Algebraic topology: course
This is an introductory, two semester course on algebraic topology and its applications. It is intended for advanced undergraduate and beginning graduate students.
The links below are outdated. The source of material is currently in a draft of a book called Topology Illustrated.
Part 1. Introduction to algebraic topology
- Topology: an introduction
- Homology as an equivalence relation
- Cell decomposition of digital images
- The algorithm for computing homology of 2D binary images
- Cubical complexes
- Homology of cubical complexes
- Homology and algebra
- Topological invariants:
Part 2. Complexes
- The topology of the Euclidean space
- Realizations of cubical complexes
- Continuity of functions of several variables
- Maps and homology
- Chain complexes
- Chain maps
- New complexes from old:
Part 3. Overview of point-set topology
- Topology in calculus
- Introduction to point-set topology
- Neighborhoods and topologies
- Open and closed sets in Rn
- Relative topology and topological spaces
- Continuous functions (maps)
- Fixed points
- Separation axioms
- New topological spaces from old:
Part 4. Homology groups
- The algebra of chains
- Cell complexes and simplicial complexes
- Manifolds and surfaces
- Homology in dimension 1, Homology in dimension 2
- Chain complexes, cycle groups, boundary groups as vector spaces
- Review of quotients of vector spaces
- Homology as a vector space
- Boundary operator
- Properties of homology groups
- Homology of surfaces
- Euler-Poincare formula
- Cell maps, simplicial maps and their homology maps
- Homology as a group
I would call the course "Applied algebraic topology" if I didn't think algebraic topology is applied enough as it is.
The content is based on the complete set of lecture notes for a course taught by Peter Saveliev in Fall 2009/Spring 2010 at Marshall University. Some of it was inspired by this book:
- Bredon, Geometry and Topology.
- Kaczynski, Mischaikow, and Mrozek, Computational Homology.
- Computational topology
- Hatcher, Algebraic Topology.
You can also reach this page by typing AlgebraicTopology.org.