Transcript CLASSICAL INFORMATION THEORY

MA5209 Algebraic Topology
Lecture 4. Category Theory
and Homotopy Theory
(1, 4 September 2009)
Wayne Lawton
Department of Mathematics
National University of Singapore
S14-04-04, [email protected]
http://math.nus.edu.sg/~matwml
Categories
A category C consists of the following three mathematical entities:
•A class ob(C), whose elements are called objects;
•A class hom(C), whose elements are called morphisms or maps or arrows.
Each morphism f has a unique source object a and target object b.
We write f: a → b, and we say "f is a morphism from a to b".
We write Hom(a, b), or Mor(a, b), to denote the hom-class of all morphisms from a to b.
•A binary operation, called composition of morphisms, such that for any three objects a, b, and c,
we have Hom(a, b) × Hom(b, c) → Hom(a, c).
The composition of f: a → b and g: b → c is written as gf and is governed by two axioms:
Associativity: If f : a → b, g : b → c and h : c → d then h(gf) = (hg)f, and
Identity: For every object x, there exists a morphism 1x : x → x
called the identity morphism for x, such that for every morphism f : a → b, we have 1b f  f  f 1a
ob(C) = Sets, Mor(a,b) = { functions f : a  b}
ob(C) = Groups, Mor(a,b) = { homomorphisms f : a  b}
ob(C) = Top Spaces, Mor(a,b) = { maps f : a  b}
ob(C) = Pointed Top Spaces,
Mor((X,p),(Y,q)) ={ maps f : a  b such that f(p) = q}
ob(C) = Top Spaces, Mor(a,b) = [a,b] = homotopy
classes of maps f : a  b with [f] [g] = [f g]
Commutative Diagram for
Composition of Morphisms
Properties of Morphisms
A morphism f : a → b is:
a monomorphism (or monic) if f g1 = f g2 implies g1 = g2
for all morphisms g1, g2 : x → a.
an epimorphism (or epic) if g1 f = g2 f implies g1 = g2
for all morphisms g1, g2 : b → x.
an isomorphism if there exists a morphism g : b → a
with f g = 1b and g f = 1a.
an endomorphism if a = b. End(a) denotes the class of endomorphisms of a.
an automorphism if f is both an endomorphism and an isomorphism. Aut(a)
denotes the class of automorphisms of a.
Homotopy Equivalence
Definition Topological Spaces X and Y are homotopy equivalent if
[X, Y] contains an isomorphism. This means that there exist maps
f : X  Y and g : Y  X such that g f  1X and f g  1Y
Question 1. Explain what isomorphisms are in the following categories:
sets, topological spaces with maps as morphisms, groups.
Question 2. Show that in any category the composition of isomorphisms
is an isomorphism.
Question 3. Use Q2 to prove that homotopy equivalence is an equivalence
relation on the class of topological spaces.
Question 4. Prove that
and
are homotopy equivalent.
Relative Homotopy
Definition Let X, Y be topological spaces and
A  X. Maps f, g : X  Y with f |A  g | A are
homotopic relative to A, written f  g rel A,
if there exists a homotopy F : X  [0,1]  Y
from f to g such that
F (a, t )  f (a)  g (a), (a, t )  A  [0,1]
Definition A groupoid is a category in which
every morphism is an isomorphism.
Question 5 Show that a group is a groupoid
with exactly one object.
Contractible Spaces
Definition A topological space X is contractible
if it is homotopy equivalent to a single point
space.
Lemma X is contractible iff there exists x0  X
and a map F : X  [0,1]  X such that
F ( x,0)  x, F ( x,1)  x0 , x  X
Theorem If X is contractible and f, g : [0,1]  X
with f(0) = g(0) and f(1) = g(1) then f  g rel {0,1}
Proof Construct H : [0,1]  [0,1]  X by
t [ 12 ,1]
t [0, 12 ]
H (s, t )  F ( f (0),2s), s [0, t ]
H (s, t )  F ( g (0),2s), s [0,1  t ]
H (s, t )  F ( f (1s2tt ),2t ), s  (t,1  t ) H (s, t )  F ( g (1s212tt ),2  2t ), s  (1  t, t )
H (s, t )  F ( f (1),2  2s), s [1  t ,1] H (s, t )  F ( g (1),2  2s), s [t ,1]
Geometry of Theorem
f
F ( f (0),0)  f (0)  g (0)

