Department Syllabus
MAT 215C: Topology
| When taught: |
Spring, alternate years |
| Suggested text ($): |
Hatcher, Dmitry Fuchs' handouts |
| Units/lectures: |
4 units; lecture/term paper or discussion section |
| Prerequisites: |
Graduate standing or consent of instructor. |
Catalog description
Fundamental group and covering space theory. Homology and cohomology. Manifolds and duality. CW complexes. Fixed point theorems.
Suggested schedule
| Prepared by: |
Dmitry Fuchs, Greg Kuperberg |
| Posted: |
May 2009 |
| Lectures | Sections |
Topics/Comments |
|---|
| Week 1 | Sec. 3.1 |
Definition of cohomology and properties: homotopy invariance,
sequences of pairs and triples, refinement, excision
|
| Week 2 | — |
Obstruction theory, e.g., for maps to classifying spaces and spheres
|
| Week 3 | Sec. 3.1 |
Ext functor, cohomology universal coefficents
|
| Week 4 | Sec. 3.2 |
Cup products, outer products, Hopf's invariant, cap products
|
| Week 5 | Sec. 3.3 |
Pseudo-manifolds, fundamental classes, homological manifolds,
Poincare duality |
| Week 6 | — |
Intersection products as the Poincare dual of cup products,
The Lefschetz number as a count of fixed points
|
| Week 7 | Sec. 3.3 |
Relative Poincare duality, oriented cobordism
|
| Week 8 | — |
Alexander duality
|
Again, there are two extra weeks; the listed pacing is approximate and
probably too fast. Additional topics: Fiber bundles, classification
of lens spaces, Twisted Poincare duality for non-orientable manifolds,
statements of manifold classification results.