Homotopy Theory and The De Rahm

Cohomology

Badis Ydri

09/14/98

Abstract

This review is based completely on the book by Nash and Sen

” Topology and Geometry For Physicists ” , and it introduces two

major concepts from differential geometry the first one is Homotopy

and The Fundamental Groups and the second one is Homology groups

and their dual Cohomolgy groups .

0

Contents

1 Definitions 3

1.1 Topological space . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Boundary and Interior . . . . . . . . . . . . . . . . . . . . . . 41.7 compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.7.1 open covering . . . . . . . . . . . . . . . . . . . . . . . 41.7.2 compactness . . . . . . . . . . . . . . . . . . . . . . . . 41.7.3 the case of Rn . . . . . . . . . . . . . . . . . . . . . . 4

1.8 connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.9 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 topological invariants . . . . . . . . . . . . . . . . . . . . . . . 51.11 homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Homotopy 6

2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Definition of Homotopy . . . . . . . . . . . . . . . . . . . . . . 82.4 Definition of the Fundamental Group . . . . . . . . . . . . . . 82.5 Topological Invariance of the Fundamental Group . . . . . . . 102.6 Simplexes and the Calculating Theorem . . . . . . . . . . . . 11

2.6.1 The Calculating Theorem . . . . . . . . . . . . . . . . 132.6.2 Triangulation of a Space . . . . . . . . . . . . . . . . . 132.6.3 Triangulation of the Circle . . . . . . . . . . . . . . . . 132.6.4 Triangulation of the Disc . . . . . . . . . . . . . . . . . 14

2.7 Fundamental Group of a Product X×Y . . . . . . . . . . . . . 15

3 The Homology Group 15

3.1 Definition of the Homology Groups . . . . . . . . . . . . . . . 163.1.1 Oriented Simplicial Complex . . . . . . . . . . . . . . . 163.1.2 Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.3 Boundary Operator . . . . . . . . . . . . . . . . . . . . 163.1.4 Cycles and Boundaries . . . . . . . . . . . . . . . . . . 17

1

3.1.5 Homology Groups . . . . . . . . . . . . . . . . . . . . . 183.1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.7 Betti Numbers . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Relative Homology Groups . . . . . . . . . . . . . . . . . . . . 243.2.1 The Relative Chain Group . . . . . . . . . . . . . . . . 243.2.2 The Relative Boundary Operator . . . . . . . . . . . . 243.2.3 The Relative Cycle Groups . . . . . . . . . . . . . . . . 243.2.4 The Relative Boundary Groups . . . . . . . . . . . . . 243.2.5 The Relative Homology Groups . . . . . . . . . . . . . 253.2.6 The Excision Theorem . . . . . . . . . . . . . . . . . . 253.2.7 Homology Sequence . . . . . . . . . . . . . . . . . . . . 273.2.8 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.9 Singular Homology . . . . . . . . . . . . . . . . . . . . 31

3.3 The De Rham Comology . . . . . . . . . . . . . . . . . . . . . 323.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.2 Poincare’s Lemma . . . . . . . . . . . . . . . . . . . . 35

3.4 Calculation Of Hp(M ; R) . . . . . . . . . . . . . . . . . . . . . 363.4.1 Hp(Rn; R) . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.2 Hp(Sn, R) , p = 0, ..., n . . . . . . . . . . . . . . . . . . 363.4.3 T n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.4 General Remarks . . . . . . . . . . . . . . . . . . . . . 37

2

1 Definitions

1.1 Topological space

Let X be a set and Y = Xa a collection ( finite or infinite ) of subsets of X .the pair (X, Y ) form a topological space if the Xa and Y satisfy :

a)φ∈Y , X∈Y

b)any finite or infinite subcollection Za of the xa has the property that⋃

Za∈Y .

c)any finite subcollection Za1, Za2, ..., Zan of the xa has the property that⋂

Za∈Y .

X is called a topological space , Xa are called open sets . the choice ofY = xa satisfying the above requirements gives a topology to X . if Y is thecollection of all subsets of x then it is called the discrete topology of X . ifY = φ, X then it is called the trivial topology .

1.2 Continuity

a function f mapping from the topological space X to the topological spaceY is continuous if the inverse image of an open set in Y is an open set in X .

1.3 Neighbourhoods

given a topology T = Xa on X , then N is a neighbourhood of a point x∈Xif N is a subset of X and N contains some open set Xa to which x belongs .

clearly N itself is not an open set , however all open sets Xa which containx are neighbourhoods of x since they are contained in themselves , thusneighbourhoods are a little bit more general than open sets .

1.4 closed sets

let T = Xa be a topology on X , then any subset U of X is closed if itscomplement in X , X - U , is an open set .

this definition applied to the sets X and φ shows that they are both openand closed regardless of the details of the topology T .

3

1.5 closure of a set

consider a set U , in general there will be many closed sets which contain U. let us denote them by Fa . then the intersection of all the Fa :

⋂Fa = U is

called the closure of U and it is the smallest closed set which contains U .

1.6 Boundary and Interior

The interior of a set U is the union of all open subsets Oa of U and is written: U0 =

⋃Oa . it is the largest open subset of U . U0 = U if and only if U is

open .

The boundary of a set U , b(U) , is the complement of the interior of Uin the closure of U : b(U) = U − U0 .

open sets are always disjoint from their boundaries : U∩b(U) = φ . inthe other hand closed sets always contain their boundaries : U∩b(U) = b(U).

A set U is dense in X if its closure U is X .

1.7 compactness

1.7.1 open covering

given a family of a sets F = Fa , then F is a cover of U if U⊂⋃

Fa . if all theFa happen to be open sets the cover is called an open cover .

1.7.2 compactness

Consider the set U and all its possible open coverings . the set U is com-pact if for every open covering Fa with U⊂

⋃Fa there always exists a finite

subcovering F1, F2, ..., Fn of U such that : U⊂⋃n

1Fa .

1.7.3 the case of Rn

if X⊂Rn , then X is compact if and only if X is closed and bounded. theboundedness condition is related to the notion of finite volume .

4

1.8 connectedness

a set X is connected if it cannot be written as : X = X1∪X2 . where X1 andX2 are both open and X1∩X2 = φ .

1.9 Homeomorphisms

consider two topological spaces T1 and T2 , a map α from T1 to T2 is calleda homeomorphism if it is both continuous and has an inverse which is alsocontinuous . we say then that T1 is homeomorphic to T2 and it means exactlythat we can continuously deformed T1 into T2 and vice versa .

if α is homeomorphism , then so is α−1 . if T1 is homeomorphic to T2 andT2 homeomorphic to T3 , then by composing the two homeomorphisms T1 ishomeomorphic to T3 . this means that we immediately are able to divide alltopological spaces up into equivalence classes : a pair of spaces T1 and T2

belong to the same equivalence class if they are homeomorphic .

