Francis Su

Topology (Math 147) Harvey Mudd College, Spring 2019. Point-set topology with fundamental groups. Pre-requisite: one semester in real analysis.

Prerequisites: Math 131 (analysis). Math 171 (algebra) is recommended as a co-requisite.

INQUIRY BASED LEARNING: this course will be taught in an IBL (inquiry-based learning) format. You will receive handouts containing theorems, definitions, examples. Your goal is to prove all the theorems by yourselves in a guided discovery process: with collaboration of classmates and guidance from me.

You will take turns presenting proofs of theorems in class, while other students will determine if it is correct. I will provide perspective on the material, and motivating examples if you get stuck.

TEXT: The text for the course is the draft of my book with Starbird: “Topology through Inquiry”. Please do NOT distribute this book in any form to anyone without my permission.

NO OUTSIDE SOURCES: As is customary in any IBL course, you are forbidden from consulting any outside sources, including textbooks or the internet, to solve these problems. [The exceptions are sources that you are required to use for other courses.] The fun of the course is in struggling with the ideas and enjoying challenging problems that will stretch your mind.

NOTEBOOK: You should acquire a loose-leaf binder, in which you will save all course notes. As you prove theorems in this course (or see them proved in class), you will write up these proofs and add them to this notebook. In a sense, you are writing your own book on the subject, filled with your own proofs. As part of your grade for the course, these notebooks will be evaluated. More importantly, a nice notebook will be something you can be very proud of years from now, as you look back on your experience.

HOMEWORK: You will be asked to write up selected theorems to be handed in weekly. Thus it is important to pay attention to proofs of theorems presented in class, since you will write these up for credit.

EXAMS: There will be no midterm, but there will be a final examination.

GRADES: I’d rather you not worry about grades. I hope you will concentrate on learning. But since we must, grades will be determined in the following ways. 70% of your grade will be specifications-based. This means that you must meet each of the following specifications to pass the course:

• Present in class 15 times over the course of the semester, including 5 productive failures.
• Ask 15 questions in class over the course of the semester.
• Contribute 15 proofs in the Common Solution Set [posted in the Resources tab]
• Edit 15 proofs in the Common Solution Set (to improve the solutions of others).
• Keep a well-organized notebook, which will be turned in 2/3 of way through the semester, these will be graded exceptional pass/pass/no pass.

Homeworks, and Final Exam, will count the remaining 30% of your grade.

HONOR CODE: Cooperation is encouraged, but solutions should be written up individually. You may not consult outside mathematical sources without my permission, unless required for some other course.

Expect this course to be challenging, but also very rewarding. The value of the IBL format is that when you prove theorems by yourself, you will never forget the proofs you came up with, and you will gain confidence in your abilities as mathematicians!

WHAT SHOULD BE IN YOUR NOTEBOOK:
– Summary Sheet and Evaluation that tracks your progress
– The Reflection Exercise
– Proofs or sketches for Theorems and Exercises
– Any other notes you’re taking for the course
– Any returned homeworks
How you organize the notebook is up to you. The proofs in your notebook may be handwritten or electronic, either is fine. When it comes time to turn in your notebook, you may turn it in electronically if you wish.

The rule of thumb is that this notebook should something you look back on with pride 10 years from now. So it should contain enough details for you to reconstruct your thoughts. I encourage you to write up theorems carefully: see my Guidelines for Good Mathematical Writing.

SUMMARY SHEET and EVALUATION. Your Summary sheet should list:

• 15 proofs you presented in class (including 5 productive failures) — list theorem numbers only
• 15 questions you asked in class (list theorem numbers)
• 15 proofs you contributed you’ve made to the Common Solution Set: you’ll record them in that document but list theorem here too.
• 15 problems you’ve edited in the Common Solution Set: you’ll record them in that document but list theorem here too.
• An evaluation of your notebook according to the following rubric.

Both you and I will evaluate your notebook. Use the following rubric, and include on your Summary sheet an explanation of the grade you would assign yourself with this rubric.

Exceptional Pass = every theorem and exercise discussed in class has complete correct proofs or proof sketches. There are many more proofs completed even though they were not assigned for class discussion or homework. Proofs and proof sketches are well-written. Pictures are abundant. Notation is well-chosen. Subtle points in proofs are acknowledged. Reflection Exercise included. Indicate why your Notebook should receive an Exceptional Pass.

