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Math 396. Bundle Pullback and Transition Matrices 1. Motivation Let F : X → X Be a C P Mapping Between Cp Premanifolds with Co

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Math 396. Bundle Pullback and Transition Matrices 1. Motivation Let F : X → X Be a C P Mapping Between Cp Premanifolds with Co Math 396. Bundle pullback and transition matrices 1. Motivation p p Let f : X0 X be a C mapping between C premanifolds with corners (X; O) and (X0; O0), ! p p 0 p . Let π : E X and π0 : E0 X0 be C vector bundles. Consider a C bundle morphism≤ ≤ 1 ! ! T E0 / E π0 π X / X 0 f so for each x X we get R-linear maps T : E (x) E(f(x )). A basic example is the case 0 0 x0 0 0 when X, X , and2 f are of class Cp+1 and Tj = df : TX! TX is the induced total derivative 0 0 ! mapping that is the old tangent map df(x):Tx (X ) T (X) on fibers. 0 0 ! f(x0) For each x0 X0 we may use E and f to obtain a vector space E(f(x0)) determined by f and E, and it is natural2 to ask if these can be \glued" together to be the fibers of some Cp vector bundle f (E) X equipped with a bundle morphism f : f E E over f : X X that is ∗ ! 0 ∗ ! 0 ! the identity map (f E)(x ) = E(f(x )) E(f(x )) on fibers.e (Strictly speaking, the notation f is ∗ 0 0 ! 0 abusive since it depends on E and not just f; hopefully this will not cause confusion.) Such a paire (f ∗E X0; f) will be called a pullback bundle when it satisfies the universal property that for any p ! p C bundle morphisme T : E0 E over f : X0 X there is a unique C vector bundle morphism T : E f E over X giving! a factorization ! 0 0 ! ∗ 0 T ( E0 / f ∗E / E T 0 f π0 e π X X / X 0 0 f so T : E (x ) (f E)(x ) = E(f(x )) is exactly the map T . (The content is that the set- 0 x0 0 0 ∗ 0 0 x0 j ! p j theoretic mapping T 0 is a C mapping.) In other words, the pullback bundle should promote bundle morphisms T between bundles over different base spaces to bundle morphisms T 0 between bundles over a common base space. Note that it does not make sense to go in the reverse direction: if we are given a vector bundle on X0 then most points in X are either not hit by any point of X0 or are hit by more than one point (or perhaps infinitely many points) of X0, so there is no reasonable way to use f to associate vector spaces to points of X by means of a vector bundle on X0. Our first aim in this handout is to develop the \pullback" construction, and to give some examples. The second aim of this handout is to provide a very classical description of Cp vector bundles of constant rank in terms of what are called \transition matrices". This is used quite a lot in the more advanced study of vector bundles, and is conveniefnt for explicitly describing many \linear algebra" bundle constructions via operations on matrices. 2. Pullback of bundles p p Let f : X0 X be a C mapping between C premanifolds with corners, 0 p . Let ! p p ≤ ≤ 1 π : E X be a C vector bundle. We want to construct a C vector bundle f ∗(π): f ∗E X0 ! 1 ! 2 equipped with a Cp vector bundle morphism f f ∗E e / E f ∗(π) π X / X 0 f p that is universal in the following sense: for any C vector bundle π0 : E0 X0 and any bundle morphism T : E E over f : X X there is a unique way to fill in a commutative! diagram 0 ! 0 ! T ( E0 / f ∗E / E T 0 f π0 f ∗(π)e π X X / X 0 0 f p with T 0 a C bundle morphism over X0. We define f ∗E to be the disjoint union of sets f E = E(f(x ))): ∗ a 0 x X 02 0 A typical point in f E is denoted (x ; v) with v E(f(x )). There may be many points x X ∗ 0 2 0 0 2 0 with the same image x X, and the associated vector space E(f(x0)) sitting in f ∗E is abstractly 2 1 \the same" for all such x0 f − (x). For this reason, we must keep track of the indexing parameter x X and not just the2 \bare" vector space E(f(x )) when describing operations on points in 0 2 0 0 f ∗E. We define the map f ∗(π): f ∗E X0 by (x0; v) x0 X0. ! p 7! 