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164 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)} topicsdiffgeom.tex; November 24, 2014; 14:38; p. 161 6.5. REDUCTION OF BUNDLES 165 • 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 . topicsdiffgeom.tex; November 24, 2014; 14:38; p. 162 166 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. topicsdiffgeom.tex; November 24, 2014; 14:38; p. 163 6.5. REDUCTION OF BUNDLES 167 • 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. topicsdiffgeom.tex; November 24, 2014; 14:38; p. 164 168 CHAPTER 6. FIBRE BUNDLES • 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. topicsdiffgeom.tex; November 24, 2014; 14:38; p. 165