Mathematics 132 Pset 7 Solutions
April 7, 2012
1. Solution to 3.5.3 by Paul VanKoughnett If X = Rk cfw_0 where 0 is the zero of the
vector eld, then all tangent vectors to X also lie in Rk cfw_0, and in particular, the derivative
of v at 0 sends T0 X to T
Math 132
Problem Set 5
Spring, 2016
This problem set is due on Friday, Feb. 26th Please make your answers as complete and clear as
possible. You are allowed to discuss these problems with others in the class, but your writing should be your
own.
1. Suppos
Math 132
Problem Set 2
Spring, 2016
This problem set is due on Friday, Feb. 5. Please make your answers as complete and clear as possible.
You are allowed to discuss these problems with others in the class, but your writing should be your own.
1. (This is
Math 132
Problem Set 11
Spring, 2016
This problem set is due on Wednesday, April 27th (but will still be accepted on Wednesday, May
4th). Please make your answers as complete and clear as possible. You are allowed to discuss these problems
with others in
Math 132
Problem Set 1
Spring, 2016
This problem set is due on Friday, Jan. 29th, 2016. Please make your answers as complete and clear
as possible. You are allowed to discuss these problems with others in the class, but your writing should be
your own.
Th
Math 132
Problem Set 10
Spring, 2016
This problem set is due on Friday, April 8th Please make your answers as complete and clear as
possible. You are allowed to discuss these problems with others in the class, but your writing should be your
own.
1. Suppo
Math 132
Problem Set 3
Spring, 2016
This problem set is due on Friday, Feb. 12 Please make your answers as complete and clear as possible.
You are allowed to discuss these problems with others in the class, but your writing should be your own.
1. Using th
MATH 132
TWO LECTURES:
CLASSIFICATION OF ONE MANIFOLDS
PARTITIONS OF UNITY
MARCH 5, 2016
M. J. HOPKINS
1. The classification of 1-manifolds
The main result is the following.
Theorem 1.1. If M is a compact smooth 1-manifold, then M is diffeomorphic to
a di
MATH 132
BORDISM HOMOLOGY
MARCH 18TH, 2016
M. J. HOPKINS
1. Gluing
Lets begin with something that we have used frequently. Suppose M = U V
is a manifold, written as a union of two open sets U and V , and N is another
manifold. A function f : M N is smooth
MATH 132
DEFINITION OF SMOOTH MANIFOLD
FEB. 19, 2016
M. J. HOPKINS
1. The Definition of a manifold
Differential topology is the study of a class of spaces on which one can do calculus.
1.1. Topological manifolds.
Definition 1.1. A topological manifold of
MATH 132
BORDISM HOMOLOGY AND SURFACES
SPRING 2016
M. J. HOPKINS
Contents
1. Gluing
2. Cobordism
3. Manifolds over a space
4. Surgery
5. A moving lemma
6. Computations
7. The neighborhood retract theorem
8. Bilinear forms
9. Special position
10. Separatio
MATH 132
Spring 2016
MWF 1:00 - 2:00
Administrative
Instructor:
Office:
email:
Office Hours:
Michael Hopkins
Science Center 508
mjh@math.harvard.edu
Wednesdays, 2:00-3:00 and by appointment
Course Assistant: Adam Al-Natsheh
email: aalnatsheh@college.harva
Math 132
Problem Set 4
Spring, 2016
This problem set is due on Friday, Feb. 19 Please make your answers as complete and clear as possible.
You are allowed to discuss these problems with others in the class, but your writing should be your own.
All the pro
Math 132
Problem Set 7
Spring, 2016
This problem set is due on Friday, Mar. 11th Please make your answers as complete and clear as
possible. You are allowed to discuss these problems with others in the class, but your writing should be your
own.
1. The fo
Math 132
Problem Set 9
Spring, 2016
This problem set is due on Friday, April 1st Please make your answers as complete and clear as
possible. You are allowed to discuss these problems with others in the class, but your writing should be your
own.
Ive updat
Mathematics 132 Pset 9 Solutions
April 25, 2012
1. Solution to 4.2.4 by Paul VanKoughnett.
This is an obvious formal consequence of exercises 1 and 3.
To further elaborate, let T be an alternating p-form and v1 , . . . , vp linearly dependent vectors.
Say
Mathematics 132 Pset 10 Solutions
April 27, 2012
1. Solution to 4.4.10 by Paul VanKoughnett
The easiest way to do this is to use problem 12. Let be a 1-form on S 1 whose integral
around S 1 is zero. Any closed curve in S 1 is a continuous map S 1 S 1 , an
Mathematics 132 Pset 11 Solutions
April 27, 2012
1. Solution to 4.7.1 by Paul VanKoughnett
A smooth 0-form is just a function, so consider a function f : [a, b] R. Stokes theorem
says that
df.
f=
[a,b]
[a,b]
Of course, the boundary of [a, b] is just the
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 7 Solutions
April 7, 2012
1. Solution to 3.2.26 by Paul VanKoughnett Let X be a simply connected k -manifold,
and choose an orientation for some Tx (X ). As described in the text, for any y X and
path from X to y , cover with open Euc
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
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 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 ) :
MATH 132: SMOOTH MANIFOLDS
Math 132: The goal of this course is to introduce the fundamental notions and tools that
are used to study questions about topological spaces with differentiable functions.
My contact information:
Cliff Taubes
Science Center roo
Math 132
Problem Set 6
Spring, 2016
This problem set is due on Friday, Mar. 4th Please make your answers as complete and clear as
possible. You are allowed to discuss these problems with others in the class, but your writing should be your
own.
1. Work GP
Math 132
Problem Set 8
Spring, 2016
This problem set is due on Friday, Mar. 25th Please make your answers as complete and clear as
possible. You are allowed to discuss these problems with others in the class, but your writing should be your
own.
Work all