108_MA5209 Algebraic Topology

Algebraic InvariantsWhen we speak of a topological space we equivalence class of all topological spaces that are homeomorphic to that space.. will often mean perhaps implicitly the Algeb

MA5209 Algebraic Topology

Wayne Lawton

Department of Mathematics National University of Singapore S14-04-04, matwml@nus.edu.sg

http://math.nus.edu.sg/~matwml

Lecture 1 Simplicial Concepts

(11, 14 August 2009)

C R,

Q,

Z, ,

} {1,2,3,

N =

natural, integer (Zahlen), rational (quotient),

real, and complex numbers

Hint: use an equivalence relation on

denote the

Question 1 How are these numbers and their Algebraic operations constructed from (N, + ) ?

N}

N b)

(a, :

b)]

{[(a,

n} b

n a

: N N

n) {(m,

b)]

d)]

b c, [(a

d)]

[(c, b)]

[(1,1)] 1)],0

k [(1, k

1,1)], [(k

k

N ∋ ↔ + − ↔ + ↔

N

N ×

What is a Topological Space ?

) ,

Definition A topological space is a pair

where

collection of subsets of X (called open subsets)

2

Top X

φ

that satisfies the following three properties:

1

(called a topology on X), is a

Top

Top B

A Top

B Top

3 The union the elements in each subset of

Top is in Top

Question 2 Express condition 3 using set theory

What is a Quotient Space ?

) ,

A topological space ,a set

we associate a

Definition If

Y

Top

topology

surjection

on

Top O

g Top

O ∈ f ⇔ −1( ) ∈

topology, by called the quotient

,

Y

, : X Y

,

Y

) ,

( X Top X is a topological space and ~ is an equivalence relation on X ,

the associated quotient topology is ( Y , TopY )

where Y is the set of equivalence subsets and

Y X

g : → is given by g(x) = [x] = {w∈ X : x ~ w}

Question 3 Describe the quotient topology if

equiv.

only 3

~ 2

~ top.,1 usual

], 3 , 1 [

=

X

http://en.wikipedia.org/wiki/Quotient_space

Euler’s Homomorphism

} 1

| :|

{ T

R : ) 2

)) i

(2 kernel(exp

/ R Z

/ R

induces, by the first homomorphism theorem

for groups, an isomorphism between

c

T

This isomorphism is also a homeomorphism

between the topological space Tc ,regarded as

a subspace of C with its usual topology, and the

quotient topology on T = R / Z induced by the canonical homomorphism π : R → T = R / Z

,

) ( x = x + Z x ∈ R

π

that is defined by

Algebraic Invariants

When we speak of a topological space we

equivalence class of all topological spaces

that are homeomorphic to that space

will often mean (perhaps implicitly) the

Algebraic topology studies topological spaces

by associating algebraic invariants to spaces

#cc(X) = number of connected components

Question 4 Is homeomorphic to

?

Affine Space of Dimension n

set of points of the affine space

on group action of

(an abelian group under addition) this means that

A

A V

A × →

:

τ

V real vector space of dimension n

A p

p

p , 0 ) = , ∀ ∈

(

τ

V v

u A p

v u

p v

u

p , ), ) = ( , + ), ∀ ∈ , , ∈

(

τ

Question 5 Prove τ (τ ( p,u),−u) = p,∀p ∈ A,u ∈V

The group action is both free and transitive

Question 7 Show that every finite dimensional real vector space is an affine space

http://en.wikipedia.org/wiki/Affine_space

Question 6 What does free, transitive mean?

Affine Combinations

for and we observe that the last condition on the the preceding page ensures that

) ,

( v p

τ

Convention: we will write

p

q − A v V p v q

q

We define to be that unique v ∈ V

Definition For

v

p +

R r

r A p

p1, , k ∈ , 1, , k ∈

Question 8 Show this point is independent

A p

p r

p p

r

p1 + 2 ( 2 − 1) +  + k ( k − 1) ∈

the affine combination with r 1 + + rk = 1

of the ordering of the elements p , ,1 pk

denote the point

k

k p r

p

r1 1 +  +

Affine Maps

and are affine spaces, a map

If

B A

f : →

is affine if it preserves affine combinations, i.e

A

Question 9 Prove that if

) (

) (

) ( r1 p1 rk pk r1 f p1 rk f pk

spaces then a map

B

b ∈

B

A and B are vector

B A

f : → is affine iff there exists a linear map L : A → B and

such that f ( a ) = La + b , ∀ a ∈ A

Question 10 Show that an affine space A

has a unique topology such that there exists

an affine bijection & homeomorphism with Rn

Convex Combinations and Simplices

An convex combination is an affine combination whose coefficients are nonnegative

