MATH 380: Homework Assignments

Turn in your homework assignment to me (in class or my office) before 4:00 PM on the due date.

  • To receive full credit, homework sets must be handed in to me on time (either in class or my office.) Turn in as much of the homework as you can by this deadline to receive partial credit. If you have a legitimate conflict you must tell me ahead of time.
  • You are encouraged to work with other students on the assignments as much as possible, but each student must write up the answers in her or his own words. You are expected to be working on the homework assignments throughout the week. DO NOT WAIT UNTIL THE DAY BEFORE IT IS DUE TO START THE ASSIGNMENT!
  • Reading the textbook is required, NOT optional. Your chances of getting a good grade in this course are infinitesimally small unless you read the textbook in addition to attending lectures.
UPDATED: Wednesday, April 25, 2018 12:05
Set Homework Problems Reading Assignment Due Date
1
 Sec. 1.1: 2(a,b), 3, 5, 6, 7, 10 (Correction on #5: a ≈ b iff |a-b| ≤ 1.)
Sec. 1.2: 6, 7, 9, 17, 18, 19
Clarification: "Give a geometric description" means to describe in words, but you can use a picture to help with your description. To "illustrate a simple bijection" I would be happy with a well labeled picture, indicating where various points are mapped to.
Extra Fun Problems that you don't have to do: Sec. 1.1: 4, 9; Sec. 1.2: 1, 3, 4, 5, 10, 11, 13, 15, 21
Sections 1.1-1.4
Wednesday, January 24
2
 Sec. 1.3: 10 (Hint: A well-drawn, well-labeled picture is effective and could make this problem incredibly simple. Use some words to describe your illustration.)
Sec. 1.4: 1, 2, 3, 7, 10, 13, 14a (Can you do #6?)
Clarification: When given or giving a formula for a function/bijection/homeomorphism, you should be able to prove if it is bijective or not, but you do not need to prove continuity. However, you should be able to explain why it is or is not continuous, without proof. When intuitively describing or drawing the function, try to provide an intuitive description of why it is or is not bijective or continuous.
Extra Fun Problems that you don't have to do: Sec. 1.4: 4, 8, 11, 12, 15
Sections 1.5-1.6
Wednesday, January 31
3
 Sec. 1.5: 4, 7, 8, 9, 11, 12, 13 [Carefully read Theorem 1.44 and Example 1.45. As a hint for #4, the given union of sets can have an order defined on it (i.e. you can define it as an ordered set).]
Extra Fun Problems that you don't have to do: Sec. 1.5: 1, 2, 5, 6, 10
Sections 2.1
Thursday, February 8
4
 Sec. 1.6: 3, 4, 5, 7, 9, 10
For most of the ambient isotopies, you will not be writing a formula. Instead you may need to draw a series of pictures to illustrate your isotopy, with a simple explanation where necessary. Overzealous students may want to actually create the animation itself using computer technology. (Automatic A if you do....)
(Read #11 and #12 about "triangulation". You don't have to do them, but it is a very important concept that we may make use of later.)
Extra Fun Problems that you don't have to do: Sec. 1.6: 1, 2, 6, 8
Sections 2.2-2.3
Wednesday, February 14
5
 Sec. 2.1: 2, 3(b), 4, 9 (Don't read this until after you do 9(a),(b). Here is Exer. 15 from Rolfsen.)
Sec. 2.2: 3, 5, 6 (prove your answer), 8, 14, 15
(Read 11, 12, and 13 of Sec. 2.2 about "unknotting and unlinking". You don't have to do them, but they are interesting.)
Extra Fun Problems that you don't have to do: Sec. 2.1: 1, 5; Sec. 2.2: 1, 4, 11, 12, 13
Sections 2.4
Wednesday, February 21
Exam
 Exam Questions: The exam coming up will cover Sections 1.1-1.6 and 2.1-2.4 of the textbook. Each student must submit a possible exam question coming from the material of Chapter 1 or Section 2.1-2.4. Submit one question by Thursday, February 22, and if I choose to include your question then you will receive 5 bonus points on the exam.
