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Lecture 1(20.01.09)
Fibre Bundles and Homotopy Exact Sequence
Definition 1 A fibre bundle is a quadruple (E, B, F, p) where E (the total space of a
bundle), F (the fibre) and B (the base) are topological spaces, p : E → B is a continuous
map (the projection) satisfying to the local triviality condition that for every point b ∈ B
there is a neighbourhood U, b ∈ U ∈ B, with a homeomorphism φ : U × F → p−1 (U)
which commutes with the projection p:
p(φ(x, f )) = x,
x ∈ U.
There are numerous examples of fibre bundles in geometry and topology and, in fact,
in the next lectures we shall prove that almost any continuous map f : X → Y maybe
considered as a fibre bundle in the appropriate sense. But first let us discuss some more
natural examples.
Example 1 Cartesian product E = B × F is the trivial fibre bundle.
Example 2 Finite covers studied extensively in PII Algebraic topology are the first examples of non-trivial fibre bundles: z 2 : S 1 → S 1 , a sphere with g ≥ 2 handles covers the
Z2
pretzel (the sphere with two handles), π : S n −→
RP n , n = 0, 1, 2, . . . , ∞ is the most
important double cover. These are fibre bundles whose fibres are finite (or more generally
discrete)
Example 3 The Mobius strip E is the total space of a fibre bundle π : E → S 1 with fibre
[0, 1], it is not trivial. The set of all lines on a two dimensional plane (∼
= R2 ) is an infinite
Mobius strip and the projection on to the angle with x-axes gives it a structure of a fibre
bundle over S 1 with the fibre R1 . This is the first example of a vector bundle - a fibre
bundle whose fibre has a structure of a vector space in such a way that the addition and
the multiplication by scalars vary continuously when the base point varies continuously
in the base space.
Like this we can remind lots of fibre bundles. If M is a smooth manifold say embedded
into Rm , then the set of all tangent vectors form the total space of the tangent bundle
τ (M) of M. It is a fibre bundle over M and even a vector bundle. The fibre of this bundle
at a point x ∈ M is the tangent space τx to M at the point x. Similarly, for an embedding
of two manifolds i : L ⊂ M the subset
νi (L) = {(x, v)| x ∈ L,
v ∈ τx (M),
v⊥τx (L)}
of all vectors normal to the tangent space of L at some point of L is the normal bundle
of embedding i. It is also a vector bundle.
In all previous examples it is relatively straightforward to check the local triviality
condition (left as an exercise). In some cases such verification requires a more serious
routine work.
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Example 4 Let G be a Lie group and i : H ⊂ G be a compact Lie subgroup, i.e., H is
a Lie group and mono morphism i is a C ∞ -embedding. Let p : G → G/H be a factor
map on the set of co-sets. We introduce a push–forward topology on G/H: a subset
U ⊂ G/H is open if and only if p−1 (G/H) is open in G. G/H with this push–forward
topology is a Hausdorff space and even a C ∞ -manifold where p : G → G/H is a C ∞ -map.
Check that (G, G/H, H, p) is a fibre bundle. A generalisation of this example is a bundle
(E, E/G, G, p) obtained as a factorisation of a topological space E by a “proper” free
action of a topological group G.
Example 5 For a bundle (E, B, F, p) and a continuous map f : X → B we define a pull
back bundle (f ∗ (E), X, F, f ∗(p)) with the total space given as
f ∗ (E) = {(x, e) ∈ X × E|f (x) = p(e)}.
If X is a subspace of B and f is an embedding we obtain the restriction of E on X.
As you can see fibre bundles are quite natural objects in geometry but the first application to homotopy groups of spheres I planning to touch, does not look as obvious as
the constructions just discussed. To clarify situation I would remind you that Algebraic
Topology does not study all spaces, this is the subject of General Topology. Algebraic
Topology focuses on the so called CW-complexes.
Definition 2 A CW complex is a topological space which is represented as a disjoint
union
∞
K=
G G
eqi
q=0 i∈Iq
of sets (cells) eqe , if there exists a family of continuous maps fjq : D q → K, such that the
restriction of fiq on Int(D q ) is a homeomorphism Int(D q ) ≈ eqi and
fiq (∂D q )
⊂
q G
G
epi .
p=0 i∈Ip
In addition to that we require the following conditions to be satisfied:
(C) the closure of each cell meets only a finite number of other cells.
