15
1.3. TANGENT VECTORS AND MAPPINGS
1.3.2
Denition 1.3.1 A tangent vector at a point p0 M of a manifold
M is a map that assigns to each coordinate chart (U , x ) about p0 an
1
n
ordered n-tuple (X , . . . , X ) such that
i
n
x
i
j
X =
j ( p0 ) X .
j=
13
1.2. MANIFOLDS
Let the transition functions
f = 1 : (U U ) Cn ,
dened by
k
zk = f (z1 , . . . , zn ) ,
k
k
where zk = x + iyk and zk = x + iyk , k = 1, . . . , n, be complex analytic.
That is, they satisfy Cauchy-Riemann conditions
k
x
j
x
or
=
yk
j
11
1.2. MANIFOLDS
The set of oriented lines through the origin of Rn+1 is S n .
Topologically RPn is the sphere S n in Rn+1 with the antipodal points
identied, which is the unit ball in Rn with the antipodal points on the
boundary (which is a unit spher
9
1.2. MANIFOLDS
5. Thus F ( M ) is compact.
Theorem 1.2.3 A continuous real-valued function f : M R on a
compact topological space M is bounded.
Proof :
1. f ( M ) is compact in Rn .
2. Thus f ( M ) is closed and bounded.
1.2.2
Idea of a Manifold
A mani
7
1.2. MANIFOLDS
Denition 1.2.1 A general topological space is a set M together with
a collection of subsets of M, called open sets, that satisfy the following
properties
1. M and are open,
2. the intersection of any nite number of open sets is open,
3. t
5
1.1. SUBMANIFOLDS OF EUCLIDEAN SPACE
1.1.3
Main Theorem on Submanifolds of Rm
The matrix of the linear transformation F is exactly the Jacobian matrix.
Therefore, the dierential F at a point x0 is a surjective (onto) map if and
only if m r and the Jac
3
1.1. SUBMANIFOLDS OF EUCLIDEAN SPACE
be a subset of the Euclidean space Rm described by the locus of r equations
F ( x 1 , . . . , x m ) = y ,
0
= 1, . . . , r .
Suppose that M is non-empty and let x0 M , that is
F ( x0 ) = y0 .
Then the Implicit Func
7.6. RELATIVE HOMOLOGY AND MORSE THEORY
205
Corollary 7.6.1 There hold:
1. Weak Morse inequalities
Mp Bp ,
2. In particular, the total number of critical points is bounded below
by the sum of all Betti numbers
n
n
Bp .
Mp
p=0
p=0
3. Strong Morse inequali
7.6. RELATIVE HOMOLOGY AND MORSE THEORY
203
The number of critical points of index p is called the p-th Morse type number and denoted by M p .
Let t be a formal variable. The polynomial
n
M (t) =
Mpt p
p=0
is called the Morse polynomial.
For each real
Chapter 1
Manifolds
1.1
Submanifolds of Euclidean Space
Idea: Manifold is a general space that looks locally like a Euclidean space
of the same dimension. This allows to develop the dierential and integral
calculus.
Let n N be a positive integer. The Eu
7.6. RELATIVE HOMOLOGY AND MORSE THEORY
201
A relative boundary (mod M ) is a sum of an absolute boundary and a
chain that lies on M .
Example.
The relative homology group is the quotient group of relative cycles modulo the relative boundaries
H p ( M,
7.6. RELATIVE HOMOLOGY AND MORSE THEORY
199
Proposition 7.6.1 Let M be a compact Riemannian manifold with
smooth boundary. Let and be two p-forms on M, which are either
both normal to the boundary or both tangent to the boundary. Then
(d, ) = (, ) .
That
7.6. RELATIVE HOMOLOGY AND MORSE THEORY
7.6
7.6.1
197
Relative Homology and Morse Theory
Relative Homology
Let M be a compact Riemannian n-dimensional manifold with smooth boundary M and
i : M M
be the inclusion map.
Let xi , i = 1, . . . , (n 1), be th
195
7.5. HARMONIC FORMS
Corollary 7.5.1 Let M be a closed Riemannian manifold. Then any
closed p-form is a sum of an exact form d and a harmonic form h,
that is,
= d + h .
Proof :
1.
Corollary 7.5.2 Let M be a closed Riemannian manifold. Then each
de Rha
193
7.5. HARMONIC FORMS
The Hodge Laplacian on p-forms is the operator
L : C ( p ( M ) C ( p ( M )
dened by
L = d + d = (d + )2 .
The operators d and commute with the Hodge Laplacian, i.e.
dL = Ld ,
L = L .
