Marathe K. Topics in Physical Mathematics

Подождите немного. Документ загружается.

388 Appendix B

the most well known of these equivalent statements is: “The sum of the

angles of a triangle is π.” If we take the surface Σ of constant negative

curvature as a model of a plane then its straight lines are the geodesics.

Gauss’s theorem applied to a geodesic triangle implies that the sum of

the angles of a triangle is less than π. All other postulates of Euclid hold

in this model. In this geometry the parallel postulate is replaced by the

statement:

Let L be a straight line in a plane P . Through every point x ∈ P

not lying on the line L, there pass at least two straight lines that are

parallel to L.

The geometry based on the above parallel postulate is called non-

Euclidean geometry. It was developed independently by the Russian

mathematician Lobachevski and the Hungarian mathematician J. Bolyai.

Gauss discovered this geometry earlier but he never published his work, al-

though he had indicated it in his letter to F. Bolyai, J. Bolyai’s father. Eu-

clidean and non-Euclidean geometries became special cases of Riemannian

geometry whose foundations were unveiled in Riemann’s famous lecture

“

Uber de Hypothesen welche der Geometrie zu grunde liegen” in G¨ottingen

delivered in the presence of his examiner Gauss. Riemann’s work paved

the way for Einstein’s theory of gravity generalizing his special theory of

relativity.

6. Platonic Solids

A regular convex polyhedron in R

all of whose faces are congruent regu-

lar polygons is called a Platonic solid. The icosahedron, which we met in

Chapter 2, is one of the ﬁve Platonic solids. These ﬁve regular polyhedra

have been known since antiquity. Their names derive from the number of

faces they have: tetrahedron (4), hexahedron (6) (more commonly called

a cube), octahedron (8), dodecahedron (12), and icosahedron (20). Taking

the centers of faces of a polyhedron as vertices generates its dual poly-

hedron. A tetrahedron is self-dual since taking the centers of its faces

as vertices generates again a tetrahedron. Note that the cube and the oc-

tahedron are dual as are the dodecahedron and the icosahedron. Each of

the Platonic solids is homeomorphic to S

and so has Euler characteristic

χ = 2. Euler gave a combinatorial deﬁnition of χ by χ = v −e + f ,wherev

is the number of vertices, e the number of edges, and f the number of faces

of the given polyhedron. In table B.1 we also list b, the number of edges

bounding a face, and n,thevalence, or the number of edges meeting a

vertex. The Platonic solids exhaust all the possibilities of pairs (b, n)that

can occur in a regular polyhedron in R

Symmetry groups of the Platonic solids occur in many applications. As

we have seen in Chapter 2, the moduli space of icosahedra inscribed in a

sphere turns out to be a homology 3-sphere that is not homeomorphic to

. This example led Poincar´e to reformulate his famous conjecture (now

On the hypotheses which lie at the foundations of geometry

Appendix B 389

Tab le B. 1 Platonic solids or regular polyhedra in R

Name v e f b n

Tetrahedron 4 6 4 3 3

Cube 8 12 6 4 3

Octahedron 6 12 8 3 4

Dodecahedron 20 30 12 5 3

Icosahedron 12 30 20 3 5

a theorem). The interest in these solids named after Plato (around 360

BCE) has continued to this day. In fact, the title of the 2009 Reimar L¨ust

lecture in Bonn delivered on June 12 by Curtis McMullen (Fields Medal,

ICM 1998, Berlin) was “From Platonic Solids to Quantum Topology.”

7. Ramanujan

India’s greatest contribution to modern mathematics came in the last cen-

tury in the work of Srinivasa Ramanujan. Its story is one of the most

fascinating chapters in the history of mathematics. It began about a hun-

dred years ago with a long letter from Ramanujan, a poor unknown clerk

in Madras to Prof. G. H. Hardy, a well known English mathematician. It

was a desperate cry for help. Ramanujan wrote that he had no university

education but was striking a new path for himself in mathematics research.

