MATH 221A: TOPOLOGY III
Instructor: Adam Simon Levine, levinea@brandeis.edu
Office hours: TBA
Textbooks: Most of the material in the course can be found in John Milnor and
James Stasheff’s classic book Characteristic Classes (Princeton U.P., 1974), and
Allen Hatcher’s online notes Vector Bundles and K-Theory, which can be found
at http://www.math.cornell.edu/~ hatcher/VBKT/VBpage.html. We will also use
Hatcher’s Algebraic Topology for some topics. Exercises will be taken from all three
of these sources.
Prerequisites: Math 121ab (Topology I-II) and 110a (Geometric Analysis). 110a may
be taken concurrently.
Assignments: There will be regular homework assignments and a final exam, date
TBA.
Course outline:
(1) Introduction to higher homotopy groups.
(2) Fiber bundles and vector bundles. Basic examples: tangent bundles, principal
bundles, Hopf bundles. Constructions such as pull-backs, Whitney sum, tensor
product, associated bundles, and projectivization.
(3) Classifying spaces and Grassmannians.
(4) Cohomology of fibrations, Leray-Hirsch theorem, Thom isomorphism theorem,
Gysin sequence.
(5) Construction of Stiefel-Whitney and Chern classes via the Leray-Hirsch theorem.
(6) Euler and Pontryagin classes.
(7) Obstruction theory.
(8) Further topics as time permits.
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