OFFICE HOURS: MW 3pm -4pm ; Friday 10-11am in SEO 508e
(If I'm late for the MW 3pm you might find me in SEO 341, DGS office.)
Also, please feel free to ask me questions before, during
or after class or make an appointment
by email.
OFFICIAL TEXTBOOKS:
1: "Algebraic Topology" by Allen Hatcher
This book is available for downloading
here
or you can actually buy a (cheap) bound version as well.
2.
"Lecture notes in algebraic topology" by James F. Davis and Paul Kirk
(Graduate studies in mathematics, v. 35, AMS)
FALL 2010 TOPICS: (modifications may be made depending on student interest)
Higher homotopy groups, Hurewicz theorem, fiber bundles, fibrations,
spectral sequences, generalized cohomology theories, Eilenberg-Mac Lane
spaces, classifying spaces
HOMEWORK: Each probelm will be worth 5 points (unless otherwise noted.)
Due every other week.
Late homework will NOT be graded.
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.
PRESENTATIONS: Each of you will be asked to give a presentation
about a project related to the course.
See below for a list of possible projects. Many other topics are
listed as projects at the end of each chapter in Davis & Kirk. I encourage you to also talk to me about your interests to
find other possible projects.
These projects have two main steps. Of course, you need to learn the material
for your project. Equally importantly though, you need to figure out how to
best present an appropriate amount of the material to the other students.
Requirements:
1. First you need to have your topic approved (source material should also be discussed). (Guideline: by October 8, but please consider doing this much sooner.)
2. Before your presentation can be scheduled, we need to discuss the material
you have learned and which parts will be included in your presentation.
(Guideline: by October 29, again, sooner is better.)
3. Give at least one practice presentation with at least one other student from this course a few days before your class presentation.
4. Give your presentation (30 to 40 minutes) in class or at a specially arranged time outside of class.
Presentations will be worth 50% of your final grade.
Topics:
Lefschetz Fix Point Theorem
Existence of Division Algebras
Cech cohomology
DeRham cohomology (a short introduction)
Morse Theory
See also the projects listed in Davis & Kirk.
Math Writing:
One link that has something useful to say about writing
good solutions to math problems is
here. If you find another site you like, let me know and I'll put a link
here.
ASSIGNMENTS: Late homework will NOT be graded.
First Assignment: Due Monday, August 30, in class.
Calculate the cohomology ring of the nth complex projective space
(CP^n) and the infinite complex projective space (CP^{\infty}).
Explain your calculation and provide references. (This problem
will help me gauge the level of preparation for this course, since Math 547,
Algebraic Topology I, covered more material than usual in spring 2010.)
Second Assignment: Due Monday, September 13, in class.
Hatcher, Section 4.1: 4, 8
Third Assignment: Due Monday, September 27, in class.
Davis & Kirk, Chapter 4: 57, 60, 61 (can use 57)
(Extensions to 9/29 might be possible for this homework due to my travel.)
Fourth Assignment: Due Monday, October 11, in class.
Davis & Kirk, Chapter 4: 66, 70; Hatcher p. 184: 2
Fifth Assignment: Due Monday, October 25, in class.
Hatcher, Section 3.3: p. 260: #26;
Munkres 68 #7: Let X be a connected closed 7-manifold. Suppose that H_7(X) = Z, H_6(X) = Z, H_5(X) = Z/2 and H_4(X) = Z + Z/3. Give al the information you can about the cohomology groups and ring of X (H^*(X)).
We may or may not come back later to some of the other Poincare problems listed below...
Sixth Assignment: Due Monday, November 8, in class.
Hatcher, Section 4.1: p. 358: #11, 14; Section 4.2: p. 389: #12.
(See also extra problems below under "Whitehead and Hurewicz Theorems.")
Seventh Assignment: Due Monday, November 22, in class.
1a.
Show that the suspension map (see p. 360, Cor. 4.24, Hatcher, but
define it by taking [f:S^i to X ] to [Sf: S^{i+1} to SX] ) is a natural
transformation. This will involve showing that it induces a homomorphism
for each X and checking naturality.
The suspension functor is defined on p. 8, Hatcher, but note that
in part (b) the definition as two cones as in the proof of Cor. 4.24 is used.
(An easier version, worth 4 points, is to instead prove this for the
reduced suspension, see bottom paragraph of p. 223 Hatcher. In the
reduced suspension the interval cross the base point in X is also
identified to a point.)
1b.
Show that the natural transformation defined in 1a.
agrees for each space X with the map given on p. 360.
Hint: choose functorial inverses "on the space level" of both of the isomorphisms that appear. (The defining maps
of these isomorphisms come from long exact sequences for the homotopy
of a pair, but both go the "wrong" way, or the opposite direction of
the suspension map.)
I will discuss this in class on Monday Nov. 15.
2a.
Calculate the direct limit of the sequence defined by F(i) = Z
and F(i \to i+1) is multipllication by i+1.
2a.
Show that this direct limit has the universal property (that it is
initial among abelian groups with commuting systems of maps out of this
sequence.)
ANNOUNCEMENTS:
1. We will have presentations at 10am every Friday
in SEO 512 starting October 22.
2. Class is cancelled for Nov. 1 and Nov. 3.
Extra Problems to think about: These are problems that
you should consider to make sure you are keeping up with the lectures. Some of them will be assigned to be turned in (stay tuned, starred ones are more likely to be assigned...) Others you should
think about and perhaps ask questions about in office hours.
Hatcher, Section 4.1: 2, 4*, 5, 6, 8*
Davis & Kirk, Chapter 4: 57*, 58, 59, 60*, 61*, 65*, 66*, 69, 70*
Hatcher, Lefschetz fix points, p. 184: 1, 2, 4, 5, 6
Poincare Duality problems: there are lots of great problems here, see also old Algebraic Topology prelim exams (before 2009).
Hatcher p. 258: 6, 7, 8, 9, 16, 25, 26, 32
Munkres 68 #5: (Partially a "redo" of Hatcher p. 229 #6).
a) Compute the cohomology rings of H^*(CP^n) and H^*(CP^\infty).
b)Show that for any f:CP^n \to CP^n, the degree of f is a^n for some integer a. (Degree is defined in Hatcher p. 258 #7.)
c) Show that every map f:CP^2n \to CP^2n has a fixed point.
d)Show that there is no map f:CP^2n \to CP^2n of degree -1.
e) If f: CP^{2n+1} \to CP^{2n+1} has no fixed point, what is the degree of f?
Munkres 68 #7: Let X be a connected closed 7-manifold. Suppose that H_7(X) = Z, H_6(X) = Z, H_5(X) = Z/2 and H_4(X) = Z + Z/3. Give al the information you can about the cohomology groups and ring of X (H^*(X)).
Munkres, 65 #1:
Let M, N be compact, connected n-manifolds. Let f: M -> N be a continuous map. Show that if M is orientable and N is non-orientable, then the induced map in the nth homology group with coefficients in Z/2 is trivial (f_*: H_n(M; Z/2) \to H_n(N;Z/2) = 0)
Munkres, 68 #8: Let M be a compact, orientable manifold of dimension 4n + 2. Show that the 2n+1 homology of M cannot be isomorphic to the integers.
Whitehead and Hurewicz Theorems
Hatcher 4.1, p. 358: #10, #15; Section 4.2, p. 389: #15, #21