Algebraic Topology

Algebraic topology is study of topological spaces, usually via the tools of abstract algebra, such as groups, rings and fields. As with many fields of mathematics, the goal is to classify the objects of interest up to some equivalence relation, in this case we wish to classify topological spaces up to homeomorphism (although we will find that the best we can often do is homotopic equivalence).

• The fundamental group
• The fundamental group and covering spaces
• Simplicial homology
• Singular homology
• Cellular homology