The letter contained over 100 results in mathematics that Ramanujan had

obtained on his own without any formal university education in mathemat-

ics. His earlier attempts to communicate his work to other mathematicians

had been unsuccessful. Hardy decided to show the letter to his close friend

J. E. Littlewood who was also an eminent mathematician. They immedi-

ately realized that the writer had no exposure to standard mathematics.

The letter contained some well known results and some that were known

to be false. Some of the results seemed familiar. But it took Hardy far

more time to prove them than he anticipated. The letter also contained

several results the likes of which they had never seen before. Ramanujan

had given no indication of how he obtained these results. After a few hours

of work, they could not prove any of these new results and they were the

best in the ﬁeld. Hardy wrote later: A single look at them is enough to

show that they could only be written down by a mathematician of the

highest class. We concluded that these strange looking results must be

true since no one would have had the imagination to invent them. Hardy

replied immediately: I was exceedingly interested by your letter and by the

theorems you state. However, I must see the proofs of some of your asser-

tions. Ramanujan wrote back: I have found in you a friend who views my

labors sympathetically. I am a half starving man who needs nourishment

to preserve my brains.

390 Appendix B

Hardy was shocked to ﬁnd out that Ramanujan had failed the ﬁrst Arts

examination of Indian universities. He could not get him a satisfactory

position in India to continue his research in mathematics. He therefore

used all his inﬂuence to get him a position at Cambridge University which

involved no teaching duties. In March 1914 Ramanujan sailed to England

to take up his new position. In 1916 he was awarded BSc by research (the

degree was later called Phd). Winter months were very diﬃcult for his

health. But he continued to produce ﬁrst rate mathematics working with

Hardy almost every day that he was not ill. In 1917 he fell seriously ill and

his doctors felt that he did not have much time. Hardy worked tirelessly to

make sure that Ramanujan received the highest honor in Britain, namely

election as a fellow of the Royal Society of London in 1918. Hardy wrote:

He will return to India with a scientiﬁc standing and reputation such as no

Indian has enjoyed before, and I am conﬁdent that India will regard him

as the treasure he is. Ramanujan returned to India in March 1919 in poor

health. His illness was never properly diagnosed and his health continued

to deteriorate. On April 26, 1920 at age 32 Ramanujan passed away.

Ramanujan rarely gave detailed and rigorous proofs of his formulas

and this led him sometimes to wrong results. In his notebooks Ramanu-

jan listed a large number of results. Some of these notebooks were lost

and were found quite accidentally. They have now been published by the

American Mathematical Society. Ramanujan discovered nearly 4000 re-

sults by himself. Most of the new results are now proved but there are

still several which are unproved. In the last year of his life spent in India,

Ramanujan’s health continued to deteriorate but his mathematical abili-

ties remained undiminished. This is clearly shown by Ramanujan’s famous

last letter to Hardy written shortly before his death. In this letter he an-

nounced his discovery of the mock theta functions and discussed some of

their remarkable properties. We are just beginning to understand these

functions 90 years after their discovery. Recently, my friend Prof. Don Za-

gier, director of the Max Planck Institute for Mathematics in Bonn gave a

research seminar on the mock theta functions and their surprising relation

to string theory in Physics. Ramanujan would not have been surprised by

this. He always felt that all his results were part of universal knowledge.

If they appear where we don’t expect to see them, then this is because of

our limited vision. Let us hope and pray that we improve our record of

recognizing great visionaries and helping them fulﬁll their destiny.

