Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 215C: Topology
Approved: 2009-05-01, Dmitry Fuchs, Greg Kuperberg
Units/Lecture:
Spring, alternate years; 4 units; lecture/term paper or discussion section
Suggested Textbook: (actual textbook varies by instructor; check your
instructor)
Algebraic Topology, Allen Hatcher, Cambridge Univ, ($30), Dmitry Fuchs' handouts
Search by ISBN on Amazon: 0521795400
Search by ISBN on Amazon: 0521795400
Prerequisites:
Graduate standing or consent of instructor.
Course Description:
Fundamental group and covering space theory. Homology and cohomology. Manifolds and duality. CW complexes. Fixed point theorems.
Suggested Schedule:
| 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 |
Additional Notes:
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.