Spring 2026, Prof. Manuel Rivera, Purdue University
This course will be an introduction to algebraic topology. The main topics to be covered will be:
1) singular homology and Eilenberg-Steenrod axioms
2) computational methods based on cellular homology of CW complexes and homological algebra
4) Cohomology ring
5) Poincaré duality of manifolds
Along the way will introduce some basic category theory, simplicial techniques, and develop the necessary homological algebra. These tools were born in the context of algebraic topology but are widely used throughout modern mathematics.
Instructor: Prof. Manuel Rivera (manuelr at purdue dot edu)
Textbooks: The lectures will not strictly follow any textbook. They will be roughly based on these notes written by Haynes Miller:
http://math.mit.edu/~hrm/papers/lectures-905-906.pdf
There are lots of other textbooks that treat algebraic topology more or less at the level of this course.
Bredon, Glen E. Topology and Geometry
Davis, James F., and Paul Kirk. Lecture Notes in Algebraic Topology
Dold, Albrecht. Lectures on Algebraic Topology
Hatcher, Allen. Algebraic Topology. (Chatty)
May, J. Peter. A Concise Course in Algebraic Topology
Munkres, James R. Elements of Algebraic Topology
Shastri, Anant R. Basic Algebraic Topology
Spanier, Edwin H. Algebraic Topology
Vick, James W. Homology Theory: An Introduction to Algebraic Topology
Pre-requisites: We will assume familiarity with basic point set topology, covering spaces, and the fundamental group, as discussed in MA571. We will also assume some basic results in algebra such as the classification theorem of finitely generated abelian groups.
Grading: Homework 75 %, one exam 25%
riveramanuel.com 