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

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.