Let me conclude this brief biography of Ramanujan by recounting the

most well known story about Ramanujan’s friendship with numbers. Hardy

went to visit Ramanujan when he was hostitalized. Hardy said that he

came in a taxi bearing a rather dull and uninteresting number and he took

it as an omen that he would ﬁnd Ramanujan in poor health. On hearing

the number, Ramanujan’s eyes lit up. He said: No Mr. Hardy, in fact it is

one of the most interesting numbers that I know. It is the smallest number

that can be written as the sum of two cubes in two diﬀerent ways. This

Appendix B 391

number 1729 is now known as the Ramanujan number. For a fascinating

account of Ramanujan’s life and work I refer the reader to Kanigel’s book

“The man who knew inﬁnity.” It is a deﬁnitive biography of Ramanujan.

Appendix C

Categories and Chain Complexes

C.1 Categories

In modern mathematics the language of category theory is often used to

discuss properties of diﬀerent mathematical structures in a uniﬁed way. In

recent years category theory and categorical constructions have been used in

algebraic and geometric theories and have also found surprising applications

to invariants of low-dimensional manifolds. The axiomatic development of

TQFT is also based on a special category. Furthermore, applications to string

theory and symplectic ﬁeld theory have developed into an extensive area of

research. We give here a mere outline of the basic ideas of category theory.

A category C consists of

1. aclassOb(C) whose elements are called the objects of C;

2. a correspondence that associates to eacth ordered pair (A, B) of objects

of C asetMor

(A, B)(orsimplyMor (A, B)) such that

A = C orB= D ⇒ Mor (A, B) ∩ Mor(C, D)=∅.

The elements of Mor

(A, B) are called C-morphisms (or simply mor-

phisms or arrows)ofA to B;

3. a correspondence that associates to each ordered triple (A, B, C) of objects

of C amap◦, called the composition law,

◦ : Mor (B, C) ×Mor(A, B) → Mor (A, C),

such that (f, g) → f ◦ g, satisfying the following two properties:

a. Existence of identities: ∀A ∈ Ob(C), there exists an identity mor-

phism 1

∈ Mor (A, A) such that, ∀B ∈ Ob(C), ∀f ∈ Mor (A, B), ∀g ∈

Mor(B,A),

f ◦ 1

= f and 1

◦ g = g;

393

394 Appendix C

b. Associativity: (h ◦ g) ◦ f = h ◦ (g ◦ f), ∀(f, g,h) ∈ Mor (A, B) ×

Mor(B,C) × Mor(C, D).

As is customary we have used the same notation for morphisms as for maps.

A morphism f ∈ Mor

(A, B)issaidtobeaC-isomorphism,orsimply

an isomorphism, of A with B if there exists a morphism g ∈ Mor

(B,A)

such that

f ◦ g =1

and g ◦ f =1

Two objects A, B are said to be isomorphic if there exists an isomorphism

of A with B. One frequently refers to only Ob(C), when speaking of a cat-

egory whose morphisms and composition law are understood. For example,

we speak of the category Gr of groups with the understanding that the mor-

phisms are the homomorphisms of groups with the usual composition law.

A similar convention holds for other categories such as the category Top of

topological spaces whose morphisms are continuous functions and Top

pointed topological spaces whose morphisms are base point-preserving con-

tinuous functions.

A category A is said to be a subcategory (resp., full subcategory)

of the category B if Ob(A) ⊂ Ob(B)andforeachX, Y ∈ Ob(A)wehave

Mor

(X, Y ) ⊂ Mor

(X, Y )(resp.,Mor

(X, Y )=Mor

(X, Y )). As an ex-

ample we note that the category Ab of Abelian groups is a full subcategory

of Gr, while the category DIFF of diﬀerential manifolds is a subcategory of

the category TOP of topological manifolds which is not a full subcategory.

Let C be a category. An object A ∈ Ob(C) is called a universal ini-

tial (resp. universal ﬁnal)orsimplyinitial (resp., ﬁnal) object of C if,

∀B ∈ Ob(C), Mor (A, B)(resp.Mor(B,A)) contains only one element. Uni-

versal initial (resp. ﬁnal) objects of a category, if they exist, are isomorphic.

