MAT 531 (Spring 1996) Topology/Geometry II

Week 4
1. Reconstructing a bundle from the transition functions. First note that the way g_ab is defined in terms of H_a o H_b^(-1) the set of transition functions automatically satisfies the "cocycle condition" g_ab x g_bc = g_ac (the multiplication in Gl(n,R).
Proposition Given X, a covering {U_a} and a set of maps g_ab : U_a intersect U_b --> Gl(n,R) satisfying the cocycle condition on every non-empty triple intersection, then there exists an n-dimensional coordinate vector bundle pi : E --> X with local trivializations H_a : pi^(-1)U_a --> U_a x R^n related by the g_ab; any two such bundles are equivalent.
Proof The total space E is defined as {disjoint union U_a x R^n} modulo the following equivalence relation (x, t_a) ~ (x,t_b) <=> t_a = g_ab(x)t_b. The cocycle condition (which clearly implies both g_aa = I and g_ab = g_ba^(-1) ) ensures that this is in fact an equivalence relation. E gets the quotient topology. The map pi is defined by sending [(x, t_a)] ([]="equivalence class of") to x, clearly well-defined. The local trivialization H_a is defined by sending [(x,t_a)] to (x,t_a), for x in U_a. This can be checked to be a homeomorphism. Clearly the change of coordinates is (x,t_b) <--> (x,t_a) if and only if t_a = g_ab(x)t_b.
Suppose two coordinate bundles have the same base X, and have the same transition functions with respect to the covering {U_a}. If E has local trivializations {H_a} and F has local trivializations {K_a}, define Phi_a: pi_E^(-1)U_a --> pi_F^(-1)U_a by Phi_a(v) = K_a^(-1) o H_a(v); i.e if Phi_a(v) = w, then H_a(v) = K_a(w) = (x,t), say. Now if If pi_E(v) is also in U_b, then H_b(v) = (x, g_ba (x)t) = K_b(w), since the transition functions are the same. So Phi_a does not in fact depend on a, and extends to an isomprphism Phi : E --> F.
So in particular if we start with a coordinate vector bundle, and throw away everything except the base, the covering and the transition functions, then the reconstruction gives a bundle isomorphic to the original one.

Examples The Möbius strip as a 1-dimensional bundle over the circle S^1. We take the covering by two overlapping intervals U_0 and U_1. They intersect in two intervals A and B; we define g_01 : U_0 intersect U_1 --> GL(1,R) by g_01(x) = 1 if x in A, -1 if x in B. This gives a non-trivial bundle (Class exercise: prove that the Möbius Strip is not homeomorphic the the cylinder S^1 x R).
This simple bundle is very important; whenever students learn a new fact about bundles, they should check it on this one (and on the tangent bundle to S^2). The Möbius strip is one example of a useful family of bundles: the canonical line bundles over projective spaces (both real and complex). They are defined as follows. The base is projective space P^n; the covering is the canonical one: U_i = {[x_0: ... :x_n] with x_i not 0}; the transition function g_ij : U_i intersect U_j --> K* (here K = R or C, and the * means non-zero, so K* = Gl(1,K)) is defined by g_ij([x_0: ... :x_n]) = x_i/x_j. These functions are well-defined, non-zero and satisfy the cocycle condition, patently.
As an example, we check that the canonical bundle over RP^1 = S^1 is in fact the Möbius strip, although with a different transition function from the one we used earlier.

The projective space P^(n-1) is embedded in P^n by the embedding of K^n in K^(n+1). A useful fact about the canonical bundles is that if a point is removed from P^n, what is left is the total space of the canonical bundle over P^(n-1). This is not hard to prove in general; in the case of RP^1 in RP^2, it can be proved by dissecting P^2 - a disc and reassembling it into a Möbius strip.

