1
Topological Manifolds
Definition and properties of topological manifolds as locally Euclidean spaces
2
Homology
Definition and properties of a simplex
3
Homotopy
Classifying spaces via topological invariants and the fundamental group
4
Covering Spaces
Equivalent conditions for simply connected spaces, covering spaces, and Seifert-van Kampen theorem
5
Computing Homology
Practical computation of homology via relative homology and Mayer-Vietoris
6
Cohomology
Definition of cohomology and the universal coefficient theorem
7
Acyclic Models Theorem
Acyclic models theorem on a category with models and its applications
8
Cup Product
Exterior product of cohomology, cup product definition, and ring structure
9
Poincaré Duality
Duality between homology and cohomology via the orientation sheaf and fundamental class
10
Stiefel-Whitney Characteristic Classes
Vector bundles, Stiefel-Whitney characteristic classes, and the infinite Grassmannian
11
Characteristic Classes of Vector Bundles
Euler, Chern, and Pontryagin characteristic classes
12
Classifying Spaces
Classification of principal G-bundles and construction of the classifying space BG