Chapters 1 and 2
Thursday, February 22
6
 Sec. 2.3: 3, 4, 6, 8, 11
Sec. 2.4: 1, 2, 10
Extra Fun Problems that you don't have to do: Sec. 2.3: 7, 10, 12; Sec. 2.4: 5
Sections 2.5-2.6
Wednesday, February 28
Exam
 Exam 1 will be given on Thursday, March 1: The exam will be a take-home exam and will cover all of Sections 1.1-1.6 and 2.1-2.4 of the textbook. You will NOT be allowed to use your textbook, notes or any other aid, and you must work alone. You must hand in the exam by 4:00 on Monday, March 5. It will be similar to the homework. To practice for the exam, review the homework problems, do some extra problems similar to the homework problems, and know ALL of the relevant definitions and theorems.
Chapters 1 and 2
Start Thursday, March 1
---
Hand in Monday, March 5
7
 Sec. 2.5: 1, 2, 5, 7, 14(a). Do #8 also, but only apply it to the unknot, trefoil knot, and the knots in #1 and #2. When p=3, this is a 3-coloring. Bonus Question: Who is Charles Livingston?
Sec. 2.6: 2, 4, 6, 7, 9, 10 Hint: Reading the last paragraph of Section 2.6 may help you with #9 and #10.
Extra Fun Problems that you don't have to do: Sec. 2.5: 3, 4, 16;
Sections 2.7-2.8
Thursday, March 8
Break
Spring Break: No classes, March 12-16.
STUDY ALL WEEK!
Have a great and safe break!
8
 Sec. 2.7: 3, 5, 8, 10 (You will use exercise 4, but you do not have to prove it.)
Mosaic Knots: Homework Problems, Lomonaco-Kauffman Paper, Example Mosaics
Extra Fun Problems that you don't have to do: Sec. 2.7: 2, 4, 9
Sections 3.1-3.2
Thursday, March 22
Exam
Corrections
  1. All of you are allowed to do exam corrections. These corrections will add points back to your exam, 50% of the points taken off.
  2. You need to make corrections for all of the mistakes you made on the exam. Please provide the correct solution to EVERY problem that you did not get completely correct.
  3. In addition to giving the correct solution, you must GIVE AN EXPLANATION for why your original solution is incorrect.
  4. The exam corrections must be submitted to me orally. Make an appointment or come by my office hours by Thursday, March 29, but try to do them as soon as possible.
DUE by
Thursday, March 29
9
 Sec. 3.1: 2, 3, 4, 5
Sec. 3.2: 4, 6, 7, 8.
Extra Fun Problems that you don't have to do: Sec. 3.1: 1, 6, 10, 11, 12; Sec. 3.2: 2, 3, 5, 10, 12
Sections 3.3
Thursday, March 29
10
 Sec. 3.3: 1, 3, 4, 6(a,b), 7, 11 (Read 6(c,d), 8, and 10. You don't have to do them, but they are interesting and basic ideas that all topologists should know.)
Extra Fun Problems that you don't have to do: Sec. 3.3: 5, 9
Sections 3.4-3.5
Wednesday, April 4
11
 Sec. 3.4: 1, 3, 4, 6, 10, 11
Sec. 3.5: 1 (make sure you read the last paragraph of this section), 2, 4. Also compute the four knot invariants that we obtained using the Seifert matrix for the second knot with five crossings in Figure 2.41 and the first knot of six crossings in Figure 2.42.
Extra Fun Problems that you don't have to do: Sec. 3.4: 2, 5, 7, 9; Sec. 3.5: 5
Sections 4.1-4.2
Thursday, April 12
Exam
 Exam Questions: The exam coming up will cover Sections 2.5-2.7 and 3.1-3.5 of the textbook and the algebraic knot invariants related to the Seifert Matrix. Each student must submit a possible exam question coming from the material of these sections. Submit one question by Thursday, April 12, and if I choose to include your question then you will receive 5 bonus points on the exam.