(W) a subset F ⊂ K is closed if and only if for each eqi the pre-image (fiq )−1 (F ) ⊂ D q
is closed in D q .
A CW complex is finite if it consists of finitely many cells. A subcomplex of a complex
K is a CW complex contained in K as a closed subset, which cells are cells of K as well.
The n-skeleton Skn (K) of a complex K is the union of all of its cells of dimension ≤ n
Example 6 The complex projective space CP n can be equipped with a structure of a CW
complex such that there is exactly one cell in each even dimension ≤ n and with no
odd-dimensional cells. The 2k-cell is the set (z1 : z2 : · · · : zk : 1 : 0 : · · · : 0).
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In order to describe a space we thus need to understand the attaching maps
fiq (∂D q ) ⊂
q G
G
epi .
p=0 i∈Ip
involved and the most important information is given by the homotopy classes of the
push-forward of fiq on epi /∂epi , i.e. by a collection of maps S q−1 → S p . These are so
called homotopy groups of spheres. In general we have a very limited information on
their structure.
Let I = [0, 1] and I n = I × · · · × I be the unit n-cube, its boundary is the set
∂I = {(x1 , . . . , xn ) ∈ I n | ∃i : xi = 0, 1}
A pointed space (X, x0 ) is a space X with a base point x0 , a pair (X, A) of spaces is
a pointed space X and its pointed subspace A: x0 ∈ A ∈ X. A continuous map of
pairs f : (X, A) → (Y, B) is a continuous map f : X → Y such that f (A) ⊂ B and
f (x0 ) = y0 . Two maps f1 , f2 : (X, A) → (Y, B) are homotopic if there is a continuous
map F : (X × I, A × I) → (Y, B) such that F (x, 0) = f1 (x), F (x, 1) = f2 (x). Homotopy
is an equivalence relation and it makes sense to talk about the set [(X, A), (Y, B)] of
homotopy classes of maps.
Definition 3 Relative homotopy group of a pair (X, A) is
πn (X, A) = [(I n , ∂I n ), (X, A)]
It is indeed a group for n ≥ 1 (check as an exercise) with zero given by f : (I n , ∂I n ) →
A ⊂ X which is abelian always if n > 2. It is also abelian for n = 2 if A is a single point
(A = x0 ). (DRAW A PICTURE!!!)
Exercise 1 Let (X, A) be a topological pair. Embeddings i : A ⊂ X, j : (X, x0 ) → (X, A)
induce homotopy group homomorphisms: i∗ : πn (A, x0 ) → πn (X, x0 ), j∗ : πn (X, x0 ) →
πn (X, A) as well as the restriction mapping ∂(I n ) ≈ S n−1 ≈ (I n−1 , ∂I n−1 ) induces a
group homomorphism ∂∗ : πn (X, A, x0 ) → πn−1 (A, x0 ). Prove that the homotopy sequence
of a pair
i
j∗
∂
∗
∗
πn−1 (A, x0 ) → · · ·
πn (X, x0 ) −→ πn (X, A, x0 ) −→
· · · → πn (A, x0 ) −→
· · · → π1 (X, A, x0 ) → π0 (A, x0 ) → π0 (X, x0 )
is exact, i.e. Ker(j∗ ) = Im(i∗ ), Ker(∂∗ ) = Im(j∗ ), Ker(i∗ ) = Im(∂∗ ).
Remark 1 I always ask every one what should be a proper generalisation of the homotopy
group of pair to the next stage, say to the case of ”properly defined” triples of spaces. I
think that it hides the key for understanding homotopy groups of spheres.
Let us now find out how the homotopies of pairs help computing some homotopy
groups of spheres. The ”main tool” is the homotopy sequence of a fibre bundle.
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Lemma 1 Let (E, B, F, p) be a fibre bundle and the base points e0 ∈ E and b0 ∈ B are
chosen in order that p(e0 ) = b0 , then p∗ : πn (E, F, e0 ) ∼
= πn (B, b0 ) is an isomorphism.
