Download Chapter 5 Homotopy Theory

Chapter 5
Homotopy Theory
5.1
Homotopy Groups
• Let X and Y be topological spaces and
f0 : X → Y,
f1 : X → Y
be two continuous maps. Then the map f0 is said to be homotopic to the
map f1 if there is a continuous map
F : X × [0, 1] → Y
such that for any x ∈ X
F(x, 0) = f0 (x),
F(x, 1) = f1 (x).
• In other words, if one can continuously deform the map f0 to the map f1 .
• Homotopy is an equivalence relation.
• We fix a point x0 ∈ X. Let f : X → Y and y0 = f (x0 ) ∈ Y. The homotopy
class of f at the basepoint y0 is the set of all maps g : X → Y homotopic to
f such that g(x0 ) = y0
[ f, y0 ] = {g : X → Y | g ∼ f, g(x0 ) = y0 }
• The homotopy classes are invariant under homeomorphisms and are topological invariants.
135
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CHAPTER 5. HOMOTOPY THEORY
• The set [X, Y; y0 ] of all homotopy classes of maps from X to Y with the
basepoint y0 is a topological invariant.
• Usually one fixes a standard manifold X = S n and studies the homotopy
classes of continuous maps f : S n → Y. These are topological invariants of
Y.
• The set of homotopy classes of maps f : S n → Y is called the n-th homotopy group πn (Y) of Y,
πn (Y, y0 ) = [S n , Y; y0 ]
• The set of homotopy classes of maps f : S 1 → Y is called the fundamental
group π1 (Y, y0 ) of Y.
5.2
Fundamental Group
• Let X be a topological space. A path with an initial point x0 and an endpoint
x1 is a continuous map
α : [0, 1] → X
such that
α(0) = x0 ,
α(1) = x1 .
• A topological space X is arcwise connected if for any two points x, y ∈ X
there is a path α with the initial point x and the endpoint y.
• A loop with a base point x0 is a closed path such that
α(0) = α(1) = x0
• A constant path c x0 is defined by
c x0 (t) = x0 ,
t ∈ [0, 1].
• A topological space is simply connected if any loop in it can be continuously shrunk to a point.
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5.2. FUNDAMENTAL GROUP
137
• The product γ = α ∗ β of two paths, α and β, such that α(1) = β(0), is a
path defined by
�
α(2t),
0 ≤ t ≤ 12
γ(t) =
β(2t − 1), 12 ≤ t ≤ 1
• The inverse α−1 of a path α is the path defined by,
α−1 (t) = α(1 − t),
0 ≤ t ≤ 1.
• Two loops α0 , α1 with a base point x0 are said to be homotopic, α0 ∼ α1 ,
if there is a continuous map F : [0, 1] × [0, 1] → X (called a homotopy
between α0 and α1 ) such that
F(t, 0) = α0 (t),
0 ≤ t ≤ 1,
F(t, 1) = α1 (t),
0 ≤ t ≤ 1,
F(0, s) = F(1, s) = x0 ,
0 ≤ s ≤ 1,
• Proposition. The loop homotopy is an equivalence relation.
• Proof.
• The equivalence classes of loops are denoted by [α] and are called homotopy classes.
• The set π1 (X, x0 ) of homotopy classes of loops with base point x0 is called
the fundamental group (or the first homotopy group) of X at x0 .
• The product of homotopy classes is defined by
[α] ∗ [β] = [α ∗ β]
• The inverse of the homotopy class is defined by
[α]−1 = [α−1 ]
• The homotopy class [c x0 ] defines the identity element.
• Proposition. Let α0 , α1 , β0 , β1 be loops based at x0 ∈ X. Suppose that
α0 ∼ α1 and β0 ∼ β1 . Then
−1
α−1
0 ∼ α1
α0 ∗ β0 ∼ α1 ∗ β1
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CHAPTER 5. HOMOTOPY THEORY
• Proof.
• Proposition. The product and the inverse of homotopy classes do not depend on the representatives.
• Proof.
• Theorem. The fundamental group is a group.
• Proof. Prove 1) associativity, 2) unit element, 3) inverse.
• A fundamental group is trivial if π1 (X, x0 ) = {e}.