F ( f (0), t )

F ( f ( s), t )
F ( g (s), t )
x0

F ( g (0), t )
g

f (1)  g (1)
Connectedness
Definition A topological space X is disconnected
if any of the following equivalent conditions hold:
1. A, B  X  A   , B   , A open, B open,
X  A  B and A  B  
2. A, B  X  A   , B   , A closed, B closed,
X  A  B and A  B  
3. A  X  A   , A  X , A closed and A closed
otherwise it is connected. A subset Y of a
topological space X is d. / c.. if it is d. / c. when it
is given the subspace topology.
Result The connected subsets of R with the usual
topology are exactly the intervals. The image of a
connected set under a map is connected. The
product of connected subsets is connected.
Path Connectedness
Definition In a topological space X a point p is
path connected to a point q if there exists a map
f : [0,1]  X such that f(0) = p and f(1) = q.
Lemma Path conn. is an equivalence relation.
Proof. p is connected to p by the constant map
f : [0,1]  X defined by f(x) = p, x in X. If p is
connected to q by a path f : [0,1]  X then q is
connected to p by the path g : [0,1]  X defined
by g(s) = f(1-s), s in [0,1]. If p is connected to q by
a path a and q is connected to r by a path b then
p is connected to r by the path ba defined by
(ba)(s)  a(2s) for s [0, 12 ], (ba)(s)  b(2s 1) for s [ 12 ,1].
Definition X is path connected if X is an equiv. class
Local Path Connectedness
Definition X is locally path connected if for every x
in X and every open O containing x there exists
an open U containing x and U subset O such that
every pair of points in U is path connected.
Theorem In a locally path connected space path
connected equiv. classes are open and closed.
Proof Let E be an equiv. class and x  E.
Then x  X and X is open hence there exists
an open set O with x  O and each point in O
is path connected to x and hence O  E  E open
If x  E there exists a path connected open O
with x  O hence E  O    O  E  x  E  E closed
Corollary If X is locally path connected
and connected then X is path connected.
Fundamental Groupoid (X )
of a pathwise connected topological space X is
the following category: the objects are the points
of X; for p, q in X the morphisms Mor(p,q) are the
equivalence classes of maps f : [0,1]  X with f(0)
= p, f(1) = q with respect to homotopy relative to
{0,1}, and for p, q, r in X the composition of [a] in
Mor(p,q) with [b] in Mor(q,r) (a, b : [0,1] X with
a(0) = p, a(1) = b(0) = q and b(1) = r), is
given by [b][a] = [ba] where [b] and [a] denote the
homotopy (relative to {0,1}) equivalence classes
containing a and b respectively.
Question 6. Prove that (X ) is a groupoid.
Remark The objects are not sets!
Fundamental Group
Fix a topological space X.
For every p in X, the set Mor(p,p) together with
the composition defined in the groupoid on the
previous page, is a group, called the fundamental
group of X based at p, and denoted by  1 ( X , p)
Question 7. Show that if p,q in X, then every
[a] in Mor(p,q) defines an isomorphism
[a] : 1 ( X , p)  1 ( X , q)
1
by the formula [a] ([b])  [aba ]
Covariant Functors
Let C and D be categories. A covariant functor F from C
to D is a function that associates to each object X in C
an object F(X) and to each morphism f : X  Y in C a
morphism F(f) : F(X)  F(Y) such that if
X
Y 
 Z
F (1X )  1F ( X ) and F ( g f )  F ( g ) F ( f )
f
then
g
Example Fix a topological space X and let C be the
associated groupoid category, let D be the category of
groups, and for p, q in X and [a] in Mor(p,q) define
F ( p)   1 ( X , p)
F ([a])  [a]  Mor( F ( p), F (q))
Question 8. Show that this describes a covariant functor
from the category (X ) to the category of groups.
The Fundamental Group Functor
Definition The category of pointed topological spaces
has objects (X,p) and
Mor((X,p),(Y,q)) = { maps f : X  Y with f(p) = q }.
Consider the rule that associates to each pointed space
( X , p) the group  1 ( X , p) and to each pointed map
f : ( X , p)  (Y , q) the function 1 ( f ) : 1 ( X , p)  1 (Y , q)
defined by
1 ( f )([a])  [ f  a])
Question 9. Show that 1 ( f ) is well defined.
Question 10. Show that 1 ( f ) is a group homomorphism.
Question 11. Show that 1 is a functor.
Question 12. Show that if f : X  Y is a homotopy
equivalence then 1 ( f ) : 1 ( X , x)  1 (Y , f ( x))
is an isomorphism for all x  X .
Category of Groupoids
The objects of this category are groupoids
and for every pair of groupoids a and b,
Mor(a,b) is the set of (covariant) functors
from a to b.
Question 13. Explain how to define composition
of morphisms in this category and show that it
actually produces a category.
The Groupoid Functor 
Definition For each topological space X let (X )
be its fundamental groupoid and for each map
f : X  Y let ( f ) be the functor from the
category (X ) to the category  (Y )
defined by: ( f )(x)  f ( y), x  X .
( f )([a])  [ f  a], [a]  Mor( x, y)
(here a : [0,1]  X , a(0)  x, a(1)  y )
Question 14. Show this defines a functor from the
CAT = topological spaces to CAT = groupoids
Question 15. Show that f  g : X  Y  functors
( f ) and  (g ) are homotopic (naturally equiv.)
Simply Connected Spaces
Theorem The trivial group is the one having a
single element, namely, its identity element.
Lemma If a space X is path connected, then
1 ( X , p)  1 ( X , q), p, q  X . ( is omorphism)
Question 16. Prove this.
Definition A space is simply connected if it is path
connected and its fundamental group is trivial.
Lemma Every contractible space is simply conn.
Question 17. Prove this and then show that every
convex subspace of an affine space and the
n
n
subspaces S \ { p}, n  0, p  S are simp. conn.
Simply Connected Spaces
Theorem If X is a topological space and
U ,V  X are open subsets that are simply
connected (regarded as subspaces) and
such that U  V is path connected and
X  U  V then X  U  V .
Corollary S n , n  2 is simply connected.
n 1