1.10 topological invariants

the next stage is to produce enough mathematical criteria to characteriseany particular equivalence class , and the idea behind this characterizationis to produce enough topological invariants ( things which do not changeunder homeomorphism ) to uniquely specify each equivalence class . thesetopological invariants can be :

a)integers such as the dimension n of Rn .b)properties of topological spaces such as connectedness or compactness.c)mathematical structures such as : homotopy groups , homology and

cohomology groups .in the search for these invariants two other notions are used : isotopy and

homotopy .

1.11 homotopy

consider two continuous maps : α1 : X−→Y and α2 : X−→Y . α1 is said tobe homotopic to α2 if it can be deformed into α2 , in precise mathematicalterms we can define a continuous map F : X×[0, 1]−→Y which satisfies :F (x, 0) = α1(x) and F (x, 1) = α2(x) . in pther words as the real variable t

5

in F (x, t) varies continuously from 0 to 1 in the unit interval [0, 1] ; the mapα1 is deformed continuously into the map α2 .

homotopy is clearly an equivalence relation and it divides the space ofcontinuous maps from X to Y , C(X,Y) , into equivalence classes . sincea homeomorphism is also a continuous map , these equivalence classes areunchanged under homeomorphism of X or Y . these homotopy equivalenceclasses are therefore topological invariants of the pair of spaces X and Y .

2 Homotopy

2.1 Example

Let X1 and X2 two rectangular regions of R2 , X1 contains a hole while X2

does not . clearly any loop in X2 can be shrunk to a point , but in X1 thereare two kinds of loops : those which can be shrunk to a point , and t hosewhich contain the hole . if we introduce an equivalence between loops inwhich two loops are equivalent ( or homotopic) if one can be obtained fromthe other by a process of continuous deformation , then although there arean infinite number of loops in X1 and X2 , X2 contains just one homotopyclass of loops , and X1 contains one homotopy class hn for each integer nwhere the elements of hn are those loops which encircles the hole n times (n > 0 is taken to mean cl ockwise encirclement , n < 0 is taken to meananticlockwise encirclement , n = 0 no encirclement ). these homotopy classesof loops allow us to distinguish the space X1 from the space X2 .

this example suggests that the study of equivalence classes of loops in anytopological space X might be a way of determining the holes in the space . itturns out that we can even furnish such loops with a group structure and getwhat we call the fundam ental group π1(x) : two loops A and B starting fromthe same point x can be combined to produce a third loop C also startingfrom x , C = A ∗ B is the loop which starts from x and first goes round theloop A and then goes round the loop B , C = B ∗A means the loop that goesround B first and A second . the inverse of a loop A , A−1 starts from x andgoes round the loop A in the opposite sense . the identity loop is defined tobe the loop which stays at x all the time , ε = A ∗ A−1 is a loop based at xand it is not the identity loop , however ε is homotopic to the identity loop .the elements of the fundamental group should be homotopy classes of loops

6

rather that the loops themseleves .

2.2 General Definitions

1)A path α(t) in X from x0 to x1 is a continuous map from [0, 1] into X withα(0) = x0 , α(1) = x1.

2)A path γ(t) from x0 to x1 can be combined with a path σ(t) from x1

to x2 to give a new path γ ∗ σ(t) from x0 to x2 only if γ(1) = σ(0) , then :

γ ∗ σ(t) =

γ(2t) 0≤t≤1/2σ(2t− 1) 1/2≤t≤1

(1)

3)A topological space X is called arc-wise connected or path connected ifthere always exists a path α between any pair of points x0 and x1 in X .

this notion of path connected is stronger than the notion of connectednessin other words a space can be connected but not path connected , but a pathconnected space is always connected .

4)A closed path or loop in X at the point x0 is a path α(t) for which :α(0) = α(1) = x0 .

5)the product C of two loops A and B based at x0 is written as : C = A∗Band is defined by :

C(t) =

A(2t) 0≤t≤1/2B(2t− 1) 1/2≤t≤1

(2)

where C(0) = A(0) = x0 and C(1) = B(1) = x0 .

6)the inverse loop A−1 based at x0 is defined by :

A−1(t) = A(1− t), 0≤t≤1 (3)

7)the constant loop c based at x0 is the map : c : [0, 1]−→X such that :c(t) = x0 for all t .

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2.3 Definition of Homotopy

DefinitionTwo loops A and B based at x0 are homeomorphic , A'B , if there exists

a continuous map H from [0, 1]×[0, 1] to X : H : [0, 1]×[0, 1]−→X and Hsatisfies :

H(t, 0) = A(t), 0≤t≤1

H(t, 1) = B(t), 0≤t≤1

H(0, s) = H(1, s) = x0, 0≤s≤1

(4)

the map H is called a homotopy between A and B . for any fixed value of s ,H(t, s) = As(t) as t varies is a loop in X based at x0 which interpolates in acontinuous manner between the loops A and B .

Lemma:if A0 , B0 , C0 ,...,A1 , B1 , C1 ,...represents loops based at x0∈X then :

a) A0 'A0

b) A0 'B0=⇒B0'A0

c) A0 'B0, B0'C0=⇒A0'C0

d) A0 'B0=⇒A−10 'B−1

0

e) A0 'A1, B0'B1=⇒A0 ∗B0'A1 ∗B1

(5)

this means that homotopy is an equivalence relation , in other words thespace of loops can be partitioned into disjoint equivalence classes , membersin a given class being homotopic to each other . Let us denote by [A] theequivalence class of loops homotopic to the loop A .

2.4 Definition of the Fundamental Group

Definition

8

Let us define the set : π1(X, x0) as the set of homotopy classes of loopsbased at x0 . and let us define the product [A].[B] of the two elements [A]and [B] of π1(X, x0) by :

[A].[B] = [A ∗B] (6)

in other words the product of two homotopy classes is defined to be the classdetermined by their representative elements .

π1(X, x0) with the product law just defined is a group . the unit elementof this group is the homotopy class of the constant loop at x0 . the groupπ1(X, x0) is called the first homotopy group or the fundamental group of thepath c onnected topological space X based at x0 .

Theoremif X is a path connected topological space and x0 , x1 ∈ X then the groups

π1(X, x0) and π1(X, x1) are isomorphic .to see this let γ be the path in X from x0 to x1 , then any loop α based

at x0 can be converted into a loop based at x1 , namely the loop γ−1 ∗α ∗ γ .in the same way any loop β based at x1 can be converted into a loop basedat x0 , namely γ ∗ β ∗ γ−1 . we have then two homomorphisms:

σγ : π1(X, x0) −→π1(X, x1)

([α], x0) −→([γ−1 ∗ α ∗ γ], x1)

(7)

and

σγ−1 : π1(X, x1) −→π1(X, x0)

([β], x1) −→([γ ∗ β ∗ γ−1], x0)

(8)

it follows that σγ ∗σγ−1 and σγ−1 ∗σγ are both identity homomorphism , i.e .isomorphisms ( Homomorphisms which are one to one and onto ) . this meansthat σγ a nd σγ−1 are themselves isomorphisms , not just Homomorphisms ,which establishes the theorem .

