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CHAPTER 1
Vector Bundles
1.1 Introduction
The purpose of this chapter is to develop the basic material about vector bundles and differential operators. We actually develop more material than we will need in this book. If a reader is already fluent with the formalism of differential operators on vector bundles, then he should by all means skip this chapter, using it only as a dictionary of notation. On the other hand, if the reader is not very fluent with the concept of vector bundles, then there is enough material in this chapter to give the "feel" of the subject. We also suggest the exercises at the end of this chapter to anyone wishing to "brush up" on their understanding of vector bundles and differential operators.
1.2 Preliminary Concepts
1.2.1Definition Let X be a topological space. Let K denote either R or C (the real or complex numbers). A K-vector bundle over X is a topological space E and a continuous map p of E to X satisfying:
(1) If x is in X, then Ex = p-1(x) is a K-vector space.
(2) For each x in X there is a neighborhood U of x and a homeomorphism F of E|U = p-1(U) with U × Kn such that F(v) = (p(v), f (v)) and f is a linear isomorphism of the K-vector space Ep(v) onto Kn.
1.2.2Examples (1) Let E = X × Kn with p(x, v) = x.
(2) Let M be a differentiable manifold. A.1.2.6 says that T(M) is a vector bundle over M.
1.2.3Definition Let E and F be vector bundles over X. A homomorphism of the vector bundles E and F is a continuous map h of E to F so that h(Ex) is contained in Fx and [MATHEMATICAL EXPRESSION OMITTED] is a linear map from Ex to Fx for each x in X. A vector bundle homomorphism is called an isomorphism if it is a homeomorphism.
1.2.4 Let (E, p) be a vector bundle over X. Let U be an open covering of X so that for each U in U there is a map FU of E|U onto U × Kn satisfying 1.2.1(2). Such a covering is called a trivializing covering for E.
Consider for each U, V in U the map FV [??] F-1U of U [intersection] V × Kn onto U [intersection] V × Kn. Then for each y in X, v in Kn,
FV [??] F-1U (y, v) = (y, gV,U(y)v)
with gU,V (y) in GL(n, K) (see A.2.1.2). Furthermore, since FU and FV are homeomorphisms, the map gV,U of U [intersection] V into GL(n, K) is continuous. We also note that if y is in U [intersection] V [intersection] W then
gW,V (y) · gV,U(y) = gW,U(y). (1)
Equation (1) is called the cocycle condition. gU,V is called the cocycle with values in GL(n, K) associated with the trivialization (U, FU).
We now abstract this concept.
1.2.5Definition Let X be a topological space and let U be an open covering of X. Let g be an assignment of a continuousmap gU,V of U [intersection] V into GL(n, K) for each pair U, V of elements of U, so that the cocycle condition 1.2.4(1) is satisfied. Then g is called a U-cocycle with values in GL(n, K).
1.2.6Theorem Let U be an open covering of the topological space X and let g be a U-cocycle with values in GL(n, K). Then there is a unique (up to isomorphism) vector bundle E over X so that U is a trivializing covering for E and g is a corresponding cocycle.
Proof. Let Y = X × Kn × U where U is given the discrete topology and Y is given the product topology. Let T = {(x, v, U) | U [member of] U, x [member of] U, v [member of] Kn}. Give T the subspace topology in Y. We define an equivalence relation on T as follows: (x, v, U) [equivalent to] (y, w, V) if x = y and w = gV,U(x)v. We note that the cocycle condition is just the statement that [equivalent to] is an equivalence relation. Let E be the set of all equivalence classes in T. Let, for (x, v, U) in T, q(x, v, U) be the equivalence class of (x, v, U). Give E the topology that makes q continuous and open (this is called the quotient topology). Let [??](x, v, U) = x. Then [??] defines a continuous map of T to X. Since [??](x, v, U) depends only on q(x, v, U), [??] induces a continuous map p of E onto X.
Let x be in X and let v, w be in p-1(x). Fix U in U so that x is in X. Then v = q(x, v', U) and w = q(x, w', U). If a, b are in K we set av + bw = q(x, av' + bw', U); the cocycle condition guarantees that av + bw is defined. Thus with this vector space structure on p-1(x), E satisfies 1.2.1(1).
By definition of E the map F-1 U of U × Kn to E defined by F-1U (x, v) = q(x, v, U) is a homeomorphismof U × Kn with p-1(U). Furthermore the vector space operations on p-1(x) are defined so that FU satisfies 1.2.1(2). This proves the existence of E.
