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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