• Theorem. Let X be an arcwise connected topological space. Then for any
two points x0 , x1 ∈ X, π1 (X, x0 ) is isomorphic to π1 (X, x1 ).
• So, we can just denote it by π1 (X).
• Proof.
• Let α be a loop at x0 .
• Let γ be a path from x0 ro x1 .
• Then γ−1 ∗ α ∗ γ is a loop at x1 .
• This induces a map
σγ : π1 (X, x0 ) → π1 (X, x1 ).
defined by
σγ ([α]) = [γ−1 ∗ α ∗ γ]
• It is easy to see that this map is a homomorphism.
• This map has an inverse
σ−1
γ : π1 (X, x1 ) → π1 (X, x0 )
defined by
σγ ([β]) = [γ ∗ β ∗ γ−1 ]
• Therefore, this map is bijective and is an isomorphism.
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5.3. HOMOTOPY TYPE
5.3
139
Homotopy Type
• Let X and Y be topological spaces. We say that X and Y are of the same
homotopy type if there are two continuous maps
f : X → Y,
g:Y→X
such that the maps
f ◦ g : Y → Y,
g◦ f : X → X
are homotopic to identity maps,
f ◦ g ∼ idY ,
g ◦ f ∼ idX .
• The map f is called the homotopy equivalence and the map g is the homotopy inverse.
• Proposition. Having the same homotopy type is an equivalence relation.
• Proof.
• Lemma. Let fi : X → Y, i = 0, 1, be maps between topological spaces X
and Y and F : X × [0, 1] → Y be a homotopy between f0 and f1 . Let x0 ∈ X
and γ be a path in Y with initial point f0 (x0 ) and endpoint f1 (x0 ). Let
σγ : π1 (Y, f0 (x0 )) → π1 (Y, f1 (x0 ))
be a group isomorphism defined by
σγ ([α]) = [γ−1 ∗ α ∗ γ].
Then there are induced group homomorphisms
fi,∗ : π1 (X, x0 ) → π1 (Y, fi (x0 )),
defined by
fi,∗ ([α], x0 ) = ([ f (α)], fi (x0 )),
such that
f1,∗ = σγ ◦ f0,∗ .
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CHAPTER 5. HOMOTOPY THEORY
• Theorem. Connected topological spaces with the same homotopy type have
the same fundamental group.
In other words: Let X and Y be acrwise connected topological spaces with
the same homotopy type. Suppose there are maps
f : X → Y,
g : Y → X,
f ◦ g ∼ id Y ,
g ◦ f ∼ id X .
such that
Then
π1 (X, x0 ) � π1 (Y, f (x0 )),
π1 (Y, y0 ) � π1 (X, g(y0 )),
• Proof. We have by the Lemma,
σγ ◦ (id Y )∗ = ( f ◦ g)∗ ,
σγ−1 ◦ (id X )∗ = (g ◦ f )∗ ,
• Since σγ is an isomorphism, so are ( f ◦ g)∗ and (g ◦ f )∗ .
• Since
( f ◦ g)∗ = f∗ ◦ g∗ ,
(g ◦ f )∗ = g∗ ◦ f∗ ,
the maps f∗ and g∗ are isomorphisms, which proves the theorem.
• Corollary. The fundamental group is a topological invariant.
• Let R ⊂ X be a subspace of a topological space X. Then R is called a retract
of X if there is a continuous map f : X → R (called a retraction) such that
f |R = id R ,
that is,
f (x) = x
for any x ∈ R.
Example.
• Proposition. The retract R and the whole space X have the same fundamental group, for any x0 ∈ R
π1 (R, x0 ) � π1 (X, x0 ).
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5.3. HOMOTOPY TYPE
141
• The subspace R is called a deformation retract of X if it is a retract of X
homotopic to the identity map id X . That is, there a homotopy
H : X × [0, 1] → X
between the retraction f and the identity map id X that leaves the points in
R fixed, that is,
H(·, 0) = id X ,
H(·, 1) = f,
H(·, s)|R = id R ,
s ∈ [0, 1],
or
H(x, 0) = x,
H(x, s) = x
H(x, 1) = f (x) ∈ R,
for any x ∈ R
for any x ∈ X
and any s ∈ [0, 1]
• Examples.
