OFFICE HOURS: Monday and Wednesday 3pm (or by appointment.)
TEXTBOOKS:
"Algebraic Topology" by Allen Hatcher
This book is available for downloading
here
or you can actually buy a (cheap) bound version as well.
We will cover Chapters 0, 1 and 2 and possibly 3.
PREREQUISITES: Familiarity with abstract algebra (Math 330) and
point-set topology (Math 445).
Quotient spaces will be particularly important;
almost any introductory book on point-set topology will cover this
concept though.
GRADING:
Homework will be assigned weekly. Only a few of these problems will
be assigned to turn in for a grade
(the total will be rescaled to 100 points total).
There will also be
one "special project" assignment during the semester (100 points)
and one final project (200 points).
EXAMS: The take home projects replace midterm and final exams.
You will be allowed to use your textbooks and class notes, but
you should not discuss the exam ("project") problems with anyone
or use other sources for your solutions.
HOMEWORK ASSIGNMENTS: You are responsible for reading the text, preferably
before the material is covered in class. You are encouraged to work together
on all homework assignments, but write up your solutions separately and
credit your collaborators explicitly. You are also encouraged to work at
least a few of the problems by yourself.
Homework sets will be due approximately every two weeks, on Fridays.
Of course, in an ideal world you would do all of the exercises in Hatcher...
Week 1 (not for turning in): Hatcher, Chapter 0, pp. 18 - 20:
3, 14, 16, 23, 27 (Some other good problems too: 2, 4, 9, 10, 11, 15)
Week 4 (9/12 - 9/16, not for turning in):
Chapter 1.2: 2, 4, 7, 8, 9
Second Written Assignment: Due September 16, 1pm:
Chapter 1.1: 5, 16, 17, 20 (no more!)
Week 5 (9/19 - 9/23, not for turning in):
Chapter 1.3: 1, 2, 4, 8, 9, 10, 12, 17, 20 (this is an important section,
these may go into Week 6 as well...)
Week 6 (9/26 - 9/30, not for turning in):
Chapter 1.3: finish list above
Third Written Assignment: Due September 30, 1pm:
Chapter 1.2: 4, 8; Chapter 1.3: 4, 9, 10
Week 7 (10/3 - 10/7, not for turning in):
Chapter 2.1: 2, 3, 4, 5, 8, 9
Week 8 (10/10 - 10/14, not for turning in):
Chapter 2.1: 11, 12, 15, 16
Fourth Written Assignment: Due October 14, 1pm:
Chapter 1.3: 12, 15; 2.1: 4, 8, 9 (*** Get these done early so you
can have more time for the midterm! ***)
MIDTERM SPECIAL PROJECT (25% of total grade): Due October 21, 1pm:
You are allowed to use Hatcher's book and our class notes, but
you should not discuss these "speical project" problems with anyone
or use other sources for your solutions.
A standard extension of the due date was discussed in class, it applies to
anyone who wants the extra time.
1.
(a) Describe all connected covering spaces of RP^2 up to equivalence. (RP^2 is the real projective plane.)
(b) Describe all connected covering spaces of RP^2 x RP^2 (the product of two real projective planes) up to equivalence.
2.
Let X be the quotient space of the cylinder (S^1 x I) that is obtained by
identifying antipodal points on the top circle (S^1 x {1}) and identifying
points on the bottom circle (S^1 x {0}) which are 120 degrees apart. Find the fundamental group of X.
3. Section 2.1, #5 from Hatcher.
4. Section 2.1, #14 from Hatcher.
Week 9 (10/17 - 10/21, not for turning in):
Chapter 2.1: 20, 22, 27, 29, 30, 31
Week 10 (10/24 - 10/28, not for turning in):
Chapter 2.2: 4, 9, 10, 11, 12, 14
Week 11 (10/31 - 11/4, not for turning in):
Chapter 2.2: 17, 19, 21, 27, 28
Fifth Written Assignment: Due November 4, 1pm:
Section 2.1: 17, 29; Section 2.2: 9a, b, 12
Week 12 (11/7 - 11/11, not for turning in):
Chapter 2.2: 30, 31, 34, 36, 40, 43
Week 13 (11/14 - 11/18, not for turning in):
Chapter 3.1: 6, 8, 9, 10, 11
Sixth Written Assignment: Due November 18, 1pm:
Chapter 2.2: 28, 36; Chapter 3.1: 10, 11
One Extra Problem: a) Show that the kernel of multiplication by m on Z/n is generated by
{n/d}, where d = gcd (m, n).
(b) Show that a quotient of a cyclic group is cyclic.
(c) Show that there is an exact sequence (where the middle map is multiplication
by m):
0 \to Z/d \to Z/n \to Z/n \to Z/d \to 0 where d = gcd(m, n).
FINAL SPECIAL PROJECT (50% of total grade): Due December 6, 5pm (NOTE NEW EXTENSION TO 5pm!):
You are allowed to use Hatcher's book and our class notes, but
you should not discuss these "speical project" problems with anyone
or use other sources for your solutions.
The final problems are
here.
Comments on exam: In number 3, "integral homology" means homology with integer coefficients.
In number 4, "differential" means "boundary map."