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 , . .
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|
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 de
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
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
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. Th
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
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 equiv
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 ,
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 func
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