Fundamental group and covering space theory. Homology and cohomology. Manifolds and duality. CW complexes. Fixed point theorems.
| Lectures | Sections |
Topics/Comments |
|---|
| 1 lecture | Ch. 2 |
Motivation: Stokes' theorem and homology |
| Week 1 | Sec. 2.1 |
Singular homology; homology of a point and a wedge
|
| Week 2 | Sec. 2.1 |
Chain complexes and homology, chain maps and homotopy invariance
|
| Week 3 | Sec. 2.1 |
Exact sequences, 5-lemma, relative homology, homology sequence of a pair
|
| Week 4 | Sec. 2.1 |
The excision/collapse theorem for good pairs, proof
using refinements
|
| Week 5 | Sec. 2.2 |
Homology of spheres, bouquets, and suspensions.
Homology of CW complexes.
|
| Week 6 | Sec. 2.3 |
Eilenberg-Steenrod axioms, uniqueness, singular cubic theory
|
| Week 7 | Ch. 4.1, 4.2 |
The Hurewicz and Whitehead theorems.
|
| Week 8 | — |
The Lefschetz fixed point theorem, geometric
applications of Euler and Lefschetz numbers
|
| Week 9 | Ch. 3.4 |
Tensor and torsion products, homology with coefficients,
universal coefficent theorem, Kunneth formula
|
The pacing is approximate; there is an extra week which
should be added to the existing topics list.