Chapters 2, 3
Thursday, April 12
12
 Sec. 4.1: 2 (b-e (take your time and really try to "visualize" these)), 3(b,c), 6, 7(the answer is NOT S3!), 9 (Read 8 (b,c) for a description of projective space P3.)
(Remember that to show something is a n-manifold, you just need to find a neighborhood homeomorphic to an open n-ball around any point.)
Extra Fun Problems that you don't have to do: Sec. 4.1: 1, 4, 8, 10
Sections 4.3-4.4
Thursday, April 19
Exam
 Exam 2 will be given on Thursday, April 19: The exam will be a take-home exam and will cover all of Sections 2.5-2.7 and 3.1-3.5 of the textbook and the algebraic knot invariants related to the Seifert Matrix. You will NOT be allowed to use your textbook, notes or any other aid, and you must work alone. You must hand in the exam by 4:00 PM on Monday, April 23. It will be similar to the homework. To practice for the exam, review the homework problems, do some extra problems similar to the homework problems, and know ALL of the relevant definitions and theorems.
Chapters 2 and 3
Start Thursday, April 19
---
Hand in Monday, April 23
13
 Sec. 4.2: 3, 4, 5, 6, 8(a,b,c), 9
(SPECIAL INSTRUCTIONS: Do 8(a), then the extra problem below, THEN do 8(b) and 8(c).)
Problem 8(a'): Triangulate Bn for n=0,1,2,3,4 with the simplest possible triangulation. (A drawing for B4 is not necessary but may help.) Make a chart whose entries are the number of vertices, edges, faces, 3-cells, 4-cells, 5-cells and 6-cells for each Bn so that the k-th row would list the number of cells in Bk-1. For example, the third row would say:

B2:     3     3     1     0     0     0     0

since it has 3 vertices, 3 edges, 1 face, and no 3-cells, 4-cells, 5-cells, or 6-cells. Make an educated guess to fill in the chart to include B5 and B6. Determine the Euler characteristic of Bn for all n.
Extra Fun Problems that you don't have to do:
Sec. 4.2: 7, 8(d-f)
Sec. 4.3: 5, 6, 7, 8
Sec. 4.4: 7, 8, 9, 10
Sections 6.1, 6.2
Thursday, April 26
14
 Sec. 6.1: 2, 3, 4, 6
Sec. 6.2: 1 (1(d) is the degree function we discussed in class. Show that the degree function is well-defined by showing that if a and b are homotopic loops, then deg(a)=deg(b).)
Sec. 6.3: 3, 5, 7. Also show h : π1(X,x0) → π1(X,x1), as defined in class, is an isomorphism.
(Hint: These 6.4 problems require almost no effort!)
Sec. 6.4: 12(d) (To do this, read #9 and #12(a-c). Bing's House is described on page 109 of the textbook.)
Sec. 6.4: 13, 14, 16, 17
Extra Fun Problems that you don't have to do:
Sec. 6.1: 5
Sec. 6.3: 1, 2, 4, 8
Sec. 6.4: 1, 9, 10, 11, 15, 18
Sections 6.3, 6.4
Wednesday, May 2
Exam
Corrections
  1. All of you are allowed to do exam corrections. These corrections will add points back to your exam, 35% of the points taken off.
  2. You need to make corrections for all of the mistakes you made on the exam. Please provide the correct solution to EVERY problem that you did not get completely correct.
  3. In addition to giving the correct solution, you must GIVE AN EXPLANATION for why your original solution is incorrect.
  4. The exam corrections must be submitted to me orally. Make an appointment or come by my office hours by Wednesday, May 2, but try to do them as soon as possible.
DUE by
Wednesday, May 2
SOFI's
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Complete by
Wednesday, May 2
Final
Exam
 The Final Exam is on Monday, May 7, 12:00-2:30 pm, in class: The exam is a "nonstandard" exam. Click here for more details.
Chapters 1-7
Monday, May 7