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 215B: Topology
Approved: 2009-05-01, Dmitry Fuchs, Greg Kuperberg
Units/Lecture:
Winter, 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 |
|---|---|---|
| 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 |
Additional Notes:
The pacing is approximate; there is an extra week which should be added to the existing topics list.