MATH 380: Algebraic Topology

There is no textbook for this course. However, you might find the following textbook useful as a resource. It is available for free online:

Algebraic Topology, by Allen Hatcher.

Topics covered: We will cover various topics related to both topology and algebra, including topological spaces and equivalences, groups, free groups, homomorphisms, group presentations, fundamental group, homology, and cohomology. Topics are subject to change depending on the progress of the class.

Welcome to our course in Algebraic Topology. By the time you take this course, you should be at the point where you are fairly comfortable with various techniques of proof. Students are required to take Abstract Algebra (Math 330) prior to this course or concurrently. This course requires a certain level of mathematical sophistication. There will be a lot of new terminology you must learn, and we will be doing a significant number of proofs.

Topology is the study of spaces and sets and can be thought of as an extension of geometry. It is an investigation of both the local and the global structure of a space or set. There are several areas of topology, and most of them have some overlap. Some of these include General Topology (or Point-Set Topology), Geometric Topology, Algebraic Topology, Differential Topology, Low-Dimensional Topology, High-Dimensional Topology, and various others. The usual topology course offered at Geneseo is General Topology (Math 338), which has its foundation in set theory, and this course will be significantly different from it. This course is an introduction to some topics in algebraic topology, including the fundamental group, homology, and cohomology. For these purposes, we will also discuss various algebraic topics including group presentations, free groups, free abelian groups, and torsion groups. The course will allow students to see how algebraic concepts and techniques can be used to study topological spaces. This will help students see the interrelation between two main branches of mathematics, topology and abstract algebra.

The motivation behind topology is that some geometric problems do not depend on the exact shape of an object but on the way the object is put together. For example, the square and the circle are geometrically different, but they have many properties in common: they are both one dimensional objects and both separate the plane into two parts. Similarly, a donut and a coffee cup are topologically the same even though they look completely different. Much of the study of topology comes from setting aside our preconceived notions of "shape" involving size, length, flat, straight, or curved, and realizing that a circle and a square are really the same thing.

Upon successful completion of this course, students will be able to:

• Determine the fundamental group, homology, and cohomology of various topological spaces, and
• Use algebraic invariants to distinguish topological spaces up to homeomorphism and/or homotopy type.

Your overall grade will be determined by some magical and mysterious formula that combines assessment of your performance in each of the following categories:

• Homework
• Class Participation
• Exams

Your overall grade for the course will reflect how well you are doing and will be high if you are working hard on the homework, participating in class discussions, and doing well on the exams. Many of the questions on the exams will be in the same spirit as the homework questions. Therefore understanding how to do all the homework questions will enable you to do well on the exams.

Communication with your peers will be a focal point. In addition to working in pairs on your homework assignments, there will also be regular discussions and presentations given in class. Students will be asked to present their solutions to homework problems and proofs/examples of concepts we are learning in class. The best way to truly understand a concept is to be able to explain it to someone else. Talking about math is just as important as thinking and writing math. Our goal is to become more comfortable with the necessary communication.

Homework: There will be semi-regular homework assignments. Follow this link for a Description of Homework requirements. You will be assigned a partner that you will be working with for the semester on homework and in class. (Depending on how the class does, I may have you switch partners during the semester.) Each student should work out every problem that is assigned, but you and your partner will only turn in one combined final version of each assignment. Please note that you are welcome to work in groups of as many people as you want, but you and your partner are responsible for writing up and handing in your own pair's homework. Each pair of students should submit their own work, not a handwritten copy of someone else's. Also, you should feel free to write out your assignments by hand, but I encourage you to typeset your homework using LaTeX. Each individual student is STRONGLY ENCOURAGED to think about and attempt the problems on their own before meeting with their partner. When you are about to tackle the homework assignment, you should use this strategy: THINK, PAIR, SHARE. When working in groups, please be careful that you are actively participating in the process. Please be careful that you are able to work all of the problems on your own before the exam time arrives, with no coaching from a friend. Please use whatever resources aid you in learning the material, including computer assistance, office hours, other students, professors, other math books, etc. However, plagiarism will NOT be tolerated, and this includes coping or paraphrasing someone else's work.

Exams: There will be one or two Midterm Exams and one Final Exam. I have no idea what format the exams will be, but I will let you know ahead of time. The dates of the exams will be decided a week or two in advance.

Class Participation: Class participation will be based on your willingness to ASK and ANSWER questions in class. There will be active discussion at times, and you will also be required to present some proofs to the class. It is imperative that you come to class, pay attention, and be INVOLVED in the discussions.

It is essential not to fall behind because each lecture is based on previous work. If you have trouble with some material, SEEK HELP IMMEDIATELY in the following ways: