For a few days we will Review. For Friday do problem 8 (Problem 8 a and Problem 8 b ) from the handout.
Week 2:
The groups that arise this quarter come in to flavors "discrete" and "topological". This week we will explore the nature of the discrete groups that will arise, via Van Kampen's theorem. Section 1.2 of chapter 1 in Hatcher's book.
Due on Monday: problem 13 (Problem 13 and continue ).
Tuesday: We will be meeting tomorrow at our X-session time this week (as well as next Tuesday). (I will be gone Wednesday April 11th and Friday 13th and need to make these classes.)
Due on Tuesday: Problem 12 (and continue ) from the initial review , and read pages 39-45 of section 1.2 in Hatcher.
Due on Friday: problem Problem 2 (and continue and continue ) from the classification handout.
Week 3:
Due on Monday: Be ready to present Problem 4 (and continue as well as Problem 4 and continue ) from the classification handout.
Tuesday: We will once again be meeting at our X-session time this week, since I will be gone Friday the 13th and on Wednesday the 11th there will be an enrichment guest lecture on the fundamental groups of Knot Complements, by Professor Peter Doyle. Every one should pay close attention during this lecture since Knot complements are the next natural class of topological spaces where one would explore the "geometrization program" explored in the surface case in this course.
Due on Tuesday: Use Van Kampen's theorem to justify that the fundamental groups listed in the table on page 57 of Hatcher's book are indeed accurate.
Week 4:
Due on Monday: Be prepared to present the problems that where due last Tuesday.
Tuesday: Algebra madness! We will explore some of the group theory we are playing with. This will be very basic and optional. In particular we will construct several views of the symmetric group and show it can be represented as the fundamental group of a topological space as well.
Due on Wednesday: From the Galois correspondence problem 13 (here are parts 5 and 2 ).
Due on Friday: From the Galois correspondence do problem 14.
Week 5:
Due on Monday: From the Galois correspondence finish problem 14 ( continue and continue ).
Tuesday: We will meet and discuss the assignment that was due on Monday.
Due on Wednesday: Explicitly describe the Deck transformations of the universal cover from Monday's assignment, attempt to find an order two subgroup of the Deck group and USE it to form a cover with fundamental group Z mod 2Z. From our discussion on Tuesday: attempt to decide which basepoint corresponds to which of the three order two subgroups.
Due on Friday: read and think carefully about pages 59-72 of Hatcher.
Week 6:
Due on Monday: The necessity of local conditions - do problems 7 and 6 from from Hatcher, and using the space in number 7 show that the Galois correspondence fails if the space is not locally path connected.
Due on Tuesday: Complete Monday's homework!
Wednesday: Here is the exam it is due Tuesday's sometime before 2 in the afternoon. The spaces (X,x) and (Y,y) are described here , while here are the partial orderings of D8 , and Q16 .
Friday: Only homework is the exam.
Week 7:
Monday: Continue working the exam, which is due on Tuesday by 2 o'clock.
Wednesday: Our glimpse into hyperbolic geometry begins, with some Here is the conformal geometry.
Friday: Read the entire conformal handout and skim Kr and Ka. The four exercises from the conformal handout will be presented ( exercise 1, exercise 5, exercise 2.
Week 8:
Monday: Read the Mobius handout and skim Kr and Ka. Do the exercise 3 from the Mobius handout exercise , exercise 7 , and part2 .
Tuesday: We will go over the exam. Here are some nice solutions. John managed to do Q16 ! Here is the first part. You can print out the next three images and put them to gether to geometrically see the Galois correspondence: left , center , right . Here is the most dramatic of all the solutions - Sam's beautiful solution to the extremely challenging number 2c: the first part, the second part, and the third part. Here is Jason's solution to 2d ( first part, the second part), where you can cut out and glue up the solution for your self! Here is Karthik's solution to the first one for D8 first part, the second part, and the third part. For those of you who used algebraic techniques to solve 2e and 2f here are John's 2e and Mihai's 2f demonstrating as particularly elegant geometric solutions.
Wednesday: Do exercise 5 from the Mobius handout. (Note this handout as been modified, and in particular this problem now exists!)
Friday: Do exercise 7 from the Mobius handout, read Kr 31-50 and Ka 1-5 and (as suggested on page 42 of Kr) prove that $d_{\rho}(x,y)$ is indeed a metric. Skim through the rest of Kr and Ka. ( exercise 7 and this .)
Week 9
Monday: Do exercises 1 and 2 from the Problems . ( exercise and this ).
Tuesday: Homework Presentation day. Please do problems 3 and 4 from the Problems for today and be ready to present last Wednesday's, Friday's, and this Monday's problems.
Wednesday: Please do problems 5,6 and 7 from the Problems for today (and be ready to present problems 3 and 4.) Finish reading the KR and KA parts of the handout.
Friday: Please do problem 9 from the Problems . Here is the final exam , due on or before 4 P.M. on June 5th. Please put it in my box or under my door, and {\bf e-mail me} after you have done this.
Week 10:
Tuesday: An optional day where we discuss the proof of the Hopf-Rinow theorem.
Wednesday: We conclude the course, here is picture 1 and picture 2 . For a very nice proof of the Hopf Rinow theorem (a little different than the one discussed in class) see Page 36 of the nice book:
John Stillwell, The Geometry of Surfaces , Springer-Verlag, 1992 (Math QA 645 .S75 1992).
This book is a nice place to explore alternate view points to what occurred in the second half of this course, as well as find a nice introduction to the generalization of these ideas to more general tesselations and their orbit spaces (orbifolds).
Exam Extension: The final exam is due by 4 P.M. on June 5th
Exam corrections:
1. Section 2 problem 3 was false as stated! Please see the corrected exam. (P.S. Thanks Sam!). (Changed on exam Sunday night).
2. By the gluing together of ideal triangles in part 3 I mean the identification space formed via a mapping sending one edge of a given ideal triangle to an edge of a second ideal triangle via an isometry, i.e. an isometry from the first edge to the second, which simply amounts to an isometry of one geodesic onto another. (changed on exam Sunday night).
3. Typos in part 1 numbers 2 and 3. Note number 3(c) has been re-phrased. (Also changed on Sunday evening).
