Algebraic topology: course

1 Description

This is an introductory, two semester course on algebraic topology and its applications. It is intended for advanced undergraduate and beginning graduate students.

2 Prerequisites

• sets, functions, etc,
• calculus (parts of),
• linear algebra and groups (for second half),
• proofs.

3 Lectures

The links below are outdated. The source of material is currently in a draft of a book called Topology Illustrated.

3.1 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:
  • Betti numbers
  • Euler characteristic

3.2 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:
  • subcomplexes
  • products of complexes
  • quotients of complexes

3.3 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:
  • subspaces,
  • product spaces
  • quotient spaces

3.4 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

4 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:

• Topology of Surfaces by Kinsey

5 Further reading

• Bredon, Geometry and Topology.
• Kaczynski, Mischaikow, and Mrozek, Computational Homology.
• Computational topology
• Hatcher, Algebraic Topology.
• Related:
  • Homology of cell complexes: course
  • Differential forms and cohomology: course.

You can also reach this page by typing AlgebraicTopology.org.