This course is for advanced bachelor students in mathematics. The aim of the course is to give an introduction to the basic concepts of algebraic topology, namely, homotopy and homology. These are methods of assigning groups (usually abelian groups) to a topological space that aid in deciding the basic question of homotopy theory: when are two topological spaces homotopy equivalent? as well as many other fundamental questions. These invariants are also important in the study of differetiable manifolds, especially through the de Rham theorem, which gives a means of computing the homology of a manifold in terms of integrals of differential forms. Algebraic topology also is an important component of modern algebraic geometry, giving both invariants useful for the study of algebraic varieties over the complex number, as well as providng motivation for many important purely algebraic constructions, such as étale cohomology.
Students interested in taking this course should have had at least Linear Algebra and Analysis I. Algebra I is not required, but is recommended. We will develop additional algebraic tools, such as some elementary homological algebra, as needed. The concepts from general topology, such as the definition of a topological space and continuous maps, compactness, product and quotient spaces and metric topologies, will be covered in the beginning of the course.