**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 R*P*^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 R*P*^1 in R*P*^2,
it can be proved by dissecting *P*^2 - a disc and reassembling
it into a Möbius strip.

2. The canonical bundle over R*P*^*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 R*P*^*n* via a continuous map of the base spaces
*f*: *X* --> R*P*^*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 C*P*^*n*
classifies all complex line bundles over complexes of
dimension < = 2*n*.

*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/D*x*_*i*
and *f*_*(*v*) = \sum_*j* *b*_*j*
D/D*y*_*j* are expressed in terms of
the bases defined by coordinate charts near *x* and *f*(*x*)
respectively, then *b*_*j* =
\sum D*f*_j/D*x*_*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*:

SoTM----->TN|f_* | pi | | pi' VfVM----->N

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*_*(*T*R^*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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