| Date | Topic |
| Review of and Crash Course in General Topology |
| Aug 28 | Topological Vocabulary:
Ch. I, Sections 2-3 |
| Aug 30 | Mostly connectedness:
Ch. I, Section 4; plus definition of Hausdorff: Ch. I, Section 5 |
| Sep 1 | Compactness, proper mappings, product topology: Ch. I, Sections 7-8 |
| Sep 6 | Quotient topology: Ch. I, Section 13 |
| Sep 8 | Homotopy: Ch. I, Section 14 |
| Sep 11 | More Homotopy: Ch. I, Section 14 |
| A brief tour of differentiable manifolds:
This material provides motivation and examples for what we will be studying.
For those interested, an undergraduate level treatment can be found in Guillemin and Pollack, Differential Topology, which is inspired by the well-known introduction by Milnor, Topology from the Differentiable Viewpoint. |
| Sep 13 | Definitions of (smooth) manifold and examples: Ch. II, Sections 2-4 |
| Sep 15 | Poincare Conjecture, Exotic Spheres, the graph of the absolute value function in
R2 as a manifold (compare with the discussion of the cone
following Definition II.4.5), manifold with boundary (Definition II.2.7),
brief discussion without proof of classification of one and two dimensional
compact connected manifolds, allowing boundary in the 1-dimensional case,
tangent vectors & differentials (Ch. II, Sec. 5),
Tangent Bundles (Ch. II, Sec. 9), and the Whitney Embedding Theorem
without proof (Ch. II, Sec. 10) |
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