Duration: 19 weeks.

Number of lectures per week: 3

Assessment:

End-of-year Examination: One 3-hour examination

The web site for this course is located at http://www.maths.tcd.ie/~dwilkins/Courses/421/. It contains lecture notes from the current and from previous years.

Description:

Michaelmas Term: survey of basic point set topology (topological spaces, continuous functions, compact and connected spaces etc.); covering maps; lifting theorems; fundamental group.

Hilary Term: simplicial complexes; simplicial homology groups; basic homological algebra; the Mayer-Vietoris exact sequence and its applications.

Algebraic topology is concerned with the study of algebraic invariants (typically groups) that can be associated to subsets of Euclidean spaces (and to more general topological spaces) and that are invariant under homeomorphism or continuous deformation. Such methods are used to attack topological classification problems (e.g., the topological classification of closed surfaces). Famous results in the subject include the Brouwer Fixed Point Theorem and related theorems which have been applied in mathematical economics to prove the existence of economic equilibria in a variety of economic models. Topological methods have also become commonplace in theoretical physics in recent years.

Dec 9, 2008