MATH 380: Low-Dimensional Topology

Topology Now!, by Robert Messer & Philip Straffin.

We will cover the first four chapters of the text but skip a few sections occasionally. We will also do our best to cover a couple of the later chapters as time permits.

Please note that we will work on developing your independent reading skills in Mathematics and your ability to learn and use definitions and theorems. I certainly won't be able to cover in class all the material you will be required to learn. As a result, you will be expected to do a lot of reading. The reading assignments will be on topics to be discussed in the following lecture to enable you to ask focused questions in the class and to better understand the material.

Topics covered: We will cover topological equivalence, deformations, knots and links, surfaces, three-dimensional manifolds, and the fundamental group. Topics are subject to change depending on the progress of the class, and various topics may be skipped due to time constraints.

By the time you take this course, most of you should be fairly comfortable with mathematical proofs. Although this course only has multivariable calculus, elementary linear algebra, and mathematical proofs as prerequisites, students are strongly encouraged to take abstract algebra (Math 330) prior to this course or concurrently. It 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 low-dimensional, geometric topology, and algebraic topology, including knots and links, surfaces and other manifolds, and the fundamental group. The course will allow students to see the exciting geometric ideas of topology and cultivate the intuitive ideas of continuity, convergence, and connectedness. 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 topologically the same. 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.

Your overall grade will be determined as follows:

• 19% - Homework and Class Participation
• 27% - Exam 1
• 27% - Exam 2
• 27% - Final Exam

Your overall grade for the course will reflect how well the class is doing, and will be high if everyone is working hard on the homework 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.

Homework: There will be regular homework assignments which must be turned in by 4:00 pm on the due date. Follow this link for a Description of Homework requirements. Please note that you are STRONGLY ENCOURAGED to WORK IN GROUPS of 2 or 3 students. However, each student is responsible for writing up his or her own homework to be handed in.

Exams: There will be two Midterm Exams and one Final Exam. Exams are closed book, closed notes, closed friends, and open brain. Calculators, cell phones, iPods, and other electronic devices will NOT be permitted in exams. 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. It is imperative that you keep up with the reading assignments. This will help you answer my questions and help you ask more essential, thought-provoking questions during the lectures.

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:

Accommodations: SUNY Geneseo will make reasonable accommodations for persons with documented physical, emotional, or cognitive disabilities. Accommodations will be made for medical conditions related to pregnancy or parenting. Students should contact Dean Buggie-Hunt in the Office of Disability Services (tbuggieh@geneseo.edu or 585-245-5112) and their faculty to discuss needed accommodations as early as possible in the semester.