Corollary 1 The homotopy sequence of a fibre bundle
· · · → πn (F, e0 ) → πn (E, e0 ) → πn (B, b0 ) → πn−1 (F, e0 ) → · · ·
· · · → π0 (F, e0 ) → π0 (E, e0 ) → π0 (B, b0 )
is exact.
The key to the proof of the lemma is the Covering Homotopy Property (CHP):
Theorem 1 Let (E, B, F, p) be a fibre bundle, Z be a CW-complex and g̃ : Z → E be a
continuous map. If G : Z × I → B is a homotopy such that G|Z×0 = p ◦ g̃ then there is a
homotopy G̃ : Z × I → E such that G̃|Z×0 = g̃ and p ◦ G̃ = G.
This is proved by ”induction”. The base of induction rests on Feldbau theorem:
Lemma 2 A fibre bundle (E, B, F, p) with a base space B = I n is trivial (is a Cartesian
product)
Proof of Feldbau Lemma. If the cube I n made of two ”bricks” I1 = {xn ≤ 1/2}
and I2 = {xn ≥ 1/2} such that the bundle is trivial over each of the bricks than we
can construct a trivialisation of the initial bundle by means of a family αxn : F → F of
T
homeomorphism of the fibre F given by the identification of E|I1 and E|I2 over I1 I2 =
{xn = 1/2}. We define trivialisation φ : E → I n × F to coincide with the trivialisation of
E over I1 , and each pair (x, y) ∈ E|I2 ≈ I2 × F we map into (x, αx−1
(y)).
n
We can now proceed by induction. We cut the cube into lots of little cubes over which
the bundle is trivial due to the local triviality property. Then we step by step extend
trivialisation from one little cube to another using previous argument.
Proof of the Covering Homotopy Property. As usual in homotopy theory it is
easier to prove such result in stronger so called relative version. We shall assume that
on CW-subspace Z ′ ⊂ Z there is already given a homotopy G̃′ : Z ′ × I → E such that
G̃′ |Z×0 = g̃|Z ′ and p ◦ G̃′ = G|Z ′×I ′ , then the homotopy on the whole space Z can be
constructed in such a way that it coincides with G̃′ over Z ′ . The advantage of the relative
version is that it makes inductive arguments easy to perform.
There are four cases to consider.
Case 1: the bundle is trivial. Then E = B × F . Any map into E is a pair of maps
into B and F . Therefore, the given map g and the given piece of homotopy G̃′ become a
pairs:
(g = G|Z×0 ; h : Z → F ), (G|Z ′×I ′ H ′ : Z ′ × I → F ).
The homotopy needed is a pair of homotopies with the first element been just G, while
the second homotopy H : Z × I → F is an extension of H ′ . Its existence is guaranteed
by “Borsuk Theorem” (DRAW A PICTURE!!!)
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Case 2: any bundle, Z = D k , Z ′ = S k−1 . Consider the pull back bundle G∗ (E, B, F, p) =
(E ′ , D k × I, F, p′ ). According to Feldbau theorem it is a trivial bundle D k × I ≈ I k+1 .
The pull back of the initial map g̃ and the homotopies G, G̃′ satisfy to all condition of
the relative CHP on (D k , S k−1 ) and, thus, G∗ (G̃′ ) can be extended on the whole Z = D k .
But E ′ = G∗ (E) is a subspace of E. Therefore, the homotopy constructed is the required
homotopy of maps Z → E.
Case 3: any bundle, Z is a finite CW-complex. Use induction and reduce question to
the case when Z − Z ′ is a single cell D k . Then apply the previous step.
Case 4: general. If Z has infinitely many cells of the same dimension then step 3
must be applied simultaneously to all of them. If there are infinitely many dimensions,
then the step 3 has to be applied infinitely many times and continuity of the homotopy
constructed follows from the property (W) of CW-complexes.
Example 7 From the Hopf bundle (S 3 , S 2 , S 1 , p) using exactness of HSFB we obtain
π3 (S 2 ) = π2 (S 2 ) ∼
=Z
π2 (S 2 ) ∼
= Z,
= π1 (S 1 ) ∼
and
πn (S 3 ) ∼
= πn (S 2 ),
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n ≥ 3.