MATH 462 Introduction to Topology
Syllabus for Spring 2013
Mon, Wed, & Fri, 9:00 – 9:50am
Gildemeister 326

Instructor: Dr. Eric Errthum              Winona Email Username: eerrthum             Office: 124A Gildemeister Hall            Office Hours:  See homepage. Or by appointment on any day.

Prerequisite: MATH210/MATH327 or any other proof writing background

Text & Calculator: Topology Now! by Messer and Straffin
No calculators will be allowed on any quiz or exam, but might be required for some homework problems.

Course Website:     http://course1.winona.edu/eerrthum/math462

About This Course:  Topology is the study of spaces/shapes when size and angle don’t matter/aren’t well-defined. In the attempt to classify and understand these spaces we will find methods of construction, dissection, and computation of invariants. We will work toward an abstraction of the familiar ideas of measurement, continuity and connectedness.

Expectations: Students who complete this course with a passing grade are expected to be able to demonstrate the following skills: (i) Mastery of prerequisite material, (ii) Computations and classifications of objects and spaces, (iii) Understand and use the algebraic tools required in analyzing topological spaces, and (iv) work with abstract definitions that provide analogies to calculus concepts.

Grading:  Homework (scaled as needed)                                    90 points---- 13.85%
                  Video Responses (3 @ 30 points)                               90 points---- 13.85%
                  Quizzes (5 @ 30 points each, drop lowest)              120 points---- 18.46%
                  Midterms (2 @ 100 points)                                       200 points---- 30.77%
                  Final (In-Class)                                                          100 points---- 15.38%

                  Final (Take-Home)                                                      50 points----- 7.69%
                                                                                                    --------------

                                                                                                     650 points total

Grades:  A = 90% (517 pts), B = 80% (460 pts), C = 70% (402 pts), D = 60% (345 pts)

Homework:    All the problems from the sections covered during each week of class are due the following week on Monday (or the first class meeting after the weekend) unless announced differently in class. The problem numbers for each section are given below in the rough schedule. Homework should be readable or it will not be graded. (If you have any desire to learn how to type your homework in LaTeX, I will gladly show you.)

Video Responses:      Students will respond to the questions for each of three videos:

·         Sphere Inside Out (Part 1) (Part 2) – Questions

·         Wind and Mr. Ug – Questions, and Mobius Strip Activities – Treat this video like a science lab and answer the questions in the video at the times it wants you to (i.e. before the cutting). Your response do not have to from a coherent narrative. Just answer the questions in the order they are asked.

·         Not Knot (Part 1) (Part 2) – Questions

·         Extra Credit Video (worth 15 points): Rubber band into Knot – Questions

The due dates are given in the schedule below. Grades will be determined by the level and depth of your responses as well as on readability/grammar. The reflection is to be written as one coherent piece, not just disjointed answers to questions.

Quizzes:    We will have a short quiz after each chapter (or set of chapters), see schedule below. The lowest quiz score will be dropped.

Exams:     There will be two in-class midterms and one final exam. The final exam will be like a third midterm with a comprehensive take-home portion. Exam dates are tentative until officially announced in class. The final exam is scheduled for Tuesday, May 6th, 8am – 10am. The take-home portion will be due by noon on Wednesday, May 7th.

Extra Credit: There will be bonus questions on quizzes and exams. If you’d like to do more extra credit, I will give a lecture (outside of regular class time, 8am?) on a section of the book or other topic we didn’t cover and assign work from that section.

Resources: You’re more than welcome to stop by during my office hours and ask questions. Really, please do.

Desire2Learn:            Some course materials and approximate grades can be found on D2L.

Late/Missed Work:   Missed quizzes will result in a score of zero. There are no make-up quizzes. Make-up exams will be given at the discretion of the instructor. If you miss class, it is your responsibility to obtain notes and assignments from fellow students. If you have an unavoidable absence, please inform the instructor beforehand.

Academic Dishonesty:           Any type of academic dishonesty (cheating, copying, etc.) will result in failure and will be reported to school authorities. This includes homework. If you are having trouble with an assignment, please see the instructor first.

Note:   This syllabus is subject to change if deemed necessary by the instructor.

