Transcript CLASSICAL INFORMATION THEORY

MA5209 Algebraic Topology
Lecture 1. Simplicial Concepts
(11, 14 August 2009)
Wayne Lawton
Department of Mathematics
National University of Singapore
S14-04-04, [email protected]
http://math.nus.edu.sg/~matwml
Algebra
N  {1,2,3,...} , Z, Q, R, C denote the
natural, integer (Zahlen), rational (quotient),
real, and complex numbers.
Question 1. How are these numbers and their
Algebraic operations constructed from (N, ) ?
Hint: use an equivalence relation on N  N
Z  {[(a,b)] : (a, b)  N  N}
[(a,b)]  {(m,n)  N  N : a  n  b  n}
[(a,b)]  [(c,d)]  [(a  c, b  d)]
N  k  [(k  1,1)], k  [(1,k  1)],0 [(1,1)]
What is a Topological Space ?
Definition A topological space is a pair ( X , Top)
where Top (called a topology on X), is a
collection of subsets of X (called open subsets)
that satisfies the following three properties:
1.
  Top, X  Top
2. A  Top and B  Top  A  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 ?
http://en.wikipedia.org/wiki/Quotient_space
A topological space ( X , TopX ) ,a set Y , and a
surjection g : X  Y , we associate a
topology TopY on Y , called the quotient
1
topology, by O Topf  g (O) Top
Definition If ( X , TopX ) 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
g : X  Y is given by g ( x)  [ x]  {w  X : x ~ w}
Question 3. Describe the quotient topology if
X  [1,3], usual top.,1~ 2 ~ 3 onlyequiv.
Euler’s Homomorphism
exp(2 i ) : R  Tc  {c  C :| c | 1}
induces, by the first homomorphism theorem
for groups, an isomorphism between
T  R / Z  R / kernel(exp(2 i)) and Tc
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  : R  T  R / Z
that is defined by  ( x)  x  Z , x  R.
Algebraic Invariants
When we speak of a topological space we
will often mean (perhaps implicitly) the
equivalence class of all topological spaces
that are homeomorphic to that space.
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
http://en.wikipedia.org/wiki/Affine_space
A set of points of the affine space
V real vector space of dimension n
(an abelian group under addition)
 : A  V  A group action of V on A
this means that  ( p,0)  p, p  A
 ( ( p, u), v)   ( p, u  v), p  A, u, v V
The group action is both free and transitive
Question 5 Prove  ( ( p, u),u)  p, p  A, u V
Question 6 What does free, transitive mean?
Question 7 Show that every finite dimensional
real vector space is an affine space.
Affine Combinations
Convention: we will write p  v for  ( p, v)
and we observe that the last condition on the
the preceding page ensures that
p, q  A, !v V  p  v  q
We define q  p to be that unique v  V
Definition For p1 ,..., pk  A, r1 ,...,rk  R
with r1    rk  1 the affine combination
r1p1  rk pk denote the point
p1  r2 ( p2  p1 )   rk ( pk  p1 )  A
Question 8. Show this point is independent
of the ordering of the elements p1 ,..., pk
Affine Maps
If
A
and
B
are affine spaces, a map
f : AB
is affine if it preserves affine combinations, i.e.
f (r1 p1   rk pk )  r1 f ( p1 )   rk f ( pk )
Question 9. Prove that if A and B are vector
spaces then a map f : A  B is affine iff
there exists a linear map L : A  B and
b  B 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 R n
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 in general position
(or geometrically independent) if the vectors
p2  p1 ,..., pk  p1 are linearly independent.
Then the set of convex combinations is called
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
(k-1)-dimensional closed ball in R
Compact Surfaces as Subspaces
Some compact surfaces are homeomorphic to
2
subspaces of R

disc
rectangle


2-simplex
annulus
others cannot but are homeomorphic to
3
subspaces of R (sphere, torus)
3
Real Projective Space RP is not homeomorphic
3
to a (topological) subspace of R
Question 12. What is real projective space?
Compact Surfaces as Quotient Spaces
annulus
 a
a
relate corresponding points on left and right sides
b
torus
 a
a
b
Question 13. What points are related to obtain a
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.
  ve f
Question 14. Compute v, e, f and  for the
Compute the quantity
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.
Barycentric Coordinates
If A is an n-dimensional affine space and
a0 ,...,ak  A are in general position then
clearly k  n and the affine subspace of
they spanned is denoted by
A
Aff (a0 ,...,ak )  {t0 a0    tk ak : ti  R, i 0 ti  1}
k
The k-simplex they span is denoted by
a0 ak  {t0a0   tk ak  Aff (a0 ,, ak ) : 0  ti }
t0 ,, tk are the barycentric coordinates of
t0a0   tk ak  a0 ak
Question 17. Show they are unique&continuous.
Boundary of a Simplex
If a0 ,...,ak  A are in general position we
define the interior of the simplex a0 ak
Int(a0 ak )  {t0a0   tk ak  a0 ak : ti  0}
and boundary a0 ak  a0 ak \ Int(a0 ak )
Question 18. Show that if
a0 ak
is regarded
as a subspace of the topological space
Aff (a0 ,, ak ), then these two concepts
coincide with the standard topological concepts
Question 19. Show that a0 ak is a disjoint
union of {a0 ,, ak } and the interiors of simplices
spanned by each subset of {a0 ,, ak }
Simplicial Maps
Question 20. Prove that if A is an affine space
and a0 ,...,ak  Aare in general position then
the set of vertices {a0 ,...,ak }is determined by
the simplex a0 ak
Definition If A and B are affine spaces and
  A,  B are simplices then a map
f :    is a simplicial map if there exists
an affine map
~
f : AB
such that
~
~
f (vertices( ))  vertices( ), f |  f
Geometric Simplicial Complexes
Definition Faces of a simplex are the simplices
spanned by its proper subsets of vertices.
a0 
faces(a0 a1a2 ) 
a2
Example
a1 
 {a0 a1 , a0a2 , a1a2 , a0 , a1 , a2 }
http://en.wikipedia.org/wiki/Simplicial_complex
Definition A geometric simplicial complex is a
collection of simplices in an affine space that
1. contains each face of each element
2. The intersection of each pair of elements
is either empty or a common face
Topological Simplicial Complexes
Definition If K is a finite geometric simplicial
complex in an affine space A we define its
polyhedron
| K |   K   A
and the associated topological simplicial complex
to the be equivalence class of topological spaces
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