Name:
M382D Midterm Exam Solutions
March 12, 2004
Problem 1: a) Suppose that X is an n-dimensional manifold, x is a point on X , and
1
n
f 1 , . . . , f n1 : X I are smooth functions such that the dierentials dfx , . . . , dfx 1 are
R
linearly independent
Dierential Topology (M382D)
Homework 11. Due April 23
Mondays lecture will be given by Dan Freed. Wednesdays class is canceled work
on your termpapers! If we dont get far enough on Monday to do all the section 4.4
problems, save the ones you cant do for n
Dierential Topology (M382D)
Homework 10. Due April 16
We have a short week, since class was cancelled on Friday. I wish they would make
those Good Friday announcements AHEAD OF TIME!
I apologize for the error in problem 3.5.1. The ow should be ht (z ) = e
Dierential Topology (M382D)
Homework 8. Due April 2
Problems in Guillemin/Pollack
Chapter 3, 2 (p. 103): 2, 7, 11, 16, 17
Note: On problem 2.7, you cannot choose the BASIS continuously on all of S 2 , but
you can choose the ORIENTATION continuously. Pick
Dierential Topology (M382D)
Homework 7. Due March 26
Problems in Guillemin/Pollack
Chapter 2, 4 (p. 82): 3, 7, 9, 11, 12, 18, 19
Computing intesection numbers: Let T be the torus obtained by taking the unit
square [0, 1] [0, 1] and identifying opposite si
Dierential Topology (M382D)
Homework 6: Due March 5
All the graded problems this week are from G&P. There are several Challenge Problems, which will not count towards your grade, at the end. See the diatribe at the end of
last weeks homework.
Problems in
Dierential Topology (M382D)
Homework 6: Due February 27
All the graded problems this week are from G&P. There are several Challenge Problems, which will not count towards your grade, at the end. See the diatribe at the end of
the page.
Problems in Guillem
Dierential Topology (M382D)
Homework 4: Due February 20
This problem set contains a lot of problems. Its OK if you dont have time to do them
all, but do as many as possible. Its only by getting your hands dirty that the theorems
weve been proving in class
Dierential Topology (M382D)
Homework 3: Due February 13
Problems in Guillemin/Pollack
Chapter 1, 2 (p. 12): 5, 7, 9, 11, 3 (p.18) 5, 7, 9, 4 (p 25) 2, 5, 6.
Problem 1. Abstract manifolds. An abstract (smooth) k-manifold is a topological Hausdor
space X su
Dierential Topology (M382D)
Homework 2: Due February 6
Problems in Guillemin/Pollack
Chapter 1, 1 (p. 6): 6, 9, 11, 13, 16, 17, 18
Problem 1. Suppose that X is a manifold and U X an open subset. Show that
U inherits the structure of a manifold.
Problem 2.
Dierential Topology
Homework 1: Due January 30
There will be weekly homework assignments due each Friday at the beginning of class.
Please work the problems neatly and staple your pages together. There is no need to copy
over the problem or hand in the pr
M382D Second Exam
May 7, 2004
This exam consists of ve problems. Do any THREE. Make it VERY clear which three you
are attempting, or I may have to pick which problems to grade at random. All answers need to be
justied. Does there exist . really means Prov
Dierential Topology (M382D)
Homework 11. Due April 30
The rst 5 problems are reruns from last week. If you didnt do them then, do them
now!
Problems in Guillemin/Pollack
Chapter 4, 4 (p. 171): 1, 3, 6, 8, 9
Chapter 4, 5 (p. 178): 1, 2 Chapter 4, 7 (p. 185