• A topological space X is contractible if a point x0 ∈ X is a deformation
retract of X.
• If X is contractible to a point x0 then there is a homotopy H : X × [0, 1] → X
(called a contraction) such that
H(x, 0) = x0 ,
H(x, 1) = id X (x) = x,
H(x0 , t) = x0
for any x ∈ X
for any t ∈ [0, 1]
• Example.
• An arcwise connected topological space is called simply connected if it has
a trivial fundamental group.
• Theorem. A contractible space is simply connected, that is, it has a trivial
fundamental group.
• Examples.
• Theorem. The fundamental group of the circle
S 1 = R/2πZ
is isomorphic to Z,
π1 (S 1 ) � Z.
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CHAPTER 5. HOMOTOPY THEORY
• One can identify the maps f : S 1 → S 1 such that f (1) = 1 with the maps
f˜ : R → R such that f˜(0) = 0 and
f˜(x + 2π) = f˜(x) + 2πn
The integer n = deg f is equal to the degree of f .
• Two maps f, g : S 1 → S 1 such that f (1) = g(1) are homotopic if and only
if they have the same degree.
• For any n ∈ Z there is a map f : S 1 → S 1 with degree n.
• Therefore, there is a bijection, in fact, isomorphism, between π1 (S 1 , 1) and
Z.
• Theorem. Let X and Y be arcwise connected topological spaces and x0 ∈
X, y0 ∈ Y. Then
π1 (X × Y, (x0 , y0 )) � π1 (X, x0 ) ⊕ π1 (Y, y0 )
• Examples.
n
π1 (T ) =
n
�
Z
i=1
π1 (S 1 × R) = Z ⊕ {e} � Z
5.4
Fundamental Groups of Polyhedra
• If the space X is triangulable then one can compute π1 (X) by a routine procedure.
• Let X be an arcwise connected topological space.
• Let f : |K| → X be a triangulation of X, where K is a simplicial complex
and |K| is a polyhedron |K|.
• Then
π1 (X) = π1 (|K|)
• There is a systematic way to compute π1 (|K|).
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5.4. FUNDAMENTAL GROUPS OF POLYHEDRA
143
• Any loop in |K| is made of 1-simplexes.
• An edge path in a simplicial complex K is a sequence of vertices v0 v1 · · · vk ,
such that each consecutive pair vi vi+1 is either a 0-simplex or a 1-simplex.
• An edge path is an edge loop at v0 if v0 = vk .
• Two edge loops are equivalent if one is obtained from another by the following operations:
1. The edge path uvw is equivalent to the path uw if u, v, w span a 2simplex.
2. The edge path uvu is equivalent to the 0-simplex u.
• the equivalence class of edge loops at a vertex v is denoted by
{vv1 · · · vk−1 v}
• We define the product of equivalence classes of edge loops at a vertex v by
{vv1 · · · vk−1 v} ∗ {vu1 · · · u j−1 v} = {vv1 · · · vk−1 vu1 · · · u j−1 v}
• The unit element is the equivalence class {v}.
• The inverse equivalence class is defined by
{vv1 · · · vk−1 v}−1 = {vvk−1 · · · v1 v}
• Theorem. The set of equivalence classes of edge loops at a vertex v forms
a group E(K, v), called the edge group of the simplicial complex K.
• Theorem. The edge group of a simplicial complex K is isomorphic to the
fundamental group of the polyhedron |K|
E(K, v) � π1 (|K|, v)
• Let L be a simplicial subcomplex of K such that L contains all vertices of K
and the polyhedron |L| is simply connected.
• For any path connected simplicial complex K there is such a subcomplex.
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CHAPTER 5. HOMOTOPY THEORY
• A one-dimensional simply connected simplicial complex is called a tree.
• A tree is called a maximal tree if it is not a proper subset of other trees.
• Proposition. A maximal tree of a simplicial complex contains all its vertices.
• Let v0 , v1 , . . . vn be vertices of K.
• For each ordered pair of vertices (vi , v j ) such that �vi v j � is a 1-simplex in K
we assign an element gi j of a group.