Thus π1(X, x0) for a path connected topological space depends , up toisomorphism , on the space X , and not on the base point selected .

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2.5 Topological Invariance of the Fundamental Group

Definitiontwo spaces X and Y are of the same homotopy type if we have continuous

maps f : X−→Y and g : Y−→X such that f.g and g.f are homotopic to theidentity maps in Y and X respectively i.e, f.g'1Y and g.f∼ eq1X .

we defined earlier that if X and Y are two homeomorphic spaces thenthere must exist two continuous maps f : X−→Y and g : Y−→X such thatf.g = 1Y and g.f = 1X . but since 1X and 1Y are homotopic to the mselvesit follows that X and Y are of the same homotopy type .

Lemmalet : F : X×I−→Y , where I = [0, 1] , be a homotopy between two maps

: f0 : (X, x0)−→(Y, y0) which is based at y0 and f1 : (X, x0)−→(Y, y1) whichis based at y1 . let γ be the path connecting y0 to y1 in Y . then the maps f0

, f1 and γ induce group homomorphisms f ∗

0 , f ∗

1 and σγ respectively definedby :

f ∗

i : π1(X, x0) −→π1(Y, yi)

([α], x0) −→([fi(α)], yi)

(9)

and :

σγ : ([α], y0)−→([γ−1 ∗ α ∗ γ], y1) (10)

such that :

σγ ∗ f ∗

0 = f ∗

1 (11)

Theoremif X and Y are two path connected topological spaces of the same homo-

topy type then π1(X, x0) is isomorphic to π1(X, y0) where : x0∈X and y0∈Y.

to see this we recall that since X and Y are of the same homotopy type ,there must exist two maps f : X−→Y and g : Y−→X and two homotopies

F and H such that : f.gF

︷︸︸︷

' 1Y and g.fH

︷︸︸︷

' 1X . using the previous lemma

10

since f.g and 1Y are two maps in the same homotopy class we should have :σγY

.1∗Y = (f.g)∗ = f ∗.g∗ . in the same way since g.f and 1X are two maps inthe same homotopy calss we have : σγX

.1∗X = (g.f)∗ = g∗.f ∗ . but in this casethe group homomorphisms σγX

and σγYare both the identity homomorphism

and clearly 1∗Y and 1∗X are isomorphic which means that f ∗g∗ and g∗f ∗ mustbe isomorphic , Hence f ∗ and g∗ are isomorphic which establishes the theorem.

Corollaryif X and Y are homeomorphic path connected topological spaces then

π1(X, x0) is isomorphic to π1(X, y0) where : x0∈X and y0∈Y .this establishesthe fundamental group as a topological invariant of a space .

Deformation Retract1)A subset A of a topological space X is called a retract of X if there

exists a continuous map , called a retraction : r : X−→A such that r(a) = afor any a ∈A .

2)A subset A of a topological space X is a deformation retract of X ifthere is a retraction r : X−→A and a homotopy H : X×[0, 1]−→X suchthat :

H(x, 0) = x

H(x, 1) = r(x)

H(a, t) = a, if, a∈A

(12)

intuitively , A is a deformation retract of X means that X can be continuouslydeformed to A without moving points of A at any stage . and in terms of thefundamental group the interest in deformation retracts lies in the followingtheorem : if X is a pat h connected topological space and A is a deformationretract of X , then π1(X, a) is isomorphic to π1(A, a) where a∈A . if wecan show that a space A is a deformation retract of a space X and we knowhow to calculate the fundamenta l group of one of the spaces we have , ineffect,determined the fundamental group of the other space .

2.6 Simplexes and the Calculating Theorem

Definitions

11

1)Let x1,x2,...,xm+1 be distinct points in Rn . the set of points x1,x2,...,xm+1

are independent if the m vectors : x2 − x1,x3 − x1,...,xm+1 − x1 are linearlyindependent vectors.

2)An m simplex , σm , is the set of points X in Rn given by :

σm = x =i=m+1∑

i=1

λixi/λi≥0,i=m+1∑

i=

λi = 1 (13)

where x1,x2,...,xm+1 are independent . we call x1,x2,...,xm+1 the verticesof the m dimensional simplex σm = [x1, x2, ..., xm+1] . the λi are called thebarycentric coordinates of the simplex . the point x of σm , which correspondsto the set of barycentric coordinates λ1,λ2,...,λm+1 , can be regarded as thecenter of mass of the system with masses λ1,λ2,...,λm+1 p

laced at the vertices x1,x2,...,xm+1 respectively . this physical analogyleads us to expect that if all the λi are non zero then the corresponding setof points x represent the interior of σm while if any λi is zero then the set ofpoints x represents a face of σm opposite to the vertex xj .

3)the set λ1x1 + λ2x2 + ... + λm+1xm+1/λj = 0 is called the jth face ofthe simplex σm . it lies opposite to the jth vertex xj . an m simplex σm is agenera lized triangle in m dimension : σ2 is a triangle , σ3 is a tetrahedron .

4)A simplicial complex K is a finite collection of simplexes in some Rn

satisfying :

i)if σp is in K , then all faces of σp belong to K.

ii)if σp and σq from K then either σp∩σq = φ or σp∩σq is a common faceof σp and σq.

the dimension of K is defined to be the maximum of the dimensions ofthe simplexes of K .

5)A simplicial complex K is path connected if for every pair of vertices uand v of K there is a sequence v0,v1,...,vn of vertices in K such that

i)v0 = u and vn = v.

ii)vivi+1 is a 1 simplex of K for all i = 0, 1, ..., n− 1 .6)The union of the members of K with the euclidean subspace topology

is called the polyhedron associated with K . a polyhedron is path connectedif the simplicial complex with which it is associated is path connected . apolyhedron can be regarded as a su bspace of some Euclidean Rn which isobtained by properly gluing together certain simplexes , the simplexes are

12

glued together in such a way that two simplexes , if they meet , have acommon vertex or edge .

7)if all the vertices of K are ordered in the form a0 < a1... < am+1 theneach simplex σn of K can be written as ai0, ai1, ..., ain where i0 < i1 < ... <in . these are the ordered simplexes of K .

2.6.1 The Calculating Theorem

Let K be a path connected polyhedron , and let L be one dimensional sub-polyhedron of K which is contractible ( has the same homotopy type as apoint ) and which contains all the vertices of K . Let G be the group gener-ated by the symbols gij one f or each ordered 1 simplex ai, aj of k , subjectto the relations

gijgjkg−1ik = 1 (14)

one for each ordered 2 simplex ai, aj , ak of K − L . if ai, aj is in Lthen gij = 1 . the theorem states that G is isomorphic to the fundamentalgroup π1(K, a0) , where a0 is a vertex of K . in order to use this theorem todetermine the fundamental group of a given topological space X we have tofind a polyhedron which is h omeomorphic to it .

