Basic Geometry and Topology

Basic Geometry and Topology (Fall Semester) – 60330

1. Point-set topology (a quick review): topological spaces, subspaces, quotients and products. Properties of topological spaces: Hausdorff, compact, connected. Examples: spheres projective spaces, homogeneous spaces, CW-complexes.
2. Basic algebraic topology: fundamental group, covering spaces, the van Kampen Theorem, the fundamendal group of a surface.
3. Smooth manifolds: tangent bundle and cotangent bundle, vector bundles, constructions with vector bundles (sums, products symmetric and exterior powers), sections of vector bundles, differential forms, the de Rham complex, inverse and implicit function theorems, transversality.

References

• Munkres, Topology.
• Hatcher, Algebraic Topology (Chapter 1).
• Lee, Introduction to Smooth Manifolds.