MATH 599 (00000R)
Algebraic and Geometric Topology
Tentative Syllabus
Fall 2008
Instructor: Francis Bonahon
Summary: The course will cover several topics in algebraic
topology. It can be seen as a continuation of the combination of MATH
540 and MATH 535a, and will discuss algebraic topology from a very
geometric point of view. The emphasis will be on explaining how
research topologists think, which is not always the same as the way
they write their proofs after the fact. In particular, we will put more
prominence on bordism than is traditional when covering this type of
mathematics.
Prerequisites: This course has as prerequisites MATH
540 and (parts of) MATH 535a, or permission from the Instructor.
Textbook: We will not follow any textbook. However, the
following can be used as references for various parts of the course.
- Glen E. Bredon, Topology and Geometry, Graduate
Texts in Mathematics, Springer-Verlag, 1993.
- Victor Guillemin, Alan Pollack, Differential Topology,
Prentice Hall, 1974.
- Allen Hatcher, Algebraic Topology, Cambridge
University Press, 2002.
- John W. Milnor, Topology from a Differentiable Viewpoint,
Princeton University Press, 1997.
- Norman Steenrod, The Topology of Fiber Bundles,
Princeton University Press, 1999.
Grading: Class participation will count for 25% of the grade,
and there will be a term paper which will account for 75% of the grade.
Topics:
- Week 1. Review of singular homology and
cohomology
- Week 2. Cohomology of manifolds;
Poincaré duality.
- Week 3. Transversality theorems in
manifolds.
- Week 4-5. Bordism groups; relation to
homology.
- Week 6-7. Cohomology products and
transversality; signature of a manifold.
- Week 8. Higher homotopy groups; relation
to homology.
- Week 9-10. Obstruction theory.
- Week 11-12. Locally trivial bundles,
characteristic classes.
- Week 13-14. Classification theorems for
bundles.
Instructor’s coordinates: Francis Bonahon, DRB 264, tel:
(213) 740-2390, email: fbonahon@math.usc.edu, web page:
http://almaak.usc.edu/~fbonahon/