A fundamental problem in topology is that of determining whether two spaces are or not topologically equivalent. The basic idea of algebraic topology is to associate algebraic objects (groups, rings, etc.) to a topological space in such a way that topologically equivalent spaces get assigned isomorphic objects. Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces. Two spaces with inequivalent invariants cannot be topologically equivalent.

We will begin the course by making this strategy precise in terms of categories and functors. One may then describe algebraic topology as the study of topology via functors to categories of algebraic objects. It turns out that this interplay yields information and interesting results in many other fields what makes this subject central to modern mathematics. After a review of the fundamental group and the theory of covering spaces we will go on to the study of homology and cohomology and discuss a number of applications.

Prerequisites

Familiarity with group theory and point-set topology. Quotient spaces will be particularly important. Some knowledge of the idea of homotopy, the fundamental group and the theory of covering spaces will be helpful, but a motivated student will be able to follow the course by referring to the material in Chapter 1 of Hatcher's textbook

http://pi.math.cornell.edu/~hatcher/AT/ATpage.html