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Algebraic topology: course
Contents
Description
This is an introductory, two semester course on algebraic topology and its applications. It is intended for advanced undergraduate and beginning graduate students.
Prerequisites
- sets, functions, etc,
- calculus (parts of),
- linear algebra and groups (for second half),
- proofs.
Lectures
The links below are outdated.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
- Compactness
- 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
- Homotopy
Introductory algebraic topology: review
Notes
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:
Further reading
- Bredon, Geometry and Topology.
- Kaczynski, Mischaikow, and Mrozek, Computational Homology.
- Computational topology
- Hatcher, Algebraic Topology.
- Related:
You can also reach this page by typing AlgebraicTopology.org.