Topological Manifolds
The definition and properties of topological manifolds as locally Euclidean spaces
Algebraic Topology
Homology
Definitions and properties of simplices
Homotopy
Classifying spaces via topological invariants and the fundamental group
Covering Spaces
Equivalent conditions for simply connected, covering spaces, and the Seifert–van Kampen theorem
Computation of Homology
Practical computation of homology via relative homology and Mayer–Vietoris
Cohomology
The definition of cohomology and the universal coefficient theorem
Acyclic models theorem
The acyclic models theorem for categories with models and its applications
Cup Product
The exterior product in cohomology, the definition of cup product, and the resulting ring structure
Poincaré Duality
Duality between homology and cohomology via orientation sheaves and fundamental classes