• Let G(K, L) be the group generated by gi j with the following relations
1.
gi j = 1
2.
if the �vi v j � ∈ L ,
gi j g jk gki = 1
if �vi v j vk � is a 2-simplex of K and there are no nontrivial loops around
vi v j vk .
• Proposition.
gii = 1
(gi j )−1 = g ji
• Theorem. The group G(K, L) is isomorphic to the edge group E(K, v) and,
therefore, to the fundamental group π1 (|K|, v),
G(K, L) � E(K, v) � π1 (|K|, v) .
• Summary. To compute the fundamental group of X we need:
1. Find a triangulation f : |K| → X.
2. Find a simply connected subcomplex L of K that contains all vertices
of K.
3. Assign a generator gi j to each 1-simplex �vi v j � ∈ K − L with i < j.
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5.4. FUNDAMENTAL GROUPS OF POLYHEDRA
145
4. Impose a relation
gi j g jk = gik
if there is a 2-simplex �vi v j vk � such that i < j < k.
If two of the vertices vi , v j , vk form a 1-simplex of L the generator
should be set equal to 1.
5. The fundamental group π1 (X) is isomorphic to the group G(K, L) generated by {gi j } with the above relations.
• The set K (p) of all i-simplexes for i = 0, 1, . . . , p of a simplicial complex K
is called a p-skeleton of K.
• It should be clear that the fundamental group of a polyhedron is determined
by its 2-skeleton
π1 (|K|) � π1 (|K (2) |)
• For any n ≥ 2, a (n + 1)-simplex σn+1 and its boundary ∂σn+1 have the same
2-skeleton.
• Examples.
• Since σn+1 is contractible and ∂σn+1 is a polyhedron of S n , we obtain
π1 (S n ) = {e}
n≥2
The n-sphere for n ≥ 2 is simply connected
• A triangulation of the circle is
K = {v1 , v2 , v3 , �v1 v2 �, �v1 v3 �, �v2 v3 �}
For the circle there is only one generator with no relations
π1 (S 1 ) � (x; ∅) � Z
• A n-bouquet M is the one-point union of n circles.
π1 (M) = (x1 , . . . , xn ; ∅)
• The 2-disk is simply connected
π1 (D2 ) � {e}
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CHAPTER 5. HOMOTOPY THEORY
• For the 2-torus there are two generators with one relation
π1 (T 2 ) � (x, y; xyx−1 y−1 ) � Z ⊕ Z
• For the surface of genus g there are 2g generators with one relation


g
�



π1 (Σg ) �  x1 , . . . xg , y1 , . . . yg ;
xi yi xi−1 y−1
i 
i=1
• For the real projective plane there is one generator with one relation
π1 (RP2 ) � (x; x2 ) � Z2
The same is true for any n > 1
π1 (RPn ) � (x; x2 ) � Z2
• Exercises.
• Show that for the Möbius strip M there is one generator with no relations
π1 (M 2 ) � (x; ∅) � Z
• Show that for the Klein bottle there are two generators with one relation
π1 (K 2 ) � (x, y; xyxy−1 )
• Theorem. Let K be a connected simplicial complex. Then the first homology group H1 (K) is isomorphic to the quotient of the fundamental group
π1 (|K|) by its commutator subgroup F
H1 (K) � π1 (|K|)/F.
• Corollary. The first homology group of a connected topological space is
isomorphic to its fundamental group
H1 (K) � π1 (|K|)
if and only if the fundamental group is commutative.
• Corollary. The first homology groups of two topological spaces X and Y of
the same homotopy type are the same,
H1 (X) = H1 (Y)
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5.5. HIGHER HOMOTOPY GROUPS
5.5
147
Higher Homotopy Groups
• Let I n be the unit cube
[0, 1] × · · · × [0, 1] = {t1 , . . . , tn | ti ∈ [0, 1]}
I n = ��������������������������������������
n
• The boundary of I n is
∂I n = {t1 , . . . , tn | ti ∈ [0, 1], t j = 0 or 1, for some j}
• A map α : I n → X is called an n-loop at a point x0 ∈ X if it maps the
boundary ∂I n to x0 .