2. The canonical bundle over RP^n is an example of a classifying bundle (definition later); in this case, "classifying" means that every 1-dimensional vector bundle E --> X over a complex X of dimension < = n-1 can be pulled back (or, induced) (definition later) from the canonical bundle over RP^n via a continuous map of the base spaces f: X --> RP^n, and that homotopy classes of maps are in 1-1 correspondance with isomorphism classes of bundles. This means in particular that any functorial question about 1-dimensional bundles over complexes of dimension < = n-1 can be settled by examining this one particular bundle.
Similarly the canonical complex line bundle over CP^n classifies all complex line bundles over complexes of dimension < = 2n.

The differential of a smooth map between manifolds. If f: M --> N is smooth, and v is a tangent vector in TM_x, then f maps v to the vector f_*(v) in TN_f(x) defined by taking v = c'(0) for a curve c: R --> M with c(0) = x, and setting f_*(v) = (f o c)'(0). This vector does not depend on the choice of c in the equivalence class defining v. The assignment v --> f_*(v) is linear; in fact an application of the chain rule shows that if v = \sum_i a_i D/Dx_i and f_*(v) = \sum_j b_j D/Dy_j are expressed in terms of the bases defined by coordinate charts near x and f(x) respectively, then b_j = \sum Df_j/Dx_i a_i. This formula also shows that f_* gives a smooth map T (U_a intersect f^(-1)V_b) --> TV_b for any coordinate neighborhoods in M and N respectively, and therefore from TM to TN. This map commutes with f and the projections TM --> M and TN --> N:

              TM  ----->  TN
               |    f_*    |
            pi |           | pi'
               V     f     V
               M  ----->   N
So f_* maps fibres to fibres, and is linear on each fibre.

Write dim M = m, dim N = n. Maps where the rank of f_* at each point is the maximum possible (i.e. min(m,n)) are of special interest topologically. They are called immersions if m < = n, and submersions if m > = n. The maps of maximum rank in the overlap (m=n) are local diffeomorphisms, by the Inverse Function Theorem.

The inverse function theorem serves also as the central ingredient in describing the local topological structure of immersions and submersions, as follows.
Proposition. If f: R^m --> R^n is smooth, takes 0 to 0 and satisfies rank(f_*)(0) = m, then there exist coordinates {x_i} near 0 in R^m, {y_j} near 0 in R^n such that f(x_1,...x_m) = (x_1,...x_m,0,...,0). [I.e. in those coordinates the map is the standard inclusion of R^m in R^n.]
Proposition. If f: R^m --> R^n is smooth, takes 0 to 0 and satisfies rank(f_*)(0) = n, then there exist coordinates {x_i} near 0 in R^m, {y_j} near 0 in R^n such that f(x_1,...x_m) = (x_1,...x_n). [I.e. in those coordinates the map is the standard projection from R^m to R^n.]
Proofs. a. In the first case, m < = n. First rotate coordinates in R^n so that f_*(TR^m_0) is spanned by the tangent vectors to the first m axes. Then define F:R^m x R^(n-m) --> R^n by F(x,z) = f(x) + (0,...,0,z). F_* has rank n at (0,0), so F maps a neighborhood of (0,0) diffeomorphically onto a neighborhood of 0 in R^n. Use this diffeomorphism to define the required coordinates in R^n.
b. In the second case, m > = n. First renumber the coordinates in R^m so that the matrix of partial derivatives (at zero) of f with respect to the first n coordinates in nonsingular. Then define F: R^m --> R^n x R^(m-n) by F(x_1,...,x_m) =(f(x_1,...,x_m),(x_(n+1),..., x_m)). Since F_* has rank m at 0, it maps a neighborhood of 0 diffeomorphically onto a neighborhood of (0,0). Use the inverse of this diffeomorphism to define the required coordinates in R^m.

A subset P of a smooth m-manifold M is called a smooth p-dimensional submanifold if each point of P lies in a neighborhood U in M with a diffeomorphism h: U --> R^n such that h(U intersect P) = h(U) intersect R^p, i.e = {x in h(U) : x_(p+1) = ...= x_m = 0}.
Proposition If f: R^m --> R^n is smooth, and if there is a point y in R^n such that rank(f_*) = n at each point in f^(-1)(y), then f^(-1)(y) is a smooth submanifold of dimension m-n.

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