Unit 1: Simplical Complexes and Homology Groups
Definition and elementary properties-Siplices, Simplical complexes and Simplical maps, homology Groups, Zero-dimensional Homology, The Homology of a cone, Homomorphism induced by Simpicial Maps, Chain Complexes and Acyclic Carriers.
Unit 2: Relative Homology and Eilenberg-Steenrod Axioms
Relative Homology, the Exact Homology Sequence, the Zig-zag Lemma, Mayer-Vietoris Sequences, the Eilenberg-Steenrod Axioms, the Axioms for Simplicial Theory. Categories and functors.
Unit 3: Singular Homology
The Singular Homology Groups, the Axioms for Singular Theory, Excision in Singular Homology, Mayer-Vietoris Sequences, the Isomorphism between Simplical and Singular Homology, More on Quotient Spaces, CW Complexes and their Homology.
Unit 4: Cohomology and Homological Algebra
The Hom functor, Simplical Cohomology Groups, Relative Cohomology, Cohomology theory, The Ext Functor, The Universal Coefficient Theorem for Cohomology, Torsion Products, The Universal Coefficient Theorem for Homology, Kunneth Theorems (for cohomology and homology proofs can be omitted).
Unit 5: Applications
Local Homology Groups and Manifolds, the Jordan Curve Theorem, Projective spaces and Lens Spaces, The Cohomology Ring of a Product Space.