Universal objects play an important role in many constructions in modern

mathematics, even though in applications it is customary to consider some

concrete realization of such objects. We now give some examples of universal

objects.

Let K be a ﬁeld and let V,W denote ﬁxed K-vector spaces and let Z, Z

,...

denote K-vector spaces. Consider the category B such that

1. the set of objects is Ob(B)={ (Z, g) | g ∈ L(V, W; Z)}, where L(V,W; Z)

is the K-vector space of all bilinear maps of V × W to Z;

2. for g

∈ L(V,W; Z

),g

∈ L(V,W; Z

) a morphism f from (Z

)to

) is a linear map f : Z

→ Z

such that f ◦ g

= g

, i.e., the

following diagram commutes:

V × WZ





Appendix C 395

It can be shown that in this category a universal initial object exists. It is

called the tensor product of V and W and is denoted by (V ⊗ W, t)or

simply by V ⊗ W . It is customary to write v ⊗ w for t(v, w) and to call it

the tensor product of v and w.IfV and W are ﬁnite-dimensional, then a

realization of V ⊗W is given by L(V

∗

; K), where v ⊗w, v ∈ V, w ∈ W

is identiﬁed with the map deﬁned by

(α, β) → α(v)β(w), ∀α, β ∈ V

∗

× W

∗

IfintheaboveexamplewetakeW = V and require g ∈ L(V,V ; Z)tobe

skew-symmetric (resp., symmetric) then in the resulting category the univer-

sal initial object is called the exterior product (resp., symmetric prod-

uct)ofV with itself and is denoted by Λ

(V )(resp.,S

(V )). The map

V ×V → Λ

(V ) is denoted by ∧ and ∧(α, β) is denoted by α∧β and is called

the exterior product of α with β.ThemapV ×V → S

(V ) is denoted by

∨,and∨(α, β) is denoted by α ∨β andiscalledthesymmetric product of

α with β. Both the above constructions can be extended to any ﬁnite number

of vector spaces. In particular, the tensor product

V ⊗ V ⊗···⊗V



 

r times

⊗V

∗

⊗ V

∗

⊗···⊗V

∗

  

s times

where V

∗

is the dual of V , is called the tensor space of type (r, s)overV

and is denoted by T

(V ). The exterior product

V ∧ V ∧···∧V



 

k times

is called the space of exterior k-forms on V and is denoted by Λ

(V ). The

symmetric product

V ∨ V ∨···∨V



 

k times

is called the space of symmetric k-tensors on V and is denoted by S

(V ).

Another important example of an initial object is the direct limit of mod-

ules, which we used to deﬁne singular cohomology with compact support. Let

I be a directed set, i.e., a set with a partial order ≤ such that, ∀i

∈ I,

there exists i ∈ I such that i

≤ i, i

≤ i. If I is a directed set, let us denote

by I

the subset of I

:= I ×I deﬁned by

:= {(i, j) ∈ I

| i ≤ j}.

A direct system in a category C relative to the directed set I is a couple

D =({A

| i ∈ I} , {f

| (i, j) ∈ I

})

396 Appendix C

such that {A

| i ∈ I} is a family of objects of C and f

: A

→ A

is a family

of C-morphisms such that

1. f

= id

, ∀i ∈ I;

2. f

◦ f

= f

, ∀i, j, k ∈ I, i ≤ j ≤ k.

Given the direct system D,letC(D) be the category such that:

Ob: The objects of C(D) are the pairs (A, {g

| i ∈ I})whereA ∈ Ob(C)

and g

: A

→ A, i ∈ I,isaC-morphism such that g

= g

◦f

, ∀(i, j) ∈ I

Mor: If A =(A, {g

| i ∈ I}), B =(B, {h

| i ∈ I}) are objects of

C(D), a morphism of A to B is a C-morphism f : A → B such that

f ◦ g

= h

, ∀i ∈ I.