2.6.2 Triangulation of a Space

A topological space X which is homeomorphic to a polhyhedron K is saidto be triangulable and the polyhedron K ( which is not unique ) is called atriangulation of X.

For a two dimensional space a triangulation actually does correspond torepresenting the space by gluing various triangles together making sure thatany two distinct triangles either are disjoint , have a single vertex , or anentire edge in common as req uired.

2.6.3 Triangulation of the Circle

since S1 is a one dimensional space , its triangulation means finding a col-lection of suitably joined 1 simplexes which is homeomorphic to S1 .

the most direct choice is a line segement where the end points are iden-tified , but this is not permitted because 1 simplex must have two distinct

13

vertices . we can just try to add an extra vertex in the middle of the linesegement but the triangulation which we get is not acceptable too becausethe two simplexes are supposed to be distinct but have identical vertices .the solution for that is to add two vert ices between the two end points ofthe line segement and by doing that we get the following triangulation :

K = 1∪2∪3∪1, 2∪1, 3∪2, 3 (15)

the contractible subpolyhedra contained in K is

L = 1, 3∪2, 3 (16)

thus the group g generated by the symbols gij , one for each ordered 1 simplexof k , is generated in this case by one element g12 = g since g13 = g23 = 1being element of L . thus π1(K, a) is isomorphic to the group G which isisomorphic to Z (the group of integeres under addition)

π1(S1, a)'Z (17)

2.6.4 Triangulation of the Disc

the disc is defined by :

D = (x1, x2) : x21 + x2

2≤1 (18)

its triangulation turns out to be the triangle itself and it is given by:

KD = K∪1, 2, 3

LD = L

(19)

we still have just one element g12 = g in the group G which is not trivial .however this time there is a 2 simplex , namely 1, 2, 3 , thus there is therelation :

g12g23g−113 = 1 (20)

14

which gives us g12 = 1 , in other words :

π1(D, a)'1 (21)

we can work out many other cases , for example we can check that π1(S2, a)'1

and π1(Mobius− strip, a)'Z .clearly from the above example , only the 1 simplex and 2 simplex struc-

ture of the polyhedra K were needed to determine π1(K, a) . in other wordsπ1(K, a) can only spot the two dimensional holes present in a space .

2.7 Fundamental Group of a Product X×Y

The fundamental group of the product of two topological spaces X and Y isisomorphic to the direct product of their fundamental groups .

π1(X×Y, x0×y0)'π1(X, x0)⊕

π1(Y, y0) (22)

the direct product denoted by G1⊕

G2 of two groups G1 and G2 is the set ofall ordered pairs (g1, g2) , g1 in G1 and g2 in G2 with multiplication definedby :

(g1, g2)×(g′

1, g′

2) = (g1g′

1, g2g′

2) (23)

3 The Homology Group

Theorem 1Let Hp(X) be the homology group associated with a topological space X

, then we have the two following theorems :Homology Groups As Topological Invariants if X and Y are two topolog-

ical spaces of the same Homotopy type then Hp(X) is isomorphic to Hp(Y )for all p . an immediate implication of this theorem is that the HomologyGroups are Topological Invariants , this is because Homeomorphic spac esare necessarily of the same Homotopy type .

Theorem 2if K1 and K2 are two triangulations of the same topological space K then

:

Hp(K1) = Hp(K2) (24)

for all p.

15

3.1 Definition of the Homology Groups

3.1.1 Oriented Simplicial Complex

An oriented p simplex (p > 0) is obtained from a p simplex σp = [v0, v1, ..., vp]by choosing an ordering for its vertices . the equivalence class of even per-mutations of the chosen ordering determines the positively oriented simplex+σp , while the equivalence class of odd permutations determines the neg-atively oriented simplex −σp . A simplicial complex whose simplexes havebeen assigned an orientation is called an oriented simplicial complex .

3.1.2 Chain

Let K be an n dimensional simplicial complex containing lp p simplexes . Thep chain of K , Cp(K) , is the free abelian group generated by the oriented psimplexes of K . an arbitrary element cp∈Cp(K) can be written as the formalsum :

cp =∑

i = 1i=lpfiσpi (25)

where fi∈Z . the simplexes σpi satisfy :

σpi + (−σp

i ) = 0i=lp∑

i=1

fiσpi +

i=lp∑

i=1

giσpi =

i=lp∑

i=1

(fi + gi)σpi

(26)

the statement that K is n dimensional simply means that p = 0, 1, 2, ... . itis often convenient to define :

Cp(K) = 0, forp > n. (27)

3.1.3 Boundary Operator

The boundary operator ∂p is the map :

∂p : Cp(K)−→Cp−1(K) (28)

16

with the following properties:1)it is linear

∂p

∑

i

(fiσpi ) =

∑

i

fi∂pσpi (29)

2)for an oriented p simplex σp = [v0, v1, ..., vp] we have :

∂[v0, ..., vp] =p

∑

j=0

(−1)j [v0, ..., vj, ..., vp] (30)

where [v0, ..., vj, ..., vp] represents the (p− 1) simplex σp−1 obtained from thep simplex σp by omitting the vertex vj .

3)The Boundary of every zero Chain is defined to be zero .theorem The boundary of a polyhedron K does not itself have a boundary

.

∂p−1.∂p = 0 (31)

3.1.4 Cycles and Boundaries

Cyclezp∈Cp(K) is called a p dimensional cycle or p cycle if

∂zp = 0 (32)

the family of p cycles is thus the kernel of the homomorphism

∂ : Cp−→Cp−1 (33)

and is a subgroup of Cp(K) . this subgroup is called the p dimensional cyclegroup of K and is denoted by Zp(K) .

Boundarybp∈Cp(K) is called a p dimensional boundary or p boundary if there is a

(p + 1) chain Cp+1 such that :

∂cp+1 = bp (34)

17

the family of p boundaries is thus the homomorphic image ∂Cp+1(K) and isa subgroup of Cp(K) . this subgroup is called the p dimensional boundarygroup of K and is denoted by Bp(K) .

clearly from the above theorem we have : if bp∈Bp(K) then bp = ∂pcp+1=⇒∂pbp =∂p∂p+1cp+1 = 0 in other words bp∈Zp(K) which means Bp(K)⊂Zp(K) .

In order to spot the (p + 1) dimensional holes we have thus to weed outthe elements belonging to Bp(K) contained in Zp(K) .

3.1.5 Homology Groups

The p dimensional homology group of k denoted by Hp(K) is the quotientgroup :

Hp(K) = Zp(K)/Bp(K) (35)

an element hp∈Hp(K) is an equivalence class [zp] defined by : z1p , z

2p∈[zp] such

that : z1p − z2

p∈Bp(K) . this equivalence relation is called homology and ifz1

p − z2p is in Bp(K) then z1

p and z2p are said to be homologous . From the

way it is defined it might seem quite remarkable that the groups Hp(K) donot depend on the triangulation of K . this is because the groups Cp(K) ,Zp(K) and Bp(K) certainly do depend on the triangulation of K . differenttria ngulations of K are certainly expected to lead to different cycle groupsand boundary groups since the number of boundaries introduced dependson triangulation . but Hp(K) which depends on the number of (p + 1)dimensional holes present in the spac e should not .

