Homology of cell complexes: course

Book

Description

This is an introductory, one semester course on algebraic topology and its applications. It is intended for advanced undergraduate and beginning graduate students. It is independent of point set topology.

Prerequisites

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

Lectures

The links below are outdated.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

Introduction to algebraic topology

• Topology: an introduction
• Quotient sets, the key construction
• Homology as an equivalence relation
• Cell decomposition of digital images
• Betti numbers
• Euler characteristic

Complexes

• Cell complexes
• Boundary operator
• Chain complexes
• New complexes from old:
  • subcomplexes
  • products of complexes
  • quotients of complexes

Homology

• The algebra of chains
• 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
• Properties of homology groups
• Euler-Poincare formula
• Chain maps and their homology maps

Homology and beyond

• Relative homology
• Cohomology
• Chains vs cochains

Further reading

• Bredon, Geometry and Topology.
• Computational topology
• Hatcher, Algebraic Topology.
• Related: Introduction to differential forms: course.