Download 6.5 Reduction of Bundles

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6.5
CHAPTER 6. FIBRE BUNDLES
Reduction of Bundles
6.5.1
Contraction of the Base Space
• Theorem. If a base space X of a bundle E is contractible to a point then the
bundle E is trivial.
• Proof. Below.
• Let E and E � be two bundles. A map
ϕ : E� → E
is called a bundle map if it maps fibers of E � onto fibers of E.
• Two bundles E and E � with the same base space X, are equivalent if there
is a homeomorphism
ϕ : E� → E
that is a bundle map, that is, it maps fibers of E � onto fibers of E.
• Let E be a fiber bundle over X with a projection π : E → X and a fiber F.
Let
ϕ : X� → X
be a continuous map.
• This defines a new bundle E � = ϕ∗ E over X � called the pullback bundle
with the same typical fiber F as follows.
• The fiber of ϕ∗ E over a point p� ∈ X � is just the fiber of E over the point
ϕ(p� ) ∈ X.
• Thus ϕ∗ E is the disjoint union of all these fibers equipped with a suitable
topology.
• Define a subset E � of X � × E by
E � = ϕ∗ E = {(p� , u) ∈ X � × E | ϕ(p� ) = π(u)}
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• We define the projection onto the first factor
π� : ϕ∗ E → X �
by
π� (p� , u) = p� ,
p� ∈ X, u ∈ E
and the projection onto the second factor
ϕ∗ : ϕ∗ E → E
by
ϕ∗ (p� , u) = u,
p� ∈ X, u ∈ E
• Then the map ϕ∗ is a bundle map such that
ϕ ◦ π � = π ◦ ϕ∗
• If X = X � and ϕ = id X is the identity map then the bundles ϕ∗ E and E are
equivalent.
• Let {Uα } be the open cover of X and the transition functions for the bundle
E be gαβ (p), p ∈ Uα ∩ Uβ .
• The coordinate charts in X � are defined by
Uα� = ϕ−1 (Uα )
and the transition functions are
g�αβ (p� ) = gαβ (ϕ(p� )),
p� ∈ Uα� ∩ Uβ�
• Theorem. Let E be a fiber bundle over X with fiber F. Let ϕ0 , ϕ1 : X � → X
be homotopic maps. Then the bundles ϕ∗0 E and ϕ∗1 E are equivalent.
• Proof in textbook.
• Corollary. If the base space X of a fiber bundle E is contractible then the
bundle E is trivial.
• Proof. Since X is contractible the identity map ϕ0 = id X is homotopic to the
constant map ϕ1 : X → x0 .
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CHAPTER 6. FIBRE BUNDLES
• Then the pullback bundle ϕ∗1 E over a point x0 is trivial and the pullback ϕ∗0 E
is equivalent to E. Thus, E is trivial.
• Examples. A bundle E over a base space X = S 1 × I is equivalent to a
pullback bundle E � over S 1 .
• That is, there is a map
ϕ : S1 → S1 × I
such that the pullback bundle E � = ϕ∗ E over S 1 is equivalent to E over
S 1 × I.
6.5.2
Reduction of the Structure Group
• Theorem. If a fiber F of bundle E is contractible then it has a section s.
• Corollary. If the structure group G of a bundle E is contractible then the
bundle E is trivial.
• Example. The structure group of the frame bundle F(M) is GL(n, R).
• The group GL(n, R) is not contractible. It has the form
GL(n, R) = O(n) × Sym+ (n),
where O(n) is the orthogonal group and Sym+ (n) is the set of all positive
definite symmetric n × n matrices.
• Proposition. The set Sym+ (n) is homeomorphic to the vector space Sym(n)
of all symmetric matrices, and, therefore, contractible.
• Proof. The homeomorphism is given by the exponential map
exp : Sym(n) → Sym+ (n).
• Therefore, the non-compact group GL(n, R) is contractible to the compact
group O(n).
• The reduction of GL(n, R) to O(n) defines a continuous assignment of an
orthogonal frame at each point p ∈ X.
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• That is, the reduction GL(n, R) to O(n) means the existence of a Riemannian
metric for X.
• If the structure group GL(n, R) may be reduced to an even smaller subgroup
G of GL(n, R) then we say that X has a G-structure.
• Examples of G Structures. For even n there are two important examples:
• Almost Hamiltonian (Symplectic) Structure
G = S p(n, R),
n = 2m is even
where S p(n, R) is the symplectic group. This is a group of n × n real
matrices A that satisfy
AT JA = J,
where
J=
�
0
Im
−Im 0
�
• Almost Complex Structure.
G = GL(n/2, C),
n = 2m is even
• The reduction of the structure group GL(n, R) to O(n) means that the transition functions gαβ (p) now take values in O(n).
• This means that the transition functions for the tangent bundle and the cotangent bundle are the same, since for any orthogonal matrix A−1 = AT .
• Therefore, the bundles T M and T ∗ M are equivalent.
• More generally, let P(X, G) be a principal fiber bundle with the structure
group G.
• If G is a connected Lie group then
G = H × D,
where H is the maximal compact subgroup of G and D is a set which is
topologically a Euclidean space, and, therefore, contractible.
• Therefore, G may be reduced to its maximal compact subgroup H.
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• The resulting bundle P(X, H) is much smaller and simpler, but it is equivalent to P(X, G).
• Example. Let M be an n-dimensional complex manifold with the frame
bundle F M = GL(n, C).
• Then GL(n, C) has the maximal compact subgroup U(n)
GL(n, C) = U(n) × Herm+ (n),
where Herm+ (n) is the set of positive definite Hermitian matrices.
• The set Herm+ (n) is homeomorphic to the set Herm(n) of all Hermitian
matrices and, therefore, contractible,
exp : Herm(n) → Herm+ (n).
• Thus, the group GL(n, C) may be reduced to its maximal compact subgroup
U(n).
• The reduction of GL(n, R) to S p(n, R) and GL(n/2, C) is not always possible. There are some topological obstructions.
• The reduction of GL(n, R) to O(n − p, p) is not always possible.
• Example. The group GL(2, R) is reducible to O(1, 1) over a closed manifold
M only when M is either the torus T 2 or the Klein bottle K 2 .
• The sphere S 2 does not admit a Lorentzian metric.
• A closed manifold M admits a Lorentzian metric, that is, the structure group
GL(n, R) is reducible to O(n − 1, 1) if its Euler characteristic vanishes
χ(M) = 0.
• A non-orientable manifold M does not admit the oriented frame bundle, that
is, the group GL(n, R) cannot be reduced to S L(n, R).
• For a manifold to admit the spin bundle it has to be orientable and satisfy
one more topological condition.
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