Theorems1)if K is contractible i.e, has the homotopy type of a single point then :

Hp(K) = 0, forp 6=0

Hp(K) = Z, forp = 0

(36)

2)if K is a contractible polyhedron then

H0(K) = Z (37)

18

3.1.6 Examples

The 2 simplex Let K be : K = σ2 = [v0, v1, v2] , then dimK = 2 .p > 2by definition Cp(K) = 0 for p > 2 this means that : Zp(K) = Cp(K) = 0

and Bp(K) = Φ so :

Hp(K) = 0 (38)

for all p > 2 .p = 0we have :

C0(K) = c0 =i=3∑

i=1

fiσ0i = c0 = a0[v0] + a1[v1] + a2[v2] (39)

where a0 , a1 and a2 are integers . since ∂[vi] = 0 then C0(K) = Z0(K) andclearly then :

∂[vi] = 0 then C0(K) = Z0(K) = Z⊕Z⊕Z = Z3. (40)

in the other hand

B0(K) = b0∈C0(K)andb0 = ∂c1wherec1∈C1(K) (41)

but :

C1(K) = c1 = e0[v0, v1] + e1[v0, v2] + e3[v1, v2] (42)

in other words

b0 = ∂c1 = (e0 − e3)[v1]− (e0 + e1)[v0] + (e1 + e3)[v2] (43)

where we have used the rule

∂[v1, v2] = [v2]− [v1] (44)

therefore any element b0∈B0(K) can be written as : b0 = a0[v0]+a1[v1]+a2[v2]where : a0 + a1 + a2 = 0 which means that B0(K) has two independentgenerators

B0(K) = Z⊕Z (45)

19

and any element h0∈H0(K) can be written as the coset :

h0 = z0 + B0(K)

= a0[v0] + b0[v1] + c0[v2] + (−a0[v0]− b0[v1] + (a0 + b0)[v2])

= (a0 + b0 + c0)[v2]

= d0[v2]

(46)

so :

H0(K) = Z (47)

H0(K) has only one generator .p = 1if c1∈C1(K) then : ∂c1 = (e0 − e3)[v1] − (e0 + e1)[v0] + (e1 + e3)[v2]

but if c1∈Z1(K) then ∂c1 = 0 and the only solution for this condition is−e1 = e0 = e2 = e Hence :

Z1(K) = z0 = e[v0, v1]− e[v0, v1] + e[v1, v2] = Z (48)

also

B1(K) = b1∈C1(K), b1 = ∂c2, where : c2∈C2(K) (49)

but:

C2(K) = c2 = e[v0, v1, v2] (50)

so: ∂c2 = e[v1, v2]− e[v0, v2] + e[v0, v1] in other words

B1(K) = Z (51)

finally any element h1 of H1(K) can then be written : h1 = z1 + B1(K) = 0which means that

H1(K) = 0 (52)

20

p = 2we have C2(K) = c2 = e[v0, v1, v2] and ∂c2 = e([v1, v2]−[v0, v2]+[v0, v1])

, so if this c2 is in Z2(K) then we must have : ∂c2 = 0 and the solution ofthis condition is e = 0 , in other words :

Z2(K) = 0 (53)

Now:

B2(K) = b2∈C2(K), b2 = ∂c3, where : c3∈C3(K) = Φ (54)

because there is no 3 simplexes in k . Hence :

H2(K) = 0 (55)

so for K = σ2 = [v0, v1, v2] we have H0(K) = Z and Hp(K) = 0 for p 6=0 .

The circle we have

S1 = ∂σ2 = [v0, v1] + [v1, v2]− [v0, v2] (56)

clearly dim(S1) = 1 .k > 1by definition Ck(S

1) = 0 for k > 1 therefore Zk(S1) = 0 and Bk(S

1) = Φwhich means that : Hk(S

1) = 0 for k > 1 .k = 0we have :

C0(S1) = c0 = e0[v0] + e1[v1] + e2[v2]

= Z⊕Z⊕Z

(57)

so;

Z0(S1) = c0, ∂c0 = 0

= C0(S1)

= Z⊕Z⊕Z

(58)

21

in the other hand :

B0(S1) = c0, c0 = ∂c1, c1∈C1(S

1) (59)

but c1 = a0[v0, v1]+a1[v0, v2]+a2[v1, v2] which leads us to ∂c1 = (a0−a3)[v1]−(a0 + a1)[v0] + (a1 + a3)[v2] therefore :

B0(S1) = c0 = e0[v0] + e1[v1] + e2[v2], where : e0 + e1 + e2 = 0

= Z⊕Z

(60)

Hence any element h0∈H0(S1) can be written h0 = z0 + B0(S

1) = a0[v0] +a1[v1]+a2[v2]−a0[v0]−a1[v1]+ (a0 +a1)[v2] = (a0 +a1 +a2)[v2] which means:

H0(S1) = Z0(S

1)/B0(S1) = Z (61)

k = 1from above an element c1∈C1(S

1) is in Z1(S1) if ∂c1 = 0 . the only

solution to this equation is a1 = −a0 = −a2 = −a which means that :

Z1(S1) = c1 = a[v0, v1]− a[v0, v2] + a[v1, v2] = Z (62)

and directly B1(S1) = c1, c1 = ∂c2, wherec2∈C2(S

1) = Φ because there isno 2 simplexes for the circle . hence

H1(S1) = Z1(S

1)/B1(S1) = Z (63)

3.1.7 Betti Numbers

Definitions 1)Let G be an abelian group . a set gi of elements G iscalled a set of generators of G if every element g∈G can be expressed in theform of a finite sum

i=k∑

i=1

nigi (64)

where ni∈Z .

22

2)The set gi freely generated G if for each g∈G the expression g =∑k

i=1 nigi is unique i.e , the elements gi are linearly independent over Z .an abelian group G which is freely generated by a set of generators is calleda free abelian group and a free generating set is called a basis .

3)An abelian group is said to be finitely generated if it has a set of gen-erators consisting of a finite number of elements . the rank of the group ( i.ethe number of elements in a basis of a free finitely generated abelian group) is independent of the choice of the basis .

Theorems 1)Let F be a free finitely generated abelian group . if R is asubgroup of F then R is free and finitely generated group .

2)Let A be a finitely generated ( not free ) abelian group generated by ngenerators . then

A'F/R = G + Zh1 + ... + Zhm (65)

where F and R are free finitely generated abelian groups with R⊂F , G is afree abelian group of rank (n −m) and Zhi is cyclic of order hi . the rank(n−m) of G and the numbers hi are uniquely determined by A . we write

A ' G + T

where : T ' Zh1⊕Zh2...⊕Zhm

(66)

T is called the torsion subgroup of A .