Tentative Schedule of Events – Math 462

(subject to change)

 

Week Starting

Monday

Wednesday

Friday

1/13

Introductions

1.1 Equivalence
HW (pg 4): 2, 5, 9, 10

1.2 Bijections
HW (pg 11): 5, 9, 11, 16, 19, 20

1.3 Continuous Functions
HW (pg 18): 5, 6abc, 7ab, 9, 10, 12

1/20

No Class
MLK Day

HW 1.1, 1.2 & 1.3 Due

7.1 General Metric Spaces

7.1, cont.

HW (pg 215): 1c, 2c, 5, 6b, 7(part 3), 8, 10, 13

1/27

SNOW DAY

7.2 General Topological Spaces

HW (pg 221): 3b, 4, 5, 9, 10, 12

Chapter 7 Activity

Basis of a Topology

2/3

HW 7.1 & 7.2 Due

1.4 Topological Equivalence
1.5 Topological Invariants

HW (pg 23): 5, 6, 7, 9, 11, 15

1.5, cont.
HW (pg 31): 3, 7, 8bc, 10, 13, 14

7.3 General Connectedness

HW (pg 226): 1, 2, 3, 6, 7

2/10

HW 1.4 & 1.5 Due

1.6 Isotopy
Video Demo
HW (pg 37): 1, 3, 4, 5cd, 9, 11, 12


“An Unshellable Triangulation of a Tetrahedron”

Chapter 1 Activity

Isotopic Letters

HW 7.3, 1.6 Due

Chapter 1 Quiz

Midterm Review

2/17

Midterm I

No Class
Assessment Day

Video Essay Due: Sphere Inside Out
(Part 1) (Part 2)

2.1 Knots, Links

Origins of Knot Theory


HW (pg 46): 2, 3, 5, 7b, 9abd, 10

2/24

2.2 Knot Diagrams
2.3 Reidemeister Moves

A tricky unknot

A really tricky unknot

Relaxing an unknot

Application of polynomial algorithm

HW (pg 53): 3, 4, 5, 8, read 11, 13, 14(just the middle one).

Knot Atlas

2.3, cont
2.4 Colorings

Two Knots, Same Coloribilty:

4_1 and 5_1

HW (pg 58): 3, 4, 6, 8, 9, 11

HW (pg 64): 2, 6, 7, 8, 10

HW 2.1 – 2.3 Due
(slide under my door)

Science Fair

Read the following:

Unknotting Unknots (Read pages 1 – 3)

Current Simplification Algorithm (Simplify Section)

Polynomial Time Unknot Recognition (Read Intro Section)

HW: Write/Type a 1-page summary/comparison of the three articles

3/3

HW Article Summary Due

2.5 The Alexander Polynomial

Two Knots, Same Alex. Poly.:
9_46 and 6_1

HW (pg 75): 1, 3, 5, 7b, 8, 11bef

Mathematica Determinant
(use “Save as..” and then open in Mathematica)

Alexander Polys

Extra Lecture:
2.6 & 2.7 Skein Relations and Jones Polynomial

Extra Credit:
 HW (pg 81): 4, 7, 9
HW (pg 87): 2, 4, 5b, 6, 9, Show Thm 2.75 works on Reidemeister Moves

---------------------

HW 2.4 & 2.5 Due

Chapter 2 Activity

Chapter 2 Quiz

3/10

No Class
Spring Break

3/17

3.1 Surface Definitions

HW (pg 95): 1, 4, 5, 7, 9


Solutions to 3.1.5b:

