This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Algebraic topology: course

### From Mathematics Is A Science

## Contents

## 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The links below are outdated.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:

### 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:

### 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
**R**^{n} - Relative topology and topological spaces
- Continuous functions (maps)
- Fixed points
- Compactness
- Separation axioms
- New topological spaces from old:

### 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:

## 5 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*.