n 1
Proof S  { ( x1 ,, xn 1 )  R : k 1 x  1 }
is clearly path connected. Let p  (0,,0,1), q   p
and U  S n \ { p}, V  S n \ {q} then show that
n
U V  S \ { p, q} is path connected.
n
2
k
Simply Connected Spaces
Proof (Theorem) Clearly X is pathwise connected.
To show that it is simply connected we choose
p  X and a loop a : [0,1]  X based at p,
namely a(0)  a(1)  p. It suffices to show that
there exists an integer n  N and loops
ai : [0,1]  X , i  1,...,n based at p, such that
each ai  c p rel{0,1} where c p (s)  p, s [0,1],
and a  an (an1 ((a3 (a2a1 )))), rel{0,1}.
Proof By Lebesgue’s lemma there exists
0  t0  t1    tn  1 with each a([t k 1 , t k ]) is contained
in U or V . Join p to each a(tk ), k  1,...,n 1 by a path
k : [0,1]  X in U ,V ,U  V if a(tk ) U ,V ,U V respectively.
Construct bk : [0,1]  X , bk (s)  a((tk  tk 1 )s  tk 1 )
1
1
1
a

b

,
a


(
b

),

,
a


(
b

and 1 1 1 2 1 2 2
n1
n2 n1 n1 ), an  n1bn .
Covering Spaces
Definition Let E and X be topological spaces.
A map p : E  X is a covering map (and E is a
covering space of X) if every x  X has an open
1
admissible neighborhood U such that p (U )
is a disjoint union of open sets Si , i  I
index
set
such that for every i  I the restriction
pi  p |Si : Si  U is a homeomorphism.
The S , i  I are called sheets over U and the
i
1
i
maps si  p : U  Si are (local) sections of
p since p  si  1U , i  I .
Covering Spaces
Lemma If p : E  X is a covering map then
1
1. Each fibre p ( x), x  X is a discrete subspace.
2. p is a local homeomorphism.
3. p is surjective and X has the quotient topology.
Question 18 Prove these statements.
Lemma Let E be a topological group, H  E
be a discrete subgroup, p : E  X  E / H be the
map p(e)  eH , e  E onto the set X of left cosets,
and give X the quotient topology. Then p is a c.m.
Question 19 Prove this lemma.
Question 20 Prove that R / Z  Tc , SU(2) /{ I }  SO(3)
Fibre Bundles
are 4-tuples ( E , X , F , p) consisting of a total
space E , a base space X , a fiber (space) F ,
and a projection p : E  X , such that there
exists an open cover  of X and for each O  
a homeomorphism O : O  F  p (O)
1
such that p  O ( x, y)  x,
( x, y)  O  F.
Example (trivial) E  X  F , p( x, y)  x, ( x, y)  X  F
n1
Example E  SO(n), X  S , F  SO(n 1), p(M )  M:,1
Example X connected, P : E  X a covering,
F  p 1 ( x) for any x  X . Question 21. Prove this.
Lifting
Y is a space,
Definition If p : E  X is a
~
then a map f : Y  E ismap,
a lift of a map f : Y  X
~
if p  f  f or, equivalently, the diagram
~
f
Y
E
f
p commutes.
X
2
Question 22. Show that if p : SO(3)  S is the
projection on to the first column of M  SO(3) then
~ 2
2
2
of
f