Suppose that E' is a vector bundle over X and that if U [member of] U there are mappings F'U of E'U to U × Kn satisfying 1.2.1(2) and such that the corresponding cocycle is g.We define a map h from T to E'
h(x, v, U) = F'-1 U (x, v).
Then by the definition of the cocycle associated with a trivialization, h respects equivalence classes and thus induces a continuous map h of E to E'. It is clear that h is a vector bundle homomorphism. Defining for v in E'|U, t (v) = q(FU(v), U). Then t is well defined and is a vector bundle homomorphism. Since t defines a right and a left inverse for h, h is a vector bundle isomorphism.
1.2.7Proposition Let g and h be U-cocycles with values in GL(n, K). Let E1 and E2 be, respectively, the corresponding vector bundles over X. Then E1 and E2 are isomorphic if and only if for each U in U there is a continuous map sU of U into GL(n, K) so that
gU,V(x) = sU (x)hU,V (x)sV (x)-1 (1)
for x in U [intersection] V.
Proof. Suppose that (1) is satisfied. Let T be as in the proof of Theorem 1.2.6. Let [MATHEMATICAL EXPRESSION OMITTED], be the equivalence relations on T corresponding to g and h, respectively. Define a map [??] of T to T by [??](x, v, U) = (x, sU (x)v, U). Then [??] is continuous. Suppose that [MATHEMATICAL EXPRESSION OMITTED]. We assert that this implies that [MATHEMATICAL EXPRESSION OMITTED]. Indeed, w = hV,U (x)v. Now
[MATHEMATICAL EXPRESSION OMITTED]
This proves the assertion. [??] therefore induces a continuous map ρ of E1 to E1, which is easily seen to be a vector bundle isomorphism.
Suppose that E1 and E2 are isomorphic. Let ρ be a corresponding isomorphism. If for each U in U, F1U and F2U are the maps satisfying 1.2.1(2) so that g and h are the corresponding cocycles, then defining, for x in U, sU (x) by F1U (ρ(F1-1U (x, v))) = (x, sU (x)v) it is easy to see that (1) is satisfied.
1.2.8 Let g and h be U-cocycleswith values in GL(n, K). Then g and h are said to be equivalent if there is for each U in U a map sU of U into GL(n, K) which is continuous and is such that g and h satisfy 1.2.7(1). The set of all equivalence classes of U-cocycles with values in GL(n, K) is denoted by H1(U; GL(n, K)). 1.2.6 and 1.2.7 combine to prove that there is a bijection between H1(U; GL(n, K)) and the isomorphism classes of K-vector bundles over X of fiber dimension n having U as a trivializing covering.
1.3 Operations on Vector Bundles
1.3.1 Let X be a topological space and let E be a vector bundle over X. Let Y be a topological space and suppose that f is a continuous mapping of Y into X. Let E × Y be given the product topology. Set
f* E = {(v, y) | v in Ef(y)}.
Give f* E the subspace topology. Taking the fiber over y to be Ef(y) × {y} = f* Ey and the natural vector space structure on f* Ey, it is easy to see that f* E is a vector bundle over Y. f* E is called the pull-back of E relative to f.
1.3.2 Let E1 and E2 be vector bundles over X. We define a vector bundle over X × X by giving E1 × E2 the product topology and the obvious projection. Let diag(x) = (x, x). Then diag maps X into X × X. We set E1 [direct sum] E2 = diag*(E1 × E2). E1 [direct sum] E2 is called the Whitney sum of E1 and E2.
1.3.3 Let E1, E2, and E3 be vector bundles over X. A bilinear map of E1 [direct sum] E2 to E3 is defined to be a continuous map that maps fibers to fibers, that is bilinear on the fibers (here we make the natural identification of (E1 [direct sum] E2)x with [MATHEMATICAL EXPRESSION OMITTED]). A tensor product of E1 with E2 is a pair (V, j) of a vector bundle V over X and a bilinear map j of E1 [direct sum] E2 into V so that if h is a bilinear map of E1 [direct sum] E2 into E3 then there exists a unique vector bundle homomorphism [??] of V to E3 so that [MATHEMATICAL EXPRESSION OMITTED]. As in the case of the tensor product of vector spaces (see A.3.1.5), if a tensor product exists it is unique up to isomorphism. We now show that a tensor product of E1 with E2 exists. Let U be a trivializing covering for E1 and E2. Let g and h be corresponding U-cocycles with values in GL(n, K) and GL(m, K), respectively. We assume that E1 and E2 have been constructed from g and h as in the proof of Theorem 1.2.6. Let, for U, W in U and x in U [intersection] W,
kU,W(x) = gU,W(x) [cross product] hU,W(x).