Given the direct system D as above, the direct limit of D is a universal

initial object of the category C(D), i.e., an object (A, {g

| i ∈ I})ofC(D)

such that, ∀(B, { h

| i ∈ I}) ∈ Ob(C(D)), there exists a unique C-morphism

f : A → B satisfying the relations f ◦ g

= h

, ∀i ∈ I. By deﬁnition,

two direct limits of a direct system D, if they exist, are isomorphic objects

of the category C(D). Hence, we can speak of “the” direct limit of D.If

(A, {g

| i ∈ I}) is a direct limit of D,wealsosaythatA is the direct limit

of the A

and write

A = lim

−→

Direct limits exist in the categories of sets, topological spaces, groups, and

R-modules. As an example we give the direct limit for a direct system D in

the category of R-modules. Let A = ⊕

i∈I

be the direct sum of the modules

and u

: A

→ A the natural injection. Let us denote by B the submodule

of A generated by all elements

) −u

), (i, j) ∈ I

∈ A

Let π : A → A/B be the quotient map and, ∀i ∈ I, g

be the map deﬁned by

:= π ◦ u

: A

→ A/B.

Then (A/B, {g

| i ∈ I}) is a direct limit of D.

Remark.IfI has a largest element m, i.e., i ≤ m, ∀i ∈ I,then

, {g

= f

| i ∈ I})

is a direct limit of D.

Let A, B be two categories. A covariant (resp., contravariant) functor

F from A to B is a correspondence which associates with each X ∈ Ob(A)

an object F (X) ∈ Ob(B)andtoeachf ∈ Mor

(X, Y ) a morphism F (f ) ∈

Mor

(F (X),F(Y )) (resp., F (f) ∈ Mor

(F (Y ),F(X))) such that

1. F (1

)=1

F (X)

Appendix C 397

2. F (g ◦ f)=F (g) ◦ F (f), (resp., F (g ◦ f )=F (f ) ◦ F (g)), ∀(f,g) ∈

Mor

(X, Y ) × Mor

(Y,Z)).

An example of a covariant functor is provided by the nth homotopy functor

: Top

→ Gr, which associates with each pointed topological space its

nth homotopy group as deﬁned in Chapter 2.

Let F : A→B,G: A→Bbe two covariant functors. A natural

transformation τ : F → G is a correspondence τ which associates to each

object X ∈Aa morphism τ(X) ∈ Mor

(F (X),G(X)) such that

G(f) ◦ τ(X)=τ(Y ) ◦ F (f), ∀f ∈ Mor

(X, Y ).

The deﬁning equation of the natural transformation τ may be indicated by

the commutativity of the following diagram:

F (Y ) G(Y )

τ(Y )

F (X) G(X)

τ(X)

F (f)

G(f)

A similar deﬁnition may be given for a natural transformation between con-

travariant functors.

C.2 Chain Complexes

Any homology (cohomology) theory of topological spaces is based on the

construction of a structure called a chain (cochain) complex. In this section we

discuss the general theory of chain complexes over a principal ideal domain.

Let A be a commutative ring with unit element. We recall that A is called

an integral domain if it has no zero divisors, i.e.,

∀a, b ∈ A, ab =0⇒ a =0orb=0.

A subring I of A is called an ideal if AI = I = IA.AnidealI is called a

principal ideal if I = aA, for some a ∈ A.Aprincipal ideal domain is

an integral domain A such that every ideal in A is principal. Clearly, every

ﬁeld is a principal ideal domain. The ring of integers Z is a principal ideal

domain that is not a ﬁeld. In what follows we shall be concerned with modules

over a ﬁxed principal ideal domain P. Recall that a module M over P or

a P-module is a generalization of the notion of a vector space with base

ﬁeld replaced by P. For example, any ideal in P is a P-module. The class

of P-modules and P-linear maps forms a category. It is called the category

of P-modules and is denoted by M

. Note that every Abelian group can be