Pass = Most theorems discussed in class have proofs or proof sketches. There are a good number of theorems completed that were not assigned for homework. Pictures are ample. If you looked at it 10 years from now, you could reconstruct most of the arguments from what you have written. Reflection Exercise included. Indicate why your Notebook should receive a Pass.

Not Yet Passing = Indicate what deficiencies you have, and what you will do to Pass.
I encourage you also to seek feedback from classmates about how they would evaluate your notebook according to this rubric.

REFLECTION EXERCISE. Reflect on the following question:
What have you learned in this class about the process of doing or creating mathematics? Weave in potential answers to the following:

• have you experienced: joy, beauty, reward?
• what have you learned about the value of struggle?
• what have you learned about the importance of community?

SUGGESTIONS FOR PROOFS. There will be many times when you are stuck on a problem. This is where the real learning occurs. Here are several ideas:

Draw pictures! For instance, unions, intersections, complements, etc. are often effectively represented by circles. Make certain you thoroughly understand the definitions involved, and work out examples! Work a special case if you cannot solve the whole problem. Is every hypothesis necessary? Construct examples to show why the theorem fails if a hypothesis is missing. This will often show you what is needed for the proof. Ask yourself if you can modify the statement to obtain a new, related theorem. When you have a proof, ask what other statements can be proved with the same proof? One of the goals of the course is to help you see mathematics more as a collection of techniques, examples, and proofs than as a collection of theorem statements.

SUGGESTIONS FOR PRESENTATIONS. Sketch. Begin by giving a brief outline of the argument, before giving details. The outline should consist of a sequence of complete, true statements, whose proofs can be explained when you give the details. Be prepared to justify the details if asked. Speak loudly. Meet with others in the class, and practice presenting your proofs to each other! Knowing a proof and presenting it are two very different things!

HOMEWORKS. Rule of Thumb. Stay 12 theorems ahead of where we end the previous time.