Then k defines a U-cocycle with values in GL(nm, K). Let V be the corresponding vector bundle over X. Let j be the map of E1 [cross product] E2 to V induced by
((x, v, U), (x, w, U)) [??] (x, v [cross product] w, U).
We leave it to the reader to check that j is well defined and bilinear and that (V, j) defines a tensor product of E1 with E2.
We use the notation E1 [cross product] E2 for the tensor product of E1 with E2. The definition of the tensor product implies that E1 [cross product] (E2 [cross product] E3) is isomorphic with (E1 [cross product] E2) [cross product] E3.
1.3.4 We leave it to the reader to formulate the universal problem to define {LAMBDA]r E, r = 0, 1, ..., fiber dimension E (see A.3.1.16).
1.3.5 Let E be a real vector bundle over X. Let C be the trivial vector bundle X × C. We note that the fiber of the bundle E [cross product] C at x is Ex [cross product] C. Thus E [cross product] C is a complex vector bundle. E [cross product] C is called the complexification of E and denoted E [cross product] C or EC.
1.3.6 Let E be a vector bundle over X. Let U be a trivializing covering for E and let g be a corresponding cocycle. Let, for U, V in U and x in U [intersection] V,
g*U,V (x) = tgU,V(x)-1
where tA denotes the transpose of the linear map A (that is, if z* is an element of (Kn)* and if z is an element of Kn then tAz*(z) = z*(Az)). Then g * is a U-cocycle with values in GL(n, K). Let E* denote the corresponding vector bundle over X. We note that up to isomorphism E* is independent of all choices used to define it. Also if x is in X then E*x can be identified with (Ex)*. E and E* are called dual vector bundles.
Using the various universal properties we have
E* [cross product] F* = (E [cross product] F)* Λr E* = (Λr E)*.
1.3.7 If E and F are vector bundles over X, let Hom(E, F) denote the vector bundle E* [cross product] F. Then there is a natural identification of (E* [cross product] F)x with E*x [cross product] Fx which is naturally identified with L(Ex, Fx) (the space of all K-linear maps from Ex to Fx). Thus Hom(E, F) is the vector bundle over X with fiber over x the space of all linear maps of Ex to Fx.
1.3.8 Let M be a differentiable manifold (resp. complex manifold). Let E be a K-(resp. complex) vector bundle over M. Then E is called a C∞ (resp. holomorphic) vector bundle if E has the structure of a C∞ (resp. complex) manifold, the projection is C∞ (resp. holomorphic), and there is a trivializing covering for E which has maps satisfying 1.2.1(2) that are C∞ (resp. holomorphic).
The basic example of a C∞ vector bundle is the tangent bundle of a C∞ manifold. A basic example of a holomorphic vector bundle is the holomorphic tangent bundle of a complex manifold. (See Appendices 1 and 2 for pertinent definitions.)
1.3.9Definition Let M be a C∞ manifold (resp. complex manifold). Let U be an open covering of M. Then a C∞ U-cocycle (resp. holomorphic U-cocycle) with values in GL(n, K) (resp. K = C) is a U-cocycle with values in GL(n, K) so that if U, V are in then gU,V is a C∞ (resp. holomorphic) map of U [intersection] V into GL(n, K).
1.3.10Theorem Let E be a vector bundle over M. Then E is a C∞ (resp. holomorphic) vector bundle over M if and only if there is a trivializing covering for E so that a corresponding cocycle is C∞ (resp. holomorphic).
Proof. The necessity is clear. Let U be an open covering of M, which is a trivializing covering for M and so that if, for each U in U, FU is a map satisfying 1.2.1(2), then the corresponding cocycle is C∞ (resp. holomorphic). We now show how to make E into a C∞ (resp. holomorphic) vector bundle. We first note that by possibly refining the covering U we may assume that for each U in U there is a C∞ (resp. holomorphic) mapping fU of U into Rn (resp. Cn) so that (U, fU) is a chart for M. We define an atlas for E by defining GU = (fU × 1) [??] FU, mapping E|U to Rm × Kn (resp. Cm × Cn). Here (fU × 1)(x, v) = (fU (x), v). The cocycle condition guarantees that GV [??] G-1U is C∞ (resp. holomorphic) on GU (E|U [intersection] E|V).
(Continues…)
Excerpted from "Harmonic Analysis on Homogeneous Spaces"
by .
Copyright © 2018 Nolan R. Wallach.
Excerpted by permission of Dover Publications, Inc..
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