Peter Teichner
Math 215B, Algebraic Topology
Spring 2006, Tu/Th 11:00-12:30 in 39 Evans
Office hours Wednesday 10-12 in 703 Evans.
Assistant: Jana Comstock, office hour Thursday 2-3 in 1047 Evans
We'll continue the qualitative study of topological spaces started in the fall. Topics will include homotopy theory, fibre bundles, spectral sequences, characteristic classes, K-theory and Bott periodicity. There will be weekly homework on this webpage and a course paper (in lieu of a final).
References include:
J. Milnor and J. Stasheff, Characteristic Classes
G. Bredon, Geometry and Topology
A. Hatcher, Algebraic Topology: [HAT]
A. Hatcher, Vector Bundles and K-Theory: [HK]
A. Hatcher, Spectral sequences in algebraic topology: [HSS]
Term papers (the first few can be downloaded thanks to the generous authours)
Serre spectral sequences and applications,
by David Brown,
Gauss-Bonnet and uniformization,
by Santiago Canez,
Delta and Simplicial Objects
and Chain Complexes,
by Matthias Goerner
Free actions of groups and cohomology,
by Tye Lidman,
An introduction to Hochschild and cyclic homology, by Hannes Thiel,
The rational bordism ring,
by Valentin Tonita.
Rational homotopy theory.
Cech and sheaf cohomology.
K-theory of operator algebras.
Conformal field theory.
Cell structure of Grassmannian.
week of Jan. 23:
Review of 215A and Outlook on 215B. Homework:
Hand in a page with your name and mathematical interests.
week of Jan. 30: HAT, chapter 4.1. Homework:
HAT, pages 358-359, numbers 4, 10, 11, 14.
week of Feb. 6: HAT, appendix pages 519-525, and chapter 4.2 pages 360-366. Homework:
HAT, page 389, numbers 2,3 and page 529, numbers 1, 4.
week of Feb. 13: HAT, chapter 4.2 pages 366-388. Homework:
HAT, pages 390-392, numbers 15, 23, 28, 35.
week of Feb. 20: HAT, chapter 4.3 pages 393-418. Homework:
HAT, page 419, numbers 1, 6, 19, 22.
week of Feb. 27: HAT, chapters 4.A-D pages 421