## Math 54 Homework Log

Week 1 HW:

From Friday's lecture: Read chapter 1 and do chapter 1 numbers 11,14,15,16. From Saturday's lecture: Do chapter 1 numbers 1,3,4,5. (These first 8 problem will be due at the first x-session. Remember that you are encouraged to work in groups , and that each group will be assigned a problem to present and to write up carefully. If you ever want to rearrange or form new groups just e-mail me about the changes before the next x-session. If you are not in any of the listed groups then you are not on my roster, and you should contact me right away.)

Week 2 HW:

From Monday's lecture: Read chapter 2 and do chapter 1 numbers 10,17,21,22. From Wednesday's lecture: Do chapter 2 numbers 1,2,5,8. From Friday's lecture: Read section 3.1 and 3.2 and do chapter 2 numbers 11,15,16,17.

Week 3 HW:

There is a holiday on Monday so I will move my Monday office hour to 10:30-12:00 on Tuesday (so that you may get last minute suggestions on the problems to be presented at noon).

From Wednesday's lecture: Read section 3.3 and do chapter 3 numbers 1,3,9,11,12. From Friday's lecture: Read section 3.4 and do chapter 3 numbers 6,12,13,14 and prove the Lemma discussed in class.

Week 4 HW:

From Monday's lecture: Read section 3.5 and do chapter 3 numbers 20,23,24,25.

From Wednesday's lecture: Read section 3.6 and do chapter 3 numbers 31,32,33,35.

From Fridays' lecture read 4.1 and 4.2 and do chapter 3 problems 39,40,41,43.

Week 5 HW:

From Monday's lecture: Re-read 4.2 and read section 4.3 and do chapter 4 numbers 1,2,5. (Warning sections 4.3 and 4.4 will be covered only lightly this quarter, in class we will discuss the relevant aspects.)

From Wednesday's lecture: Read section 4.3 and 4.4 and do chapter 4 numbers 6,7,10.

From Fridays' lecture read 4.4 and do chapter 4 problems 20,21,22.

Week 6 HW:

From Monday's lecture: Re-read 4.4 and read section 5.1 and do chapter 4 numbers 26,27, and for extra credit number 33.

From Wednesday's lecture: Re-read chapter 4 and do chapter 4 numbers 3,4,18,19,20 and find represent any glide reflection as an element of E(2).

Friday: Happy Winter Carnival!

Here are some hints .

Week 7 HW: The Exam.

Here is a list of questions I have received together with some answers which you may find useful.

1. Q: "...the concept of the smallest topology. I thought I understood it, but my understanding definitely didn't match up with the open balls."

A: As stated on the exam: "To be the smallest topology on M such that d is continuous means that any topology where d is continuous will contain all of the metric topology's open sets." In other word in problem 1 you must show that the open balls form the base to a topology and that any topology on M where d is continuous contains the open sets in this topology.

2. Q: "...in one problem we're asked to explicitly describe a metric space, and I'm not sure exactly what that entails."

A: To explicitly describe a metric space means to describe the underlying set, the metric, and the metric's topology.

3. Q:"I'm afraid I've forgotten how to use theta, phi and r to describe three dimensional space."

A: Nothing fancy. These are just the variables' names, i.e. instead of the usual variables (x,y,z) use the variables (theta,psi,r). Example: the point (1,11,3) has a theta value of 1, a psi value of 11, and an r value of 3. Notice the notation used to describe the partition in part 2 has been changed to make this correspondence clearer.

4. Q:"... in particular I'm not sure what properties about the real numbers I should assume to do this problem."

A: You should assume that the real numbers are bigger size than the integers, i.e. you can not list all real numbers in a sequence.

5. Q:"The partition in part 2 is somewhat opaque to me, are there really only six subsets? Is this the cube with or without its inside?"

A: CU is the cube including its inside (captured by the singleton points described by the sixth type of partition set). There are six types of subsets not six elements in the partition. For example the first type of partition element describes the line segment where r is fixed to be 0, theta is fixed to be some value in (0,1), and psi takes on any value in [0,1].

6. Q:"I'm confused about why the open balls form a base, theorem 2.5 suggest they can't."

A: Theorem 2.5 is not characterizing bases, it only provides a method for constructing them. The definition of a base shows up at the bottom of page 30 where it is shown to be understood in one of the following two ways:

Definition 1: A base is a collection B of open sets such that every open set can be expressed as union of elements of B.

Definition 2: A base is a collection B of open sets such that for any point x and any neighborhood N of x, there is an element of B such that x is in B and B is in N.

Week 8 HW:

Read 5.1,5.2,5.3, 5.4 and skim 10.4. Do chapter 5 numbers 1,2,3,5,9,13,15,21,22.

Starting Friday we will be slowly and with great care examining a few applications of the fundamental group ideas, some of which is not covered in our text. There have been several comments that this fundamental group stuff is a bit confusing, and going by at such a pace that it is not possible to keep track of what all the object are an why. I will help you out by identifying for you the ideas that you will need to understand in order to follow the arguments we will be exploring in class. Namely by Friday you should understand what the fundamental group is as a group, and the statements of theorem 5.7, 5.13, 10.10 and 10.11. Here is the material from Friday.

Week 9 HW:

Read 5.5,10.4. Home Work

Week 10 HW:

Think about the ZIP proof and do the Exam. The exam is due by Sunday March 11th at 12:30 (P.M.!), and can be slid under my office door.

clarifications.

1. With regards to 2(a). If f is mapping from A viewed as subspace of a space X to a space Y, then an extension of f is a mapping g from X to Y such that g(a)=f(a) for every a in A.

2.The term connected cover in part 3 refers to a connected COVERING SPACE, (not a cover in the sense of an open cover for example).

3. In part one: ANY space which is having the one point compactification process applied to is assumes to be locally compact, Hausdorff, and not compact.

4. Part 3 number 4: The only part of the ZIP proof that we know is that any surface is homeomorphic to a surface in canonical form; we do not know that the surface in reduced canonical form are distinct or that the Euler characteristic is a topological invariant. You are expected to do this problem using the fundamental group ideas from chapter 5.