Betti Numbers the boundary groups Bp(K) and the cycle groups Zp(K)are free finitely generated abelian groups while

Hp(K)'Zp(K)/Bp(K)'Gp⊕Tp (67)

where Gp is a free finitely generated abelian group and T the torsion subgroupof Hp(K) . this result is interesting , we introduced the homology groupHp(K) as an algebraic object which could spot the (p +1) dimensional holespresent in K . this feature of Hp(K) is reflected in its free finitely generatedabelian group part Gp .

23

We define the pth Betti Number of K , Rp(K) , as the rank of Gp(K) whichcounts the number of holes . but Hp(K) contains in it even more information: for even if G = 0 , Hp(K) 6=0 , if Tp 6=0 . the group Tp(K) containsinformation about the manner in which the space K is twisted .

3.2 Relative Homology Groups

3.2.1 The Relative Chain Group

Let K be a complex , and let K be a subcomplex of L . Cp(K) and Cp(L) arerespectively the chain groups . The p dimensional chain group of K moduloL or the relative p chain group with integer coefficients is the quotient group

Cp(K; L) = Cp(K)/Cp(L) (68)

for p > 0 . thus each member of Cp(K; L) is a coset cp + Cp(L) wherecp∈Cp(K) .

3.2.2 The Relative Boundary Operator

for p > 1 the relative boundary operator ∂p is the map

∂p : Cp(K; L)−→Cp−1(K; L) (69)

defined by :

∂p(cp + Cp(L)) = ∂pcp + Cp−1(L) (70)

it is easy to check that the relative boundary operator is a homomorphism .

3.2.3 The Relative Cycle Groups

the group of relative p dimensional cycles on K modulo L , denoted byZp(K; L) is the kernal of the relative boundary operator . for p = 0 ,Z0(K; L) = C0(K; L) .

3.2.4 The Relative Boundary Groups

the group of relative p dimensional boundaries on K modulo L , Bp(K; L) ,is the image of Cp+1(K; L) under the relative boundary homomorphism .

24

3.2.5 The Relative Homology Groups

the relative p dimensional homology group of K modulo L denited by Hp(K; L)is the quotient group :

Hp(K; L) = Zp(K; L)/Bp(K; L) (71)

for p > 0 . the members of Hp(K; L) are : zp + Cp(L) .Note that it is required that ∂zp be a (p− 1) dimensional chain on L not

that zp be an actual cycle , since if : zp∈Zp(K; L) then ∂pzp = 0 which meansthat : ∂pzp = −Cp−1(L) .

3.2.6 The Excision Theorem

The Theorem Let K be a complex containing a closed subcomplex L . ifL0 is an open subcomplex of L such that L0 the closure of L0 is contained inthe interior of L then

Hp(K; L) = Hp(K − L0; L− L0) (72)

for all p . let us recall the concepts of closure and interior : if Fa is a familyof closed sets with the property that L0⊂Fa . then we define the closure L0

as the intersection of all the Fa in other words the smallest closed set whichcontains L0 . and the interior L0 as the union of all open subsets Oa of L0 ,it is the largest open subset of L0 .

Relation between Hp(L) and Hp(K) since L is a subcomplex of K we canrelate L to k by means of the inclusion homomorphism on the correspondingchain groups

i : Cp(L)−→Cp(K) (73)

defined as :

i[cp] = cp (74)

where cp∈Cp(L)⊂Cp(K) . in turn this map induces the group homomorphism

i∗ : Hp(L)−→Hp(K) (75)

25

Relation between Hp(K) and Hp(K; L) this relationship is establishedby considering the homomorphism

j : Cp(K)−→Cp(K; L) (76)

defined by : j[cp] = cp + Cp(L) where cp∈Cp(K) . then j induces a homo-morphism j∗ as follow :

from the definitions we have j[∂cp] = ∂cp + Cp−1(L) from one hand andfrom the other hand we have ∂[j(cp)] = ∂[cp + Cp(L)] = ∂cp + Cp−1(L) , inother words ∂ represents the relative boundary operator . so

j∂ = ∂j (77)

now if we take any zp∈Hp(K) then j(∂zp) = ∂j(zp) but ∂zp = 0 sincezp∈Hp(K) . thus ∂j(zp) = 0 which means that j(zp)∈Zp(K; L) . in otherwords using j each class [zp]∈Hp(K) can be mapped into the class [j(zp)]∈Hp(K; L), and it is easy to check that such a mapping is a homomorphism .

j∗ : Hp(K) −→Hp(K; L)

[zp] −→[j(zp)]

(78)

Relation between Hp(K; L) and Hp−1(L) Let zp∈Hp(K; L) . by defini-tion this implies that zp∈Cp(K; L) and ∂zp = 0 which can be written :

∂zp + Cp−1(L) = 0 (79)

∂zp is some element cp−1 in Cp−1(L) . furthermore cp−1 is not just a chainbut a cycle , since ∂cp−1 = −∂2zp = 0 . we write this correspondence ( whichcan be shown to be a group homomorphism ) as :

∂∗ : Hp(K; L) −→Hp−1(L)

[zp + Cp(L)] −→[∂zp]

(80)

26

3.2.7 Homology Sequence

all of the abov inter relationships which we have described can be fittedtogether into a sequence of groups and homomorphisms called the homologysequence of the pair (K; L) .

Definitionthe homology sequence of the complex K with subcomplex L is the se-

quence of groups and homomorphisms

∂∗

−→ Hp(L)i∗−→ Hp(K)

j∗

−→ Hp(K; L)∂∗

−→ Hp−1(L) (81)

this homology sequence has the following important property:TheoremThe homology sequence of the complex K with subcomplex L is exact ,

that is , the image of each homomorphism in the sequence is equal to thekernel of the next homomorphism .

theoremsuppose that an exact sequence has a section of four groups

0f−→ A

g−→ B

h−→ 0 (82)

where 0 denotes the trivial group . then g is an isomorphism from A ontoB .

Example1Let A1 = S1 − [n] where [n] is the north pole of the unit circle S1 . and

B1 = S1 − [s] where [s] is the south pole . it is clear that : A1∪B1 = S1

. we can define also a space : X1 = A1∩B1 = R1∪R2 where R1∩R2 = Φ .consider now the following exact homology sequence for p > 0 :

−→Hp(A1)−→Hp(S

1)−→Hp(S1; A1)−→Hp−1(A

1)−→Hp−1(S1) (83)

and

−→Hp(X1)−→Hp(B

1)−→Hp(B1; X1)−→Hp−1(X

1)−→Hp−1(B1) (84)

but A1 and B1 are contractible spaces hence :

Hp(A1) = Hp(B

1) = 0 (85)

27

for all p > 0 , so :

0 −→ Hp(S1)−→Hp(S

1, A1)−→0

0 −→ Hp(B1, X1)−→Hp−1(X

1)−→0

(86)

but using the last theorem

Hp(S1) = Hp(S

1, A1)

Hp(B1, X1) = Hp−1(X

1)

(87)

we remark then that by removing a point say the south pole [s] from S1 andA1 , S1 becomes B1 and A1 becomes X1 . thus from the exicision theoremwe have :

Hp(S1, A1) = Hp(S

1 − [s], A1 − [s]) = Hp(B1, X1)

Hp(S1) = Hp−1(X

1)

(88)

for all p > 1 . finally X1 can be retracted to two points [e] and [w] say .hence by theorem 1

Hp−1(X1) = Hp−1([e]∪[w]) (89)

but the dimension of the space consisting of the two points [e] and [w] is zeroby definition , thus Cp([e]∪[w]) = 0 for all p > 0 . so that

Hp([e]∪[w]) = 0 (90)

for all p > 0 . hence

Hp(S1) = 0 (91)

for all p > 1 .

