Math 524: Algebraic Topology

Course Description

Algebraic topology uses techniques from abstract algebra to study how (topological) spaces are connected. Most often, the algebraic structures used are groups (but more elaborate structures such as rings or modules also arise). A typical approach projects continuous maps between topological spaces onto homomorphisms between the corresponding groups. This course will introduce basic concepts of algebraic topology at the first-year graduate level. We will follow mostly the book Elements of Algebraic Topology by James R.~Munkres, and cover in a fair bit of detail the topics on homology of simplicial complexes, relative homology, cohomology, and the basics of duality in manifolds (selected Sections from Chapters 1–5 and 8). We will also refer to the book Algebraic Topology by Allen Hatcher for some of the topics. But unless there is strong interest, we will not cover topics related to the fundamental group. We will stress geometric motivations as well as applications (where relevant) throughout the course. Prerequisites: Some background in general topology as well as abstract algebra, both at the undergraduate level, will be assumed. In particular, familiarity with the concepts of continuous functions, connectedness, and compactness, as well as with the concepts of groups, homomorphisms, fields, and vector spaces will be helpful to follow the course. But no particularly deep theorems from these topics will be needed. Some flexibility could be afforded as far as this background is concerned—please contact the instructor if you have doubts.