1
:
S

S
there exists a lift f : S  SO(3)
S2
iff there exists a nowhere vanishing tangent
2
S
vector field on . Then what can you conclude?
Fibrations
Definition A surjective map p : E  X is a
fibration if it satisfies the following
homotopy lifting property:
~
If h  i0  p  f then there exists h that makes
f
Y
E
Remark: h
~
h
is a homotopy
i0
p
from i0 to i1
X
Y  [0,1]
where
h
commute.
ik ( y)  ( y, k ), k  0,1
~
and h is a lift of h
Fibrations
Theorem If ( E , X , F , p) is a fiber bundle and
X is paracompact then p : E  X is a fibration.
Proof Corollary 14 on page 96 in Spanier’s
Algebraic Topology (requires much work).
Definition A paracompact space is one in which
every open cover has a locally finite refinement.
Hence every compact space is paracompact
Question 23. Show that if F is not discrete then
~
the lift h may not be uniquely determined by
h and f .
Unique Lifting
Theorem If p : E  X is a covering, Y is conn.&loc
~ ~
X and
conn. and f1 , f 2 : Y  E lift a map f : Y 
~ ~
~
~
 y1  Y such that f1 ( y1 )  f 2 ( y1 ) then f1  f 2 .
~
~
Proof Let Y1  {y Y : f1 ( y)  f 2 ( y)}. Since y1  Y1
and Y is connected, it suffices to show that Y1
is both open and closed. Since we assume that
all spaces are Hausdorff, it is closed (why?). For
1
y Y1 choose admissble open U  f ( y)  p (U )
is a disjoint union of open sheets and let S be the
sheet that contains f1 ( y)  f 2 ( y). Y loc. conn.
 conn.open y Vk  f 1 (U )  f k (Vk )  p1 (U ), k  1,2.
1
W open&closed in p (U )  f k (Vk )  S Hence p |S
injective and
~
~
~ ~
p  f1  p  f 2  f  f1  f 2
on
V1 V2 .
Coverings are Fibrations
Theorem (Covering Homotopy Theorem 3 on
page 66 in Singer and Thorps Lecture Notes …)
If p : E  X is a covering and Y is compact and
~
h  i0  p  f then  h that makes the diagram
commute. Proof. Let U1 ,...,U r be an admissible
f
Y
E open cover of
~
h(Y  [0,1]).
h
i0
p
Then cover Y
h
X V1,...,Vs and find
Y  [0,1]
0  t0  tk  1 such that each h(Va [ti , ti 1 ])  Ub
for some U b . Then use glueing and induction.
Implications of Unique Homotopy Lifting
Corollary 1.(p. 67 in S&T) If p : E  X is a covering
e  E  1 ( p) : 1 ( E, e)  1 ( X , p(e)) is injective.
Corollary 2.(p. 67 in S&T) If p : E  X is a covering
and a : [0,1]  X , e  E, p(e)  a(0) then there
~
~
a
exists unique lift a : [0,1]  E with (0)  e.
Corollary 3.(p. 68 in S&T) If p : E  X is a covering
e  E  there exists a ‘natural’ correspondence
1
between the fiber p ( p(e)) and the set of cosets
1 ( X , p(e)) / 1 ( p) 1 ( E, e)
Theorem 4.(p. 71 in S&T) X path.&loc. path.
conn.
1 ( X , x) 
loc. simply conn. H subgroup of
 coveringp : E  X  1 ( p)1 ( E, e)  H .
