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

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.