(See also web page on Courseworks.)

Time: TTh 4:10-5:25 p.m.

Place: Math 417

Textbook: Topology (second edition) by James Munkres.

Office hours: Monday, 11:00-11:59 a.m.; Tuesday, 5:30-6:30 p.m.

Teaching assistant: Thomas Peters

Help room hours: Monday 12:00-3:00 p.m.

Final exam: Tuesday Dec. 16, 4:10-7:00 p.m. in Math 417.

Announcements: see Courseworks page.

Syllabus | Problem sets | Policies | Advice |
---|

## Prerequisites.

One semester of "Introduction to Modern Algebra" (Math W4041) or equivalent is required: the second half of "Topology" will use some group theory, and we will assume some familiarity with writing proofs. A course in analysis (Math V1208 or W4061) is recommended, as well: many of the ideas of the first half of the semester are generalizations of ideas from analysis, and it is easier to develop an intuition in the more restricted setting.

## Description and goals.

Topology is, somewhat tautologically, the study of topological spaces and continuous maps. Roughly, it focuses on qualitative, as opposed to quantitative, properties of spaces and functions.

The first half of the semester will be devoted to point-set topology -- the axioms of topological spaces and what they imply. These notions are central to modern mathematics, and are core tools in almost all parts of the field. The second half of the semester will be an introduction to algebraic topology, via the fundamental group and a little covering space theory. This is the starting point for most of modern topology, and one of the first places one encounters the viewpoint of modern mathematics.

## Policies.

### Grading

Homework | 35% |

The Big Problem | 5% |

Midterm exam | 25% |

Final exam | 35% |

The lowest homework score will be dropped.

### Homework

Problem sets are due on Tuesdays at the *beginning of class*, except as noted below. If you can't make it to class, put it in my mailbox before class in Math 509 (to the right of the door when you enter).

You're welcome to work on the homework together. However, you must write up your final answers by yourself. I consider writing them up together cheating.

You are also generally welcome to use any resources you like to solve the problems. However, any resource you use other than the textbook (Munkres) *must* be cited in your homework. This includes electronic resources (including Wikipedia and Google) and human resources (including the help room and your classmates). Failure to cite sources constitutes academic misconduct.

### The Big Problem

The Big Problem is a long term assignment, a bit like a midterm paper. Details about it are here (PDF).

### Students with disabilities

Students with disabilities requiring special accommodation should contact Office of Disability Services (ODS) promptly to discuss appropriate arrangements.

### Missed exams

If you have a conflict with any of the exam dates, you must contact me ahead of time so we can make arrangements. (At least a week ahead is preferable.) If you are unable to take the exam because of a medical problem, you must go to the health center and get a note from them -- and contact me as soon as you can.

## Syllabus and schedule.

Note: "+" indicates material not discussed in the textbook. Material in parentheses will probably be omitted from class (but discussed in problem sets).

Date | Material | Textbook | Announcements |
---|---|---|---|

09/02 | Metric spaces and continuous maps. | + (Recommend: Rudin, Principles of Mathematical Analysis, §2.2) | Welcome to W4051. |

09/04 | Topological spaces and continuous maps. | §12, 13, most of 18, 20 | |

09/09 | Examples of topologies. | §13, 14, 15, 16 | Problem set 1 due. |

09/11 | Closures, interiors, the Hausdorff axiom. Infinite products of spaces. | §17, 19, (rest of 18, 20, 21) | |

09/16 | The quotient topology. CW complexes. | §22, + | Problem set 2 due. |

09/18 | Connectedness and path connectedness. | §23, 24, 25, + | |

09/23 | Compactness. | §26, 27 | Problem set 3 due. |

09/25 | More on compactness. | §28, 29 | |

09/30 | Countability axioms. | §30 | Problem set 4 due. |

10/02 | Separation axioms. | §31 | |

10/07 | Normal spaces and the Urysohn lemma. | §32, 33 | Problem set 5 due. |

10/09 | The Urysohn Metrization Theorem and the Tietze Extension Theorem. | §34, 35 | |

10/14 | Midterm |
||

10/16 | Review of groups. Homotopies. | §51, + | |

10/21 |
The fundamental group. The fundamental group of the circle (announcement). The Brouwer fixed point theorem. | §52, 55 | Problem set 6 due. |

10/23 | More applications of the fundamental group. | §56, 57 | |

10/28 | Covering spaces. | §53 | Problem set 7 due. First draft of Big Problem due. |

10/30 | The fundamental group of the circle. | §54 | |

11/4 | Election day -- go vote. | ||

11/6 | Deformation retracts and homotopy equivalence. | §58 | Problem set 8 due. |

11/11 | Free groups, amalgamated products, generators and relations. | §67-69 | |

11/13 | Statement of the van Kampen theorem. Examples. | §70, 71, + | |

11/18 | Proof of the van Kampen theorem. | §70 | Problem set 9 due. |

11/20 | The fundamental group of a CW complex. | §72, 73, + | Final draft of Big Problem due. |

11/25 | A further topic: the Jordan curve theorem; knot theory; classification of surfaces; categories and functors; higher homotopy groups; or homology. (Class vote.) | + | Problem set 10 due. |

11/27 | Thanksgiving holiday. | ||

12/2 | The fundamental group and covering spaces, I. | §79, 80 | |

12/4 | The fundamental group and covering spaces, II. | §81, 82 | Problem set 11 due. |

## Problem sets.

- Problem set 1 (PDF). Due Sept. 9.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 2 (PDF). Due Sept. 16.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 3 (PDF). Due Sept. 24.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 4 (PDF). Due Oct. 1
.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 5 (PDF). Due Oct. 8.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 6 (PDF). This problem set is
*optional*. Due Oct. 15. - Problem set 7 (PDF). Due October 22.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 8 (PDF). Due October 29.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 9 (PDF). Due November 6.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 10 (PDF). Due November 19.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 11 (PDF). Due November 26.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 12 (PDF). Due December 10.
- Solutions are now available on Courseworks, in the class files section.

- Problem set 13 (PDF). This is definitely just for fun. Don't turn it in.
- By popular request: the bonus problem (PDF) from the final exam.

## Other advice.

**Reading mathematics.** You are expected to read the sections in the textbook *before* coming to class. It's usually only a few pages, so read it carefully. Note down the questions you have; I would
expect you to have at least one per page. Read the section again
after class. See which questions you now understand. Think about the
remaining questions off and on for a day. See which you now
understand. Ask someone (e.g., me) about the questions you still have left.

**Linguistics and mathematics.** I will make a big deal about using words in the course in a grammatically correct way. For example, for a vector space V, one says "a basis for V" but "the dimension of V." This will help you: often the grammar prevents you from making statements which are incorrect (or meaningless). So pay attention to it.

**Getting help.** If you're having trouble, get help *immediately*. Everyone who works seriously on mathematics struggles. But if you don't get help promptly you will soon be completely lost. The first places to look for help are my office hours and the course TA in the help room. Talking to your other classmates can also be helpful.

**Teaching to learn.** The best way to learn mathematics is to explain it to someone. You'll find that, particularly in office hours, I'll try to get you to explain the ideas. You should also try explaining the material to each other. The person doing the explaining will generally learn more than the explainee. Another thing to try is writing explanations to yourself, in plain English or as close as you can manage, of what's going on in the course. File them somewhere, and then look back at them a few days later, to see if your understanding has changed.