Galois Theory of Covering Transformations
Definition If p : E  X a homeomorphism c : E  E
is called a covering transformation if p  c  p.
Question 24 Show that these form a group G,
that this group acts freely on each fiber.
Theorem 9.(p. 76 in S&T)* If 1 ( p) 1 ( E, e)
is a normal subgroup of 1 ( X , p(e)) then
G  1 ( E, e) \ 1 ( p) 1 ( E, e) and X is the quotient
space of E under the action of G.
Remark* E is pathwise connected, p is surjective.
Assignments
Read Handouts:
[1] Appendix: Generators and relations
[2] pages 131-140 in Armstrong’s Basic Topology
[3] pages 62-77 in Singer and Thorpe’s Lecture
Notes on Elementary Topology and Geometry
Do Problems 20-24 on page 140 of [2]. The
‘dunce cap’ in problem 22 is the polyhedron of a
2-simplex.
Learn how to compute the fundamental group of
a simplicial complex using the algorithm in [2]
and describe the group by its generators and
relations as described in [1]. Learn van
Kampen’s theorem.
Projects
Write a 5-10 page report and give a 20 minute
talk on either 5,9 October on 1 of following topics:
1. classification of 2-dimensional manifolds
2. Jordan curve theorem 3. Riemann surfaces
4. Covering spaces (and their Galois theory)
5. Differential topology theory (Milnor’s book)
6. Higher homotopy groups 7. Knot theory
8. Lie group topology 9. Gauss-Bonnet theorem
10. Fibrations and Fiber Bundles11. Graph theory
12. Groupoids and van Kampen’s theorem
or another topic related to algebraic topology
Homotopy
Problem 1. Let X be a topological space, let
a : [0,1]  X be a path from p  a(0) to q  a(1),
1
1
define a : [0,1]  X by a (s)  a(1  s), and
1
define a  a : [0,1]  X by
1
a
(
2
s
),
s

[
0
,
2]
1
a  a (s)  1
a (2s  1)  a(2  2s), s [ 12 ,1].
Construct a homotopy H RELATIVE TO {0,1}
1
from a  a to the constant path c p : [0,1]  X
defined by c p (s)  p, s [0,1].
Homotopy
Solution 1. We need to construct a map
H : [0,1]  [0,1]  X with the following 3 properties:
1
H (s,0)  a  a (s), s [0,1],
H (s,1)  c p (s)  p, s [0,1],
H (0, t )  H (1, t )  p, t [0,1].
Strategy: construct H as a composition of simple
G
a
maps [0,1] [0,1] 
[0,1] 
 X
where 2s(1  t ), s [0, 12 ], Glueing lemma  G and
G ( s, t ) 
(2  2s)(1  t ), s [ 12 ,1]. H  a  G are continuous.
s [0, 12 ]  H (s,0)  a(2s), s [ 12 ,1]  H (s,0)  a(2  2s)
s [0, 12 ]  H (s,1)  a(0)  p, s [ 12 ,1]  H (s,1)  a(0)  p,
H (0, t )  H (1, t )  p, t [0,1].
Covering Maps
Problem 2. Let E  Tc  {3,1,1,3}, X  Tc  {1,1}
have the topology as subspaces of C and define
p : E  X by
c
z T 
p( z  3)  z  1, p( z 1)  z 1, p( z  1)   z  1.
2
2
i. Draw a ‘picture’ of E and X and p.
Solution
X
1
E
b
a
1
a1
 2
2
0
b2