from Wikipedia

from “A topological picturebook”
by George K. Francis

Extra Credit 2.6 & 2.7 Due

7.5 General Quotient Spaces

3.2 Cut-and-paste Techniques

HW (pg 230): 2, 4, 5

3.2, cont.
3.3 Euler Characteristic

HW (pg 101): 2, 3, 7, 8, 9, 11

3/24

3.3 Orientability

HW (pg 108): 1, 4, 6, 9, 11

HW 3.1, 7.5, & 3.2 Due

3.4 Classification of Surfaces

HW (pg 117): 2d, 3, 4bef, 6, 7, 9, 10

Extra Lecture: 3.5

Extra Credit:
 HW (pg 124): 1b, 3, 5, 6, 7

---------------------

HW 3.3 & 3.4 Due

Chapter 3 Activity

3/31

Chapter 3 Quiz
Midterm Review

Midterm II

Video Essay Due: Wind and Mr. Ug and
Mobius Strip Activities

Section 4.1

HW (pg 129): 2, 4, 7, 8

4/7

Extra Credit 3.5 Due

Section 4.2

HW (pg 134): 1, 3, 5, 6, 8

 7.4 General Compactness

HW (pg 228): 1, 3, 5, 6b, 8

Section 4.3

HW (pg 141): 2, 3, 4, 7

HW 4.1 & 4.2 Due

4/14

HW 7.4 & 4.3 Due
Chapter 4 Activity

Chapter 4 Quiz

Section 6.1

HW (pg 172): 3, 4, 5abc, 6a

No Class
Spring Break Day

4/21

Video Essay Due: Not Knot
(Part 1) (Part 2)

Section 6.2

HW (pg 177): 1, 4, 5, 6

Section 6.4

HW (pg 189): 1b, 2, 6, 9, 11, 12ab, 14ac, 15, 16, 17

Section 6.5

HW (pg 199): 5, 7, 8, 9, 10, 13

Section 6.6

HW (pg 207): 1, 5

HW 6.1 & 6.2 Due

4/28

Extra Lecture: This pdf Document

Extra Credit:
HW (from the pdf): 1, 3, 5, 6, 7

---------------------

Extra Credit Video Essay Due:
Rubber band into Knot

 

Chapter 6 Activity

 

HW 6.4  - 6.6 Due Tuesday by 3pm

Chapter 6 Quiz

Final Review (Chaps 1 – 3)

Chapter 7 Quiz

Final Review (Chaps 4 – 6, 7)

 

Final Exam
In Class: 8 – 10am, May 6th
Take-Home: Noon, May 7th

 

 

Welcome to college math!

 

If this is your first math class taken in college, there are some important things you need to know. College math classes are run very differently from high school math classes. On the surface it may seem they are similar as you listen to the lecture and take notes, but there are significant underlying differences. Knowing these ahead of time can help you make the most of this coming semester.

 

#1: College math classes generally stay on the schedule in the syllabus. If there is one day allotted to the topic that is probably all of the class time that will be spent on it, even if “most” of the students “don’t get it.”

 

#2: It is expected that you will read the text and do the problems in order to learn the material, even if no one checks up on you. The instructor might never collect the homework, but that doesn’t mean it doesn’t affect your grade.

 

#3: You will sometimes be responsible for material in the textbook that is not covered in class. If there is a text reading and/or homework problems covering a concept that was not discussed in class, you are still expected to learn it. If you don’t understand it, make an appointment with your instructor for help.

 

#4: Some material is covered only in class, is not in the textbook, and may not have any homework problems on it. If you miss class, you may miss content that you are responsible to know. If you have an unavoidable absence, be sure to get the notes and any announcements from a classmate.

 

#5: There will be test questions that don’t look “just like the homework.” In college, you are expected to focus on learning the concepts, not just memorizing how to do certain types of problems. These concepts can – and will – appear in very different forms on tests and quizzes.

 

#6: At times you will be expected to be able to explain why a problem is done a certain way in addition to being expected to do the problem. As you work on problems in class and on homework, don’t be satisfied with getting the correct answer; ask yourself why that method is logical, and how you could explain that logic to someone else.

 

#7: Most importantly, you are responsible for your own learning. If you attend class faithfully, get the notes and announcements if you have an unavoidable absence, read the text, do the homework and question yourself (as in #6), and still don’t understand something, it is up to you to get the extra help you need. Visit the instructor during office hours or make a special appointment to ask questions, form a study group, etc. There are many resources and people willing and happy to help, but you need to take the initiative and seek out the help you need.

 

Good luck on a happy and successful semester!

 

 

Commitment to Inclusive Excellence:  WSU recognizes that our individual differences can deepen our understanding of one another and the world around us, rather than divide us. In this class, people of all ethnicities, genders, religions, ages, sexual orientations, disabilities, socioeconomic backgrounds, regions, and nationalities are strongly encouraged to share their rich array of perspectives and experiences.  If you feel your differences may in some way isolate you from WSU’s community or if you have a need of any specific accommodations, please speak with the instructor early in the semester about your concerns and what we can do together to help you become an active and engaged member of our class and community. 

 

Campus Resources (Short version):

 

Campus Resources (Long version):

 

The Standard Disclaimer applies.

© Eric Errthum, January 2014, all rights reserved.