• A constant n-loop is a map
c x0 (t1 , . . . , tn ) = x0 ,
for any ti ∈ [0, 1].
• The cube I n whose boundary ∂I n is shrunk to a point is homeomorphic to
the sphere
I n /∂I n � S n
• Two n-loops at x0 , α0 and α1 , are homotopic if there is a continuous map,
called a homotopy between α0 and α1 ,
F : In × I → X
such that
F(t, 0) = α0 (t),
F(t, s) = x0
where t = (t1 , . . . , tn ).
F(t, 1) = α1 (t)
for t ∈ ∂I n , s ∈ [0, 1]
• Homotopy is an equivalence relation.
• The corresponding equivalence classes are called the homotopy classes.
• The product of two n-loops at x0 is defined by
�
α(2t1 , t2 . . . , tn ),
(α ∗ β)(t1 , t2 , . . . , tn ) =
β(2t1 − 1, t2 . . . , tn ),
0 ≤ t1 ≤ 12
1
≤ t1 ≤ 1
2
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CHAPTER 5. HOMOTOPY THEORY
• The inverse of an n-loop is defined by
α−1 (t1 , t2 , . . . , tn ) = α(1 − t1 , t2 , . . . , tn )
• Then
α−1 ∗ α ∼ α ∗ α−1 ∼ c x0
• For any n ≥ 1 the set πn (X, x0 ) of all homotopy classes of n-loops at x0 is
called the n-th homotopy group of X at x0 .
• The product of homotopy classes is defined by
[α] ∗ [β] = [α ∗ β]
• This product does not depend on the representatives of the homotopy classes.
• Theorem. The n-th homotopy groups are groups.
• The set π0 (X) is not a group; it is rather the number of independent component of X.
• Proposition. Higher homotopy groups πn (X, x0 ), for any n ≥ 2, are Abelian.
• Proposition. For a connected space X the homotopy groups πn (X, x0 do not
depend on the choice of the point x0 .
• Proposition. If the (connected) spaces X and Y have the same homotopy
type then the their homotopy groups are isomorphic
πn (X) � πn (Y)
• Corollary. The homotopy groups of a contractible space X are trivial
πn (X) � {e}
• Proposition. For any connected topological spaces X and Y
πn (X × Y) � πn (X) ⊕ πn (Y)
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5.6. UNIVERSAL COVERING SPACES
5.6
149
Universal Covering Spaces
• A connected topological space X̃ is called the covering space of a topological space X if there is a continuous map
f : X̃ → X
such that
1. f is surjective,
2. for each x ∈ X there is a connected open set U ⊂ X containing x such
that the inverse image f −1 (U) is a disjoint union of open sets Vi in X̃,
�
Vi ,
f −1 (U) =
i
each of which is homeomorphic to U, that is,
f : Vi → U
are homeomorphisms.
• If the covering space X̃ is simply connected, then it is called the universal
covering space of X.
• If the spaces X and X̃ are topological groups and the map f is a group
homomorphism, then X̃ is called the covering group.
• The universal covering of the circle S 1 (or the group U(1)) is the real line
R.
• Theorem. Let X̃ be a universal covering space of a connected space X and
f : X̃ → X be the covering map. Let x̃0 ∈ X̃ and x0 = f ( x̃0 ) ∈ X. Then for
any n ≥ 2, the induced homomorphism
f∗ : πn (X̃, x̃0 ) → πn (X, x0 )
is an isomorphism.
• As an example,
for any n ≥ 2.
πn (S 1 ) = {e}
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CHAPTER 5. HOMOTOPY THEORY
• The sphere S n is a covering space of the real projective space RPn for n ≥ 2.
• For any n ≥ 2
πn (RPn ) = {e}
• Examples.
RP3 = S O(3)
•
S 3 = S U(2)
• Proposition. For any n ≥ 2
πn (S O(3)) = πn (RP3 ) = πn (S 3 ) = πn (S U(2))
• The universal covering group of the special orthogonal group S O(n) is
called the spin group.
• Examples.
Spin (3) = S U(2)
Spin (4) = S U(2) × S U(2)
Spin (6) = S U(4)
• Examples.
πn (S n ) = πn (RPn ) = Z,
n≥2
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