂M)of

the boundary ∂M, of a manifold with boundary M ,isempty.Moreover,if

f : M → N is a diﬀeomorphism of manifolds with boundary, then the maps

Int

:IntM → Int N and f

∂

: ∂M → ∂N induced by restriction are diﬀeo-

morphisms. We now give several examples of manifolds that appear in many

applications.

Example 3.1 The sphere S

is the subset of R

n+1

deﬁned by

:= {(x

,...,x

n+1

) | x

+ x

+ ···+ x

n+1

=1}.

Let us consider the map (stereographic projection)

φ : S

\{(0,...,0, 1)}→R

deﬁned by

φ(x

,...,x

n+1

):=(x

/(1 −x

n+1

),...,x

/(1 − x

n+1

))

and the map

ψ : S

\{(0,...,0, −1)}→R

deﬁned by

3.2 Diﬀerential Manifolds 77

ψ(x

,...,x

n+1

):=(x

/(1 + x

n+1

),...,x

/(1 + x

n+1

)).

Then, for ψ ◦ φ

−1

: R

\{(0,...,0)}→R

\{(0,...,0)} one has

ψ ◦ φ

−1

,...,y

)=(y

/(y

+ ···+ y

),...,y

/(y

+ ···+ y

)),

∀(y

,...,y

) ∈ R

\{(0,...,0)}.Thus,ψ ◦ φ

−1

is smooth and {φ, ψ} is a

smooth atlas, which makes S

into a diﬀerential manifold. Let

:= {(x

,...,x

) | x

+ x

+ ···+ x

≤ 1}

be the unit disk in R

. D

is a manifold with boundary and ∂D

= S

n−1

a manifold without boundary of dimension n −1.

Example 3.2 Let f : R

→ R be a smooth map and let S := f

−1

({0}).Let

us suppose that S = ∅ and that ∀x ∈ S, the Jacobian matrix of f at x has

rank 1. Then for each x ∈ S there exists k, 1 ≤ k ≤ n, such that

∂f

∂x

(x) =0,

and hence, by the implicit function theorem, we have a smooth bijection of a

neighborhood of x in S onto a neighborhood of (x

,...,ˆx

,...,x

) ∈ R

n−1

(the signˆover a symbol denotes deletion). This bijection is a chart at x,and

the collection of all the charts of this type is an atlas, which makes S into a

diﬀerential manifold. We observe that, for the sphere S

, we have

= f

−1

({0})

with f : R

n+1

→ R deﬁned by f(x

,...,x

n+1

)=x

+ ···+ x

n+1

− 1.One

can verify that the diﬀerential structure on S

deﬁned by the procedure of this

example coincides with the one deﬁned in the previous example.

Manifolds of the type discussed in the above example arise in many appli-

cations.

Example 3.3 Consider the equivalence relation in R

n+1

\{0} deﬁned as

follows. We say that two points x, y

x =(x

,...,x

n+1

),y =(y

,...,y

n+1

) ∈ R

n+1

\{0}

are equivalent if there exists λ ∈ R\{0} such that x

= λy

, ∀i ∈{1, 2,...,n+

1}. Let us denote by [x] the equivalence class containing x and by RP

the

set

:= {[x] | x ∈ R

n+1

\{0}}.

Let U

,i∈{1, 2,...,n+1}, be the subset of RP

deﬁned by

:= {[(x

,...,x

n+1

)] ∈ RP

| x

=0}.