**Duration:** 21 weeks.

**Number of lectures per week:** 3

**Assessment:**

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

**Description: **

**Michaelmas Term:**
survey of basic point set topology
(topological spaces, continuous functions, compact and connected
spaces etc.); covering maps; lifting theorems; winding numbers;
applications to topology in the plane (the Fundamental Theorem
of Algebra, the Brouwer Fixed Point Theorem and Hairy Ball Theorem
in the two-dimensional cases; the fundamental group.

**Hilary Term:**
simplicial complexes and homology groups; basic homological
algebra; the Mayer-Vietoris exact sequence and its applications;
closed surfaces and their topological classification.

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.

Course 212 (topology) is not an essential prerequisite for this
course, though those who have not taken Course 212 may find that
extra effort may be required of them at the beginning of the course,
where the essentials of basic point set topology (e.g., the notion of
a continuous function between topological spaces) will be briefly
and quickly reviewed.

Jun 10, 1998