Instructor: Sasha Voronov
10:10-11:00 MWF, VinH 207 (Note a room change!)
Prerequisite: Math 8301 or basic knowledge of homology or cohomology,
the fundamental group, and manifolds
Textbook: Allen Hatcher. Algebraic Topology. Cambridge University Press,
Cambridge, 2002. $29.95 paperback. Also available on-line at
http://www.math.cornell.edu/~hatcher/#ATI
Description: The goal of the one-year course is to study the powerful machinery of algebraic topology and provide necessary background for a student planning to work in the fields of algebraic topology, algebraic geometry, geometric topology, symplectic geometry, K-theory, gauge theory, mathematical physics, etc. The course may be considered an independent course in algebraic topology covering complementary topics to those studied in Math 8301-8302: Manifolds and Topology. However, the full mastery of Math 8301-8302 is not required: you just need to know some basics of homology theory and the fundamental group. We plan to study most of the following topics, depending on the interests of the students: homology and cohomology theories (including universal coefficient theorems, products in singular cohomology, fiber bundles, and homology with local coefficients), homotopy theory (including fibrations and cofibrations, homotopy groups, exact homotopy sequences, the Whitehead and Hurewicz Theorems, Whitehead products, the James construction), simplicial sets, fiber bundles and classifying spaces, Hopf algebras and graded Lie algebras, spectral sequences (including those of Bockstein, Serre and Eilenberg-Moore), localization and Bousfield-Kan completion, generalized homology and stable homotopy (including Brown's representability theorem and spectra), cohomology operations and Steenrod algebra.
Requirements: There will be homework throughout the year, but no exams. One in-class topics presentation per year will be expected.
More information: Contact the instructor at
voronov@math.umn.edu
VinH 324
624-0355