Let A be an n-dimensional affine space

Points p1, , pk ∈ A

are linearly independent

are in general position (or geometrically independent) if the vectors

Then the set of convex combinations is called

1 1

2 p , , p p

the (k-1)-simplex spanned by these points

Question 11 Show that all (k-1)-simplices are affinely isomorphic and homeomorphic to a

(k-1)-dimensional closed ball in Rk−1

Compact Surfaces as Subspaces

Some compact surfaces are homeomorphic to subspaces of

disc rectangle 2-simplex annulus

others cannot but are homeomorphic to

2

R

subspaces of R3 (sphere, torus)

Real Projective Space

to a (topological) subspace of R3

Question 12 What is real projective space?

is not homeomorphic

3

RP

Compact Surfaces as Quotient Spaces

relate corresponding points on left and right sides

torus

≈

Question 13 What points are related to obtain a

a

annulus a

b b

torus? A sphere ? A Klein bottle ? Draw figures

Euler Characteristic of Surfaces

Divide the surface of a sphere into polygonal regions having v vertices, e edges, and f faces Compute the quantity

Question 14 Compute v, e, f and for the

surfaces of each of the the five platonic solids and discuss the results

Question 15 Repeat using various triangular divisions of the sphere

Question 16 Repeat for other surfaces

Hint: use their quotient space representations

f e

v − +

=

χ

χ

Barycentric Coordinates

If

clearly

is an n-dimensional affine space and

A

A a

a0 , , k ∈ are in general position then

and the affine subspace of

The k-simplex they span is denoted by

n

they spanned is denoted by

∑ = =

∈ +

+

k t a t a t R t a

a

} 0

: ) ,

, (

0 ak t a tkak Aff a ak ti

a  = +  + ∈  ≤

k

t

t0,  , are the barycentric coordinates of

k k

t a

t0 0 +  + ∈ 0 

Question 17 Show they are unique&continuous

Boundary of a Simplex

A a

a0 , , k ∈ are in general position we

define the interior of the simplex

Question 18 Show that if

as a subspace of the topological space

} 0 :

{ )

then these two concepts

If

k

a

a 0

coincide with the standard topological concepts

k

a

a 0

), ,

,

Aff 

and boundary

is regarded

) (

0

0 ak a ak Int a ak

∂

Question 19 Show that a 0 ak is a disjoint

union of{ a 0, , ak} and the interiors of simplices spanned by each subset of { a 0, , ak}

Simplicial Maps

and

is an affine space are in general position then

A

Question 20 Prove that if

the set of vertices is determined by

A a

a0 , , k ∈

} , ,

{ a0 ak

the simplex a 0 ak

Definition If A and B are affine spaces and

B

σ , are simplices then a map

τ

σ →

:

f is a simplicial map if there exists

an affine map ~ f : A → B such that

f f

vertices vertices

f ( ( σ )) ⊆ ( τ ), ~ |σ =

~

Geometric Simplicial Complexes

Example

Definition Faces of a simplex are the simplices spanned by its proper subsets of vertices

Definition A geometric simplicial complex is a

1 contains each face of each element

•

•

•

collection of simplices in an affine space that

2 The intersection of each pair of elements

is either empty or a common face

http://en.wikipedia.org/wiki/Simplicial_complex

=

) ( a0a1a2

faces

0

a

1

} ,

, ,

, ,

{a0a1 a0a2 a1a2 a0 a1 a2

Topological Simplicial Complexes

is a finite geometric simplicial

Definition If

complex in an affine space we define its

and the associated topological simplicial complex

to the be equivalence class of topological spaces

K

A

A

K

=  σ∈ σ

|

|

that are homeomorphic to | K | with the subspace

topology

Assignment: Read pages 1-14 in WuJie and do

exercises 2.1, 2.2 on page 25 for finite complexes polyhedron