28

Example 3

Hp(Sn) = Hp−1(S

n−1) (92)

for n > 1 and p > 1 . by repeated use of this equation and a direct compu-tation of H1(S

1) it is possible to show that

Hp(Sn) = Z, forp = 0, andp = n

Hp(Sn) = 0, otherwise

(93)

3.2.8 Torsion

The Torsion the homology groups Hp(K) were obtained from the chaingroups Cp(K) in which the elements are formal linear combinations of theoriented p simplexes of K multiplied by integer coefficients . it is possible togeneralize the chain groups by introducing as coefficients of the p simplexeselements of an arbitrary abelian group G rather than the integer

cp∈Cp(K) : cp =i=lp∑

i=1

giσpi (94)

where gi∈G . these are the chain groups defined over the abelian group G ,written Cp(K, G) and the corresponding homology groups Hp(K, G) can beconstructed . However the groups Hp(K, G) are not more general than thegroups Hp(K) , in fact it is possible to prove that a knowledge of the groupsHp(K) and G is enough to completely determine the groups Hp(K, G) . ifthe integers coefficients are replaced by rational coefficients , say , then thecorresponding homology groups Hp(K, Q) are far from being more generalthan Hp(K) and actually they contain less informations than the correspond-ing homology groups Hp(K) . the reason for this is simple , since the integercoefficients homology groups Hp(K) had the general structure

Hp(K) = Gp(K)⊕Tp(K) (95)

where elements of Tp(K) , the torsion subgroup , were finite order cyclicgroup ( Zm = Z/mZ is the cyclic group of order m , elements of Zm are

29

cosets , two integers belong to the same coset if they are congruent modulom , that is if their difference is a multiple of m , Zm has m elements ) . thefact that the elements of Tp(K) are finite order cyclic group means that iftp∈Tp(k) was of order n then ntp = 0 which is the identity element . butfor chains with rational coefficients , the equation ntp = 0 would mean thattp = 0 . thus Hp(K, Q) does not have any torsion subgroup while Hp(K)does .

The Kunneth Formula we consider the case of the homology groupsHp(K, Q) where the complications due to the torsion subgroups are notpresent . the formula reads :

Hp(X×Y, Q) =⊕

k+q=pHk(X, Q)⊗Hq(Y, Q) (96)

Exampleconsider T 2 = S1×S1 . we know that

Hp(S1) = Z, forp = 0, andp = 1

Hp(S1) = 0, otherwise

(97)

so

H0(T2) = H0(S

1⊗S1) = H0(S1)×H0(S

1) = Z2

H1(T2) = H0(S

1)⊗H1(S1)⊕H1(S

1)×H0(S1) = Z2⊕Z2

H2(T2) = Z2

(98)

The Euler Poincare Formula we mentioned earlier that the rank ofHp(K) , Rp(K) , is related to the number of (p+1) dimensional holes presentin the space . this number is given by

χ(K) =n∑

p=0

(−1)pRp(K) (99)

30

since the groups Hp(K) are topological invariants the number χ(K) is atopological invariant itself . it is called Euler characteristic of K . it can bewritten also as :

χ(K) =n∑

p=0

(−1)plp(K) (100)

where lp(K) is the number of p simplexes present in k .

3.2.9 Singular Homology

if we want to define the homology groups for an arbitrary topological spaceX , we just need to start by introducing the standard p simplexes δp . theseare the set of points (x0, x1, ..., xp) in euclidean space with the property :0≤xi≤1 and

∑pi=0 xi = 1 . the singular p simplexes λp

i in X can be definedas continuous mappings

λpi : δp−→X (101)

the term singular is used to indicate that the maps λpi need not be invertible

. in terms of the singular p simplexes λpi the singular chain group Sp(X) can

be defined as the set of formal sums

Sp =∑

i

λpi gi (102)

where gi are elements of the abelian group G . the addition is defined by∑

i

λpi gi +

∑

i

λpi hi =

∑

i

λpi (gi + hi) (103)

gi , hi are in G .Next we define the boundary map ∂ by its action on a singular p simplex

λp as follow

∂λp =p

∑

r=0

(−1)rλp.F r (104)

where λp.F r denotes the rth face of the singular p simplex λp and is definedby

λp.F r : ∆p−1−→X (105)

31

and

F r : ∆p−1−→∆p (106)

where F r(x0, ..., xp−1) = (x0, ..., xr, ..., xp) . the notion xr means that thevertex xr has been removed . we can show that ∂2 = 0 and define thesingular homology group for the arbitrary topological space X as :

Hp(X, G) =Ker[∂ : Sp−→Sp−1]

Img[∂ : Sp+1−→Sp](107)

3.3 The De Rham Comology

3.3.1 Definition

Cohomolgy which is the dual of homology turns out to be a much strongerconcept . and we start our discussion of it by considering a compact differntialmanifold M . The Stokes’ Theorem for differential forms reads :

∫

Mdω =

∫

∂Mω (108)

where ω is an (n−1) form and dimM = n . Let us recall now that a p−chainc used in calculations of Hp(M ; Z) is a formal finite linear combination :

c = a1λ1 + a2λ2 + ... (109)

where the ai are all integers and the λi are singular p simplexes i.e maps

λi : ∆p−→M (110)

∆p being the standard n simplex in Rp . from now on we will require thatthe ai be real numbers rather than integers , and that the maps λi are C∞

therefore we will call the singular p chain c a C∞ p chain . the function λi

induces a map λi∗ defined by

λi∗ : To(∆p)−→Tλi(o)(M) (111)

which takes the tangent space at the point o∈∆p , To(∆p , to the tangentspace at the point λi(o)∈M , Tλi(o)(M) . its actual action on a vector

32

X∈To(∆p) is a vector λi∗X∈Tλi(o)(M) such that for any function f on Mwe have :

(λi∗X)f = Xf(λi(0)) (112)

The map λi also has consequences for forms . in fact , it induces anothermap λi

∗ defined by

λi∗ : Ω1

λi(o)(M)−→Ω1

o(∆p) (113)

which takes the space of forms at the point λi(o)∈M , Ω1λi(o)

(M) , to the

space of forms at the point o∈∆p, Ω1o(∆p) . it is defined in terms of λi∗ by :

< λi∗ω, X >=< ω, λi∗X > (114)

where ω is in Ω1λi(o)

(M) . clearly λi∗ω is a 1 form in Ω1

o(∆p) .Now if we take a p chain c and a p form ω with 0≤p≤n , then we define

the integral of ω over c using the maps λi∗ defined above as follow :

∫

cω =

k∑

i=1

ai

∫

∆p

λi∗ω (115)

this integral is a real number and thus we can regard ω as producing , fromc , a real number . so if we denote by cp the set of all C∞ p chains , then ap form ω is a map from cp to R , i.e an element of the dual of cp , in otherwords ω is a cochain .

