Geometric topology is essentially the topological study of geometric objects, particularly manifolds and related objects. The goal of this course is as follows:

- Introduce the two key approaches to the modern study of geometric topology: algebraic and differential methods.

- Exhibit connections and interactions of different fields, particularly algebra, analysis and topology in our case, which often lead us to a discovery of deep results in mathematics.

Our treatment will use ingredients from undergraduate level algebra, calculus, and general topology. The methods and expositions will be as elementary as possible.

Students are expected to understand basic definitions and theorems in general topology, and to have basic knowledge on undergraduate level group theory and multivariable calculus. Necessary facts from algebra and calculus will be reviewed in this class.

Final Project 50%
Homework, attendance, and other aspects including participation 50%
(This is subject to change.)

The course will be based on the lecture notes of the instructor. I recommend the following as useful references:

- J. Munkres, _Topology_ (2nd Edition), Pearson, 2000.

- W. Massey, _A basic course in algebraic topology_, Springer-Verlag, 1991.

- M. Spivak, _Calculus on manifolds_, Benjamin, 1965.

- J. Milnor, _Topology from the differentiable viewpoints_, Princeton University Press, 1997 (revised reprint of the 1965 original).

- V. Guillemin, A. Pollack, _Differential Topology_, American Mathematical Society, 2010 (Reprint edition).

Topics include the following:

- Review of topological spaces and continuous maps
- Definition of manifolds and some history
- Homotopy of maps
- Fundamental groups
- Covering spaces
- Seifert van-Kampen theorem
- Applications of fundamental groups
- Knots and fundamental groups
- Alexander polynomials
- Review of multivariable calculus and inverse function theorem
- Smooth manifolds
- Regular values and transversality
- Sard's theorem
- Mod 2 degree of maps
- Orientations and degree
- Applications to vector fields

The course will consists of lectures, homework and exams/projects.