• For Mon 1/28. Read the Introduction, skim Chapter 1, Read Chapter 2 introduction and Section 2.1.
• Due Wed 1/30. HW#1. Hand in Chapter 2 ( 3, 5 [finite complement only], 8, 13, 14 ).
• For Mon 2/4. Begin reading Chapter 3. Work up through the end of Chapter 2, especially focusing on theorems.
• For Wed 2/6. HW#2. Hand in Chapter 2 ( 20, 23, 26, 30 ) and Chapter 3 ( 3 ).
• For Mon 2/11. In Section 3.1, work on 9, 10. Skim Section 3.2 (subbases), work on it if you want, but we won’t spend much time on it. In Section 3.3, only work problems related to the lexicographically ordered square (20, 21). In Section 3.4, work through at least 25 – 27.
• For Wed 2/13. Continue in Section 3.4 through 37, reading the introduction to product spaces.
  HW#3. Hand in Chapter 3 ( 8, 9, 10 (part 4), 21 (example D only), 28 ).
• For Mon 2/18. Continue in Section 3.4, work 38 – 42 (think about at least 3 of them) and in Chapter 4, work 1 to 7.
• For Wed 2/20. Continue in Chapter 4, work through 17, but 12 and 13 are optional.
  HW#4 Hand in Chapter 3 ( 31(F only), 41 ) Chapter 4 ( 1, 5, 9 ).
• For Mon 2/25. Continue in Chapter 4: complete all of Section 4.1 and 4.2. Read all of Section 4.3, but focus attention on 19, 20, 23. In Section 4.4, focus attention on 29, 30, 31, 32. Read 4.5.
• For Wed 2/27. Complete 4.4 and focus on 29, 30, 31, 32. Read Chapter 5 intro, and Section 5.1, and complete all exercises and theorems except 5.7 and 5.8.
  HW#5 Hand in Chapter 4 ( 6[part 4 only], 11, 15[all but normal], 17, 23 ).
• For Mon 3/4. Work up through 5.15 (skip 7, 8, 12, 13 ). Read the rest of Chapter 5.
• For Wed 3/6. In Section 5.3 only do 15 and 18. Then work in Chapter 6 through 6.9 ( 1 through 3 should be a review from analysis).
  HW #6 Hand in Chapter 4 ( 29, 32 ) Chapter 5 ( 6, 11, 18 ).
• For Mon 3/11. Work all remaining Theorems in Section 6.1 and do all Theorems in Section 6.3 (Exercises optional.) Section 6.2 should be mostly review from Analysis, so read it, but we’ll mostly skip it. We’ll skip Sections 6.4 and 6.5. Read Section 6.6.
• For Wed 3/13. Start working Chapter 7, in particular all theorems in Section 7.1 (except 8, 14). Many are straightforward to dispense with, so we’ll fly through some of them.
  HW#7 Hand in Chapter 6 ( 5, 9, 12, 18 ) Chapter 7 ( 7 ).
• For Mon 3/25. Work these problems in Section 7.2: 15, 18, 20, 21, 22, 24, and all Theorems in Section 7.3. (You’re avoiding theorems involving Lindelöf, countably compact, 2nd countable).
• For Wed 3/27. Work all Theorems in Section 7.4, and Exercise 7.41. No HW to hand in, but please be working on your Notebooks and contributing to the Online Solution Set.
• For Mon 4/1. Read all of Section 7.5 and read all theorems and exercises, thinking about each one, but focus on these problems: 45 (especially part 3), 46, 48, 53, 54, 55. Read Section 7.6, 7.7 but do not do problems. Read Chapter 8 intro and Section 8.1, do 1, 2, 3, 4, 5.
• For Wed 4/3. In Section 8.1, do: 6, 8, 12, 13. Read Section 8.2 (don’t do any problems). In Section 8.3 do: 18, skim rest of section.
  HW#8 Hand in Chapter 7 ( 21, 24, 41, 42 ) Chapter 8 ( 4 ).
• For Mon 4/8. In Section 8.4 do: 35, 36, skim the rest of chapter. Skim Sections 9.1 and 9.2, especially Theorem 9.11 and 9.14. Read Section 9.3, and do the 2 theorems in that section: 24, 25. Read Section 12.1, do: 2, 3, 4, 5.
• For Wed 4/10. Do Section 12.1, do: 4-14. The homework is turning in notebooks. See the Tab above on ‘Notebooks’ for the Rubric and complete the instructions found, there including the Reflection Exercise. (I’ve indicated in blue small changes to the instructions on that page.) This is a preliminary notebook check. If you get a High Pass or Pass, you will complete your notebook requirement for the semester. If you get a Not Yet Passing (or get a Pass but want a High Pass), you can turn your notebook in the final Monday of the course.
• For Mon 4/15. Do 12.1 ( 11, 12, 14, 15 ) 12.2 ( 18-20, skip 21 ) 12.3 ( 22-24 )
• For Wed 4/17. Make sense of the definition of a strong deformation retract. Do 12.3 ( 25-34 ) HW#9 Hand in Chapter 12 ( 2, 3, 7, 18, 19 ). (No rewrites on this assignment.)
• For Mon 4/22. Do 12.3 ( 20, 22, 24, 26, 29, 32, 33, 34, 36 ) and be ready to present.
• For Wed 4/26. Work on 12.4 ( 40 – 48 ) on Van Kampen’s theorem.
  HW #10 Hand in Chapter 12 ( 14 [worth double], 27, 30, 40 ) (No rewrites on this). Discuss problems with each other! (In particular, if you missed class on Wed 4/17 when we discussed problem 14, see me or others to chat about it.)
• For Mon 4/29. Work on 12.4 ( 41, 42, 44 [think about why?], 45, 46, 47 ). Also, pick a few spaces of your own, and compute the fundamental group of those spaces using van Kampen’s Theorem in a couple of different ways. This should give everyone a chance to present something if you want to. Also, we will give people some time to present a question that you are now wondering about, based what you’ve seen so far of topology. You might read the last few sections 12.5 and 12.6 and 12.7 for ideas.
• For Wed 5/1. Work on 12.4 ( 49, 50, 51, 52, 53 ). Also, we will give people some time to present a question that you are now wondering about, based what you’ve seen so far of topology. HW for Wednesday is to prepare your Notebooks. The criteria are the same (see Tab above on ‘Notebooks’), but update your Reflections, and hand those in separately, since I’d like to keep them!