Final Exam

(There is also a web page on Courseworks.)

Time: TTh 4:10-5:25
Place: Math 307
Instructor office hours: TTh 5:30-6:30 in Math 613.
Teaching assistant: Kristen Hendricks
TA round table: M 4:30-5:30 in Math 528.

Textbook: Algebraic Topology by Allan Hatcher.

Some other relevant books:

Other online notes:

## Prerequisites.

A background in point-set topology (e.g., Math W4051) and abstract algebra (e.g., Math W4041 and W4042).

## Policies.

 Homework 30% Midterm exam 30% Final exam 40%

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.

You're welcome to work on the homework together. However, you must write up your final answers by yourself. Writing them up together is considered 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 must be cited in your homework. This includes electronic resources (including Wikipedia and Google) and human resources (including your classmates). Failure to cite sources constitutes academic misconduct.

### 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 the midterm exam date, 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, I may require a doctor's note.

## Syllabus and schedule.

Section / page numbers refer to Hatcher's book.

Date Material Textbook Announcements
9/06 CW complexes. Some familiar spaces.

Ch. 0, appendix

Welcome to G4307.
9/08 Homotopy and homotopy equivalence. Operations on spaces.

Ch. 0

9/13 The fundamental group: definition, basic properties. §1.1 Problem set 1 due.
9/15 The fundamental group of the circle. Applications. §1.1

9/20 Van Kampen's theorem: statement, examples §1.2 Problem set 2 due.
9/22 Proof of van Kampen's theorem. §1.2
9/27 Covering spaces: definitions, examples. Lifting lemmas. §1.3 Problem set 3 due.
9/29 Fundamental theorem of covering spaces. §1.3, Ramras's notes.
10/04 Simplicial homology: definition. §2.1 Problem set 4 due.
10/06 Singular homology: definition, homotopy invariance. §2.1

10/11 Relative homology. Long exact sequence for a pair. First computations and applications. §2.1

Problem set 5 due.

10/13 Mayer-Vietoris sequence. More computations. §2.2 (skipping cellular homology)
10/18 Applications: Euler characteristic, Jordan curve theorem, Invariance of Domain, commutative real division algebras. §2.2 (skipping cellular homology), 2.B Problem set 6 due.
10/20 Simplicial approximation. Lefschetz fixed point theorem. Fundamental group and homology. §2.A, 2.C

10/25

Midterm exam
10/27

Cellular homology. Introduction to degree theory.

§2.2

11/01

Cohomology: definition, examples.

§3.1 (skipping the universal coefficient theorem)

Problem set 7 due.
11/03

Cohomology: basic properties.

§3.1 (skipping the universal coefficient theorem)

11/08

Election day holiday - go vote.

11/10 Universal coefficient theorems. §3.1, 3.A Problem set 8 due.
11/15 Künneth theorem. §3.B

Problem set 9 due.

11/17

Cup product: definition, examples.

§3.2
11/22 Čech cohomology.   Problem set 10 due.
11/24 Thanksgiving holiday.
11/29 Orientability and the fundamental class. §3.3
12/01 Poincaré duality. §3.3
12/06 More duality. §3.3 Problem set 11 due.
12/08 Intersection Product.

## Final Exam

Due Monday, December 19 by 5:00 p.m.