Algebraic Topology

a.

Elementary Homotopy and the Fundamental Group: Basic homotopy of maps, deformation retracts, homotopy equivalences and homotopy type. The fundamental group and its main properties. Computation of the fundamental group. The theory of covering spaces and its relation to the fundamental group.

b.

Homology Theory: Construction and basic properties of singular homology theory including excision, the homotopy property and exactness. The Eilenberg-Steenrod axioms. CW complexes and cellular homology theory. Computation of homology groups. Applications of homology theory. Elementary cohomology theory.

c.

Homological Algebra: Exact sequences, chain and cochain complexes, chain homotopy and the exact homology sequence of a short exact sequence of chain complexes. Introduction to categories and functors.