This course, CRN 94034, is an introduction to algebraic topology. It will be followed in Spring 2009 by Math 6324, TS: Algebraic Topology II. The main topics will be the fundamental group, singular and simplicial homology, and cohomology. There will be weekly graded homework, but no tests or exams. The main text will be Algebraic Topology, by Allen Hatcher.

In algebraic topology, one approaches problems in topology (such as the question of when two topological spaces are homeomorphic, or when a continuous map from a topological space to itself has to have a fixed point) by attaching algebraic invariants to the spaces. The fundamental group is a simple invariant to define, but the group need not be abelian and so it can be hard to work with. In homology and cohomology theories, one gets a sequence of abelian invariants.

I will be assuming standard topological concepts from the beginning. If you haven't had Math 4324 (topology) or Math 4225-4226 (real analysis, which includes a good grounding in metric space topology) or their equivalents, you should contact me before signing up for the course. I will also be assuming algebraic concepts, say on the level of Math 4124-5114.