ω : Cp −→R

c −→ < ω, c >=∫

cω

(116)

where c∈Cp and ω∈Ωp(M) . Stokes’ theorem will then take the form

< dω, c >=∫

cdω =

∫

∂c

ω =< ω, ∂c > (117)

where ω is a (p−1) form and c is a p chain .the boundary operator ∂ and theexterior derivative operator d are formal adjoint of one another . we know

33

already that the elements of a homology class which we write as [c] belongto the space

Hp(M, R) = Zp(M)/Bp(M) (118)

where Zp(M) are all p chains c for which ∂c = 0 . and Bp(M) are all p chainsc for which c = ∂c‘ for some (p + 1) chain c‘ .the dual of the space Hp(M, R)is Hp(M, R) where

Hp(M, R) = Zp(M)/Bp(M) (119)

where Zp(M) are all cochains or p forms ω for which dω = 0 , and Bp(M)are all p forms ω for which ω = dη for some (p−1) form η . to show preciselythat the space Hp is the dual of the space Hp we must define how [ω]∈Hp

acts on [c]∈Hp to give a real number ([ω], [c]) . well this action is defined by:

([ω], [c]) =∫

cω (120)

we can show then that if [ω] = [ω‘] then ([ω], [c]) = ([ω]‘, [c]) and that if[c] = [c‘] then ([ω], [c]) = ([ω], [c‘]) . therefore Hp is the dual of the space Hp

. Hp(M.R) is called the pth homology group of M with real coefficients , andHp(M, R) is the pth cohomology group of M with real coefficients .Finallywe have the two following exact sequences

...∂p−1

←− Cp−1∂p

←− Cp

∂p+1

←− Cp+1 (121)

and

...dp−1

−→ Ωp−1dp

−→ Ωp

dp+1

−→ Ωp+1 (122)

The exactness of these two sequences follows from the properties

dp+1dp = 0

∂p∂p+1 = 0

(123)

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for any p .from the above sequences we can conclude that :

Hp(M, R) = Zp(M)/Bp(M) = Ker∂p/Img∂p+1

Hp(M, R) = Zp(M)/Bp(M) = Kerdp+1/Imgdp

(124)

as a last remark we note that each sequence has the same finite number ofterms and because we know that Ωp and Cp are non trivial spaces only forp = 0, ..., n where n = dimM the only cohomology groups which can be nontrivial are H0(M, R) , H1(M, R) ...Hn(M, R) .

This construction of cohomology groups for differentiable manifolds usingp forms ω is due to de Rham and that’s why we call Hp(M, R) the de RhamCohomology groups .

3.3.2 Poincare’s Lemma

We know that elements of Hp(M, R) are those forms ω which satisfy dω = 0but cannot be written as ω = dη . let us now introduce the following notation: we call those forms which satisfy dω = 0 closed forms ,

in other words we just call cycles closed forms . similarly we call thoseforms which satisfy ω = dη exact forms , clearly they are just the boundaries. from the construction Bp(M)⊂Zp(M) but the converse is not in generaltru

e , if it was true then all cohomolgy groups Hp(M ; R) would be tri’vial .Definition of ContractibilityM is smoothly contractible if there is a C∞ map

α : M×[0, 1] −→M

(x, t) −→x, ift = 0

(x, t) −→x0, ift = 1

(125)

so as t varies smoothly from 0 to 1 , the map α shrinks M down to the singlepoint x0 .

Poincare’s Lemma

35

if a space M is contractible to a point then all closed forms on M are alsoexact .

3.4 Calculation Of Hp(M ; R)

3.4.1 Hp(Rn; R)

since Rn is contractible all closed p forms on Rn are exact and we have

Hp(Rn; R) = 0 (126)

for p = 1, 2, ...n . however :

H0(Rn; R) = R (127)

and that’s because we have from one hand : closed 0-forms on Rn are simply-functions f satisfying df = 0=⇒f = c, c∈R which means that Z0(Rn) = R .and from the other hand there are no 1-forms which means that B0(Rn) = φ.therefore H0(Rn; R) = Z0(Rn) and from it the resultabove which can bestated as follow : to each real number c there corresponds an element of thezeroth cohomology class , i.e an element of H0(Rn; R) . Finally we remindourselves that Hp(Rn; R) is always trivial for p > dimM = n .

3.4.2 Hp(Sn, R) , p = 0, ..., n

p = 0when p = 0 we simply have ω = c where c∈R , in other words ω is the

constant function on Sn .therefore :

H0(Sn; R) = R (128)

p = 1We can show that if M is a simply connected differentiable manifold then

:

H1(M ; R) = 0 (129)

in other words every closed curve on M can be contracted to a point . Inparticular if M = Sn then :

H1(Sn; R) = 0 (130)

36

for n > 1 and that’s because Sn is simply connected for n > 1 . for the caseof n = 1 the calculation gives :

H1(S1; R) = R (131)

p > 1for p > 1 we can prove the following inductive formula :

Hp(Sn; R) = Hp−1(Sn−1; R) (132)

which with the above two results for p = 0 and p = 1 give us Hp(Sn; R)completely , the result is

Hp(Sn; R) = 0, p > n

Hp(Sn; R) = R, p = n

Hp(Sn; R) = 0, 1≤p < n

H0(Sn; R) = R

(133)

3.4.3 T n

The torus T n which is S1×S1...S1 (n factors) has in contrast to the sphereSn non trivial cohomlogy groups Hp(Sn; R) . they are given by :

Hp(T n; R) = Rα

where : α =n!

p!(n− p)!

(134)

3.4.4 General Remarks

1) if M is a compact, connected ,orientable manifold and dimM = n then :

Hn(M ; R) = R (135)

37

2)if M is a compact, connected, nonorientable manifold and dimM = nthen

Hn(M : R) = 0 (136)

an immediate application of this result are: the Mobius Strip (H2(MS; R) =0) , and the projective space P 2n which is defined to be the quotient spaceobtained from S2n by identifying antipodal points (H2n(P 2n; R) = 0) .

3)if M is a non compact connected manifold , orientable or nonorientableand dimM = n then :

Hn(M ; R) = 0 (137)

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