Theorem 7.5.3 For any p there holds
L=+W,
whe
191
7.5. HARMONIC FORMS
The coderivative is a linear map
: p p1
dened by
= 1 d = (1)(n p+1)( p1) d .
The exterior derivative and the coderivative satisfy the important conditions
d2 = 2 = 0 .
Problem. Show that in local coordinates the coderivative o
189
7.4. DE RHAM COHOMOLOGY GROUPS
Proposition 7.4.2 . Let M be a closed manifold. Let p Z p ( M ) be a
closed p-form on M such that for any p-cycle z p Z p ( M )
p, zp = 0 .
Then the p-form p is exact.
Proof : Dicult.
Theorem 7.4.1 de Rham Theorem. Let M
187
7.4. DE RHAM COHOMOLOGY GROUPS
de Rham cohomology groups are vector spaces.
Let
r
ak k
p
cp =
k =1
be a real p-chain in M , and be a p-form on M .
We dene the integral of over c p by
r
, c p =
=
cp
ak
k =1
.
k
p
Thus every p-form on M denes a line
183
7.3. SINGULAR HOMOLOGY GROUPS
Theorem 7.3.6 Let M be an n-dimensional compact manifold. Every
real p-cycle z p in H p ( M ; R) is homologous to a nite formal sum
r
zp
ak Vk
k =1
of closed oriented p-dimensional submanifolds Vk of M with real coefcien
181
7.3. SINGULAR HOMOLOGY GROUPS
3. Therefore, a multiple gp of a single point is not a boundary for any
g 0.
4. Thus, any point p M is a 0-cycle that is not a boundary.
5. Moreover, for any g G and any p M the 0-chain gp is a 0-cycle
that is not a bound
179
7.3. SINGULAR HOMOLOGY GROUPS
Let M and V be manifolds, G be a n Abelain group, F : M V be a map
and F : C p ( M ; G) C p (V ; G) be the induced homomorphism of chain
groups.
Since the induced homomorphism F commutes with the boundary homomorphism ,
177
7.3. SINGULAR HOMOLOGY GROUPS
The set of all p-boundaries in M
B p ( M ; G) = cfw_b p C p ( M ; G) | b p = c p+1 for some c p+1 C p+1 ( M ; G)
is a subgroup of the chain group C p ( M ; G) called the p-boundary group.
Obviously the p-boundary group
175
7.2. SINGULAR CHAINS
2. Then, for a singular p-simplex p
p = [ p ( p )] = p ( p ) = (0) = 0
7.2.1
Examples
Cylinder.
M bius Band.
o
digeom.tex; April 12, 2006; 17:59; p. 173
176
7.3
7.3.1
CHAPTER 7. HOMOLOGY THEORY
Singular Homology Groups
Cycles,
173
7.2. SINGULAR CHAINS
Let p : p M be a singular p-simplex in M . Then its boundary p is
the integer ( p 1)-chain in M dened by
p = p ( p ) .
In more detail
p = p ( p )
p
=
p
k
(1)
)
p (pk1 )
=
k =0
(1)k ( p fk ) .
k =0
)
Recall that p (pk1 ) = ( p
171
7.2. SINGULAR CHAINS
Examples.
The boundary of a standard simplex is not a simplex, but an integer ( p 1)chain.
Let G be an Abelian group. A singular p-chain on M with coecients in
G is a nite formal sum
r
gk k
p
cp =
k =1
of singular simplexes k :
169
7.2. SINGULAR CHAINS
7.2
Singular Chains
The standard Euclidean p-simplex in R p is the convex set p R p generated by an ordered ( p + 1)-tuple (P0 , P1 , . . . , P p ) of points in R p
P0 = (0, . . . , 0),
Pi = (0, . . . , 0, 1, 0, . . . , 0),
i = 1
167
7.1. ALGEBRAIC PRELIMINARIES
7.1.2
Finitely Generated and Free Abelian Groups
Let G be an Abelian group. Let g1 , . . . , gr G be some elements of G and
r
H = nk gk | gk G, nk Z
k =1
be a set of linear combinations of gk .
Then H is a subgroup of G.
165
7.1. ALGEBRAIC PRELIMINARIES
Theorem 7.1.1 Fundamental Theorem of Homomorphisms. Let G
and F be groups. Let F : G F be a homomorphism. Then
G/Ker F
Im F .
Proof :
1. Since
F (g + Ker F ) = F (g) .
Theorem 7.1.2 Let G and E be Abelian groups, and H G a
Chapter 7
Homology Theory
7.1
7.1.1
Algebraic Preliminaries
Groups
A group is a set G with a binary operation, , called the group multiplication,
that is,
1. associative,
2. has an identity element,
3. every element has an inverse.
A group is Abelian if