2
a3

b3
a
0
b
p(ai )  a, p(bi )  b, i  1,2,3,
1
p (0)  {2,0,2}.
Covering Maps
Problem 2. Let E  Tc  {3,1,1,3}, X  Tc  {1,1}
have the topology as subspaces of C and define
p : E  X by
c
z T 
p( z  3)  z  1, p( z 1)  z 1, p( z  1)   z  1.
2
2
ii. Is p a covering map? Solution: p is a 3-1 C.M.
b1 E a2
a1
 2
0
b2

2
a3
1

b3
X
a
0
b
p(ai )  a, p(bi )  b, i  1,2,3,
1
p (0)  {2,0,2}.
Covering Maps
iii. Compute  1 ( X ,0).
Solution Free group generated by a and b.
b1 E a2
a1
 2
0
b2

2
a3
1

b3
a
X
0
b
p(ai )  a, p(bi )  b, i  1,2,3,
1
p (0)  {2,0,2}.
Covering Maps
iv. Compute 1 ( E,2),1 ( E,0),1 ( E,2).
Solution Free groups generated by
1
2
2 3 2
1 1
2 3 3 3 2
1
and
3 3 3
1 1
2 1
1 1 2
{a1, b1b2 , b a a b , b a b a b },
1
{b2b1, b1 a1b1, a2a3 , a b a },
1
{b3 , a3a2 , a2 b2b1a2 , a b a b a } respectively.
b1 E a2
a1
 2
0
b2

2
a3
1

b3
a
X
0
b
p(ai )  a, p(bi )  b, i  1,2,3,
1
p (0)  {2,0,2}.
Covering Maps
v. Compute 1 ( p)(1 ( E,2)),1 ( p)(1 ( E,0)),1 ( p)(1 ( E,2)).
Solution Free subgroups of 1 ( X ,0) generated by
2
1 2
1 1
These 3 subgroups
{a, b , b a b, b a bab},
are conjugate to
2
1
2
1
{b , b ab, a , a ba}, and
each other!
2
1 2
1 1
{b, a , a b a, a b aba} respectively.
b1 E a2
a1
 2
0
b2

2
a3
1

b3
a
X
0
b
p(ai )  a, p(bi )  b, i  1,2,3,
1
p (0)  {2,0,2}.
Cofibrations
Definition A map i : A  X is a
cofibration if it satisfies the following
homotopy extension property: if there exist g, G
that makes the solid diagram below commute
~
g
then there exists that makes the augmented
diagram below
1A{0}
A  {0}
A  [0,1] commute.
G
Compare with
i 1{0}
i 1[0,1] fibrations on
Y
~
g
g
slide 25.
X {0}
1X {0}
X  [0,1]
Contravariant Functors
Given categories C,D . A contracovariant functor F from
C to D is a function that associates to each object X in C
an object F(X) and to each morphism f : X  Y in C a
morphism F(f) : F(Y)  F(X) such that if
X
Y 
 Z
f
g
then F (1X )  1F ( X ) and F ( g f )  F ( f ) F ( g )
Example Let C be a category, A in Obj(C) and define
F ( X )  Mor( X , A)
F ( f )(a)  a  f , f  Mor( X , Y ), a  Mor(Y , A)
Question 24. Show this describes a covariant functor
from the category C to the category SETS
Natural Transformations
Given categories C, D and functors F,G from C to D.
A natural transformation from F to G is a function
T : Obj(C )  Mor( D)
Such that if F, G are covariant (contravariant) the left
(right) diagram below commutes for every h Mor( X , Y )
F ( X ) 
 F (Y )
F ( h)
T (Y )
T (X )
G( X ) 
G(Y )
G ( h)
F ( X ) 
 F (Y )
F ( h)
T (X )
T (Y )
G( X ) 
 G(Y )
G ( h)
Question 25. Show why NT are also called homotopies.
Suggestion. Let I  {0,1} be the two object groupoid and
derive a correspondence between NT and functors H
from C  I to D such that H | C{0}  F , H | C{1}  G.