2 p We need to topologize f ∗E and give it a C -structure such that (i) f ∗(π): f ∗E X0 is a C 1 ! map, (ii) the linear structure on the fibers (f ∗(π))− (x0) = E(f(x0)) satisfies the local triviality p condition to make f ∗E a C bundle over X0, and (iii) the proposed universal property holds. Our procedure will be very similar to the method used in an earlier handout to define the vector bundle p VM associated to a locally free finite-rank O-module M (by putting a suitable topology and C - structure on the disjoint union x X M (x)). Before we get into the details, we wish to emphasize ` 2 that the local picture will be quite simple: if Ui is an open covering of X for which there are f g p ni (i) C bundle isomorphisms E Ui Ui R corresponding to a trivializing frame sk in E(Ui), j 1 ' × fp g then for the opens U 0 = f − (Ui) that cover X0 the restriction (f ∗E) U has a C trivialization i j i0 given by the sections u s(i)(f(u )) E(f(u )) = (f E)(u ) to f (π): f E X over U X . i0 7! k i0 2 i0 ∗ i0 ∗ ∗ ! 0 i0 ⊆ 0 Roughly speaking, the \only" problem is to cleanly build the right bundle f ∗E over X0 giving 1 rise to such local trivializations on fibers over f − (Ui)'s. This construction problem can be solved in several ways (all of which give answers that are uniquely isomorphic in accordance with the universal property), and in what follows we have chosen the construction that seems most elegant (in terms of minimizing non-canonical choices and the intervention of matrices) in view of our present knowledge. To define the topology on f ∗E we will use the method of gluing topologies. Consider pairs n p (φ, U) where U X is a non-empty open set and φ : E U U R is a C isomorphism of ⊆ j ' ×1 bundles. (Of course, n may depend on U.) For the open set U 0 = f − (U) in X0 we get a bijection 1 n 1 φ0 :(f ∗(π))− (U 0) U 0 R over U 0 by using the linear isomorphism (f ∗(π))− (u0) = (f ∗E)(u0) = E(f(u )) Rn defined' × by φ over each u U . We wish to use φ to transfer the topology 0 ' jf(u0) 0 2 0 0 3 of the product U Rn to a topology on (f (π)) 1(U ), and to glue these topologies to topologize 0 × ∗ − 0 f ∗E. Note that as we vary (φ, U) the opens U do cover X and hence the open preimages U 0 do cover X0. To glue, as usual we have to verify two properties: for any (φ1;U1) and (φ2;U2) such that U1 U2 1 1 1 \ is non-empty, we must prove (i) the overlap (f ∗(π))− (U10 ) (f ∗(π))− (U20 ) = (f ∗(π))− (U10 U20 ) is 1 1 \ \ open in each of (f ∗(π))− (U10 ) and (f ∗(π))− (U20 ), and (ii) this overlap inherits the same subspace topology from both. (Note that if U1 U2 is empty then so is its preimage U U under f.) Since \ 10 \ 20 U1 meets U2, the constant ranks for E on U1 and U2 are equal, say n. The open subset property n (i) just says that φi0 carries the overlap to an open subset of Ui0 R , and indeed it carries the n n × overlap to the subset (U10 U20 ) R in Ui0 R that is obviously open. The agreement of subspace topologies in (ii) is equivalent\ to× the assertion× that the transition mapping 1 n n φ0 (φ0 )− :(U 0 U 0 ) R (U 0 U 0 ) R 2 ◦ 1 1 \ 2 × ! 1 \ 2 × 1 is a homeomorphism. Explicitly, the map is given by (u ; v) (u ; (φ2 φ− ) (v)) where 0 7! 0 ◦ 1 jf(u0) 1 n n φ2 φ1− :(U1 U2) R E U1 U2 (U1 U2) R ◦ \ × ' j \ ' \ × is the transition isomorphism over U1 U2 for E. For u U this latter map is given on u-fibers by \ 2 p 1 a matrix L(u) GLn(R) such that L : U GLn(R) is a C mapping (this encodes that φ2 φ1− is a Cp mapping).2 Thus, ! ◦ 1 (1) φ0 (φ0 )− :(u0; v) (u0; ((L f)(u0))(v)): 2 ◦ 1 7! ◦ Since L is a continuous mapping and f is continuous, so is L f. Thus, the formula for evaluation of ◦ 1 n a matrix on a vector implies (via (1)) that the self-map φ20 (φ10 )− of (U10 U20 ) R has continuous component functions and so is continuous; of course, the same◦ argument\ applies× to the inverse map (swap the roles of (φ1;U1) and (φ2;U2)), so we get the homeomorphism result.
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