Mathematics 132 Pset 5 Solutions
March 22, 2012
1. Solution to 2.3.12 by Paul VanKoughnett We prove that N (Z ; Y ) is a manifold by
exhibiting a parametrization. Choose independent functions g1 , . . . , gl in a neighborhood U
= cfw_(x1 , . . . , xM ) :
Mathematics 132 Pset 1 Solutions
February 15, 2012
1. Solution to 1.1.4 by Dmitri Gekhtman
(a) Let f : Ba Rk be the map given in the problem statement and dene the map
g : Rk Ba by
ay
g(y) =
.
2 + |y|2
a
Straightforward computation shows that f g = idRk a
The inverse function theorem
These notes first state and then prove the inverse function theorem.
Inverse function theorem: Let Bk denote the ball of radius 1 about the origin in Rk and
let : Bk Rk denote a smooth map that maps the origin to itself and wh
The construction of manifolds
This note gives a recipe of sorts for constructing a smooth manifold.
a) The definition of a smooth manifold
My purpose here is to remind you of the definition of smooth manifold of a given
dimension k cfw_1, 2, . To this end
The vector field theorem
The vector field theorem stated momentarily plays a central role in much of
differential topology. By way of notation, the theorem uses to denote the constant
vector field on R with length 1 that points from the origin along the p
The tangent bundle and tangent spaces
The purpose of these notes is to say more about the tangent bundle of a smooth
manifold.
1. Tangent spaces
Let X denote a given smooth, k-dimensional manifold. The text book assigns to
each point p X an abstract, k-di
Mathematics 132 Pset 2 Solutions
February 21, 2012
1. Solution to 1.4.2 by Yale Fan
(a) Suppose that X and Y are smooth manifolds such that X is compact and Y is connected.
We rst note that if f : X Y is a submersion and U is open in X , then f (U ) is op
Mathematics 132 Pset 4 Solutions
March 18, 2012
1. Solution to 2.1.5 by Paul VanKoughnett
The intersection of these two sets is equivalently the intersection of the unit sphere with
1 2z 2 a, or equivalently z 2 1a . For a 1, this is the whole sphere. For
Mathematics 132 Pset 2 Solutions
March 6, 2012
1. Solution to 1.5.7 by Amy Huang For any point x (g f )1 (W ) X , let y = f (x)
g 1 (W ) Y and z = g (y ) W Z . Furthermore, since g transversal to W , by problem 5
we have (dgy )1 (Tz (W ) = Ty (g 1 (W )
(
Mathematics 132 Pset 6 Solutions
March 22, 2012
1. Solution to 2.6.1 by Paul VanKoughnett Let f : S k S k carry antipodal points to
antipodal points. Composing with the inclusion S k Rk+1 gives a function f satisfying
the conditions of the Borsuk-Ulam The
Math 132 midsemester project instructions
What follows are the rules for the midsemester writing project.
THE RULES
Write a lecture that to teach your classmates about one of the three topics listed
momentarily. You can also write a lecture on a topic of