432/532 - Class Outlines, Homework, and Handouts

Math 432/532 - Class Outlines, Homework, and Handouts

Class syllabus page

Last homework due Friday March 12 by 5pm.
Review problems for the final exam (they're a doozy, but you asked...)
Solutions to HW3.
We will have a review session on Sunday March 14 (Pi Day!) at 3pm, probably in our usual classroom.

Week Nine, March 1-5

Monday: We go through the calculation of the fundamental group of the projective plane more carefully, and finish discussion of the covering spaces of a wedge of two circles.
Wednesday: We discuss free groups and presentations of groups by generators and relations.
Friday: We discuss almagamated free products, state the Van Kampen theorem, and do some calculations.

Week Ten, March 8-13

Monday: We go through much of the proof of the van Kampen theorem.
Wednesday: We will apply the van Kampen theorem to calculate the fundamental groups of surfaces and finish their classification. We will also discuss Escher prints.
Friday: We will develop some more covering space theory and show that the free group on two generators contains the free group on n generators as a subgroup.

Week One, Jan 4-7

Monday: We will start talking about completions and compactifications.
Wednesday: We will wrap up discussion of the one-point compactification and start a little "digression" on compactifications of configuration spaces based on this paper of mine.
Friday: We'll finish the digression on configuration spaces.

First homework due Tuesday January 19th at 5pm (you can stop by my office on Tuesday and ask questions). Update: because people had trouble downloading the HW, it will now be due at noon on Wednesday.

Week Two, Jan 11-14

On Monday, we'll begin discussion of the Stone-Cech compactification.
On Wednesday, we'll show the Stone-Cech compactification maps to any other reasonable compactification.
On Friday, we'll "go back" to talking about separation and countability axioms.

Here are some solutions to the first homework.

Week Three, Jan 18-21

Monday is a University holiday.
On Wednesday, we'll start studying the Urysohn lemma.
On Friday, we'll use worksheets to look more closely at the Urysohn lemma.

Week Four, Jan 25-28

On Monday we'll finish the Urysohn lemma with a bit of lecture, and then start on the metrization theorem through worksheets.
On Wednesday, we'll finish the metrization theorem through lecture and then start talk about topologizing function spaces.
On Friday, we'll talk about the uniform topology on function spaces, completeness (with a detour through the Peano curve) and then start defining other topologies on function spaces.

HW due Wednesday February 3rd at 5pm, all from Munkres' book: 31: 2, 6, 7; 32: 4; 33: 1, 4, 7; 34: 1, 3.

Week Five, Feb 1-5

On Monday we'll use completeness under the uniform topology to construct the Peano curve and then start talking about the compact-open topology.
On Wednesday, we'll dig into the non-result that the point-convergence topology fails to have a continuous evaluation map. We'll then use the evaluation map to establish the "power law" for function spaces.
On Friday, we'll expand on the power law and related laws and then start discussing the fundamental group.

Week Six, Feb 8-12
On Monday we'll review and on Wednesday we'll have our midterm.
Here is a first review for the midterm and a second review

Week Seven, Feb 15-19

On Monday we start in earnest on the fundamental group (with Nick Proudfoot substituting).
On Wednesday, we apply the calculations of the fundamental groups of the circle and the disk to show the non-existence of a retraction of the disk onto the circle.
On Friday, we establish that the fundamental groups of convex subspaces of Euclidean spaces are trivial, and then we present the main idea in the calculation of the fundamental group of the circle.

Week Eight, Feb 22-26

On Monday we establish the lifting lemma for paths, state the general lifting lemma, and use it to establish that the fundamental group of the circle is the integers.
On Wednesday, we prove the general lifting lemma and start to discuss other covering spaces.
On Friday, we discuss covering spaces of the circle, the torus, the real projective plane and a wedge of circles.

Here is the third homework, due Monday March 1 at 5pm (was originally due Friday February 26). Note that in problem 6 you may assume that the covering p is finite (that is, p^{-1} of some U is homeo to U x {a finite set}). The statement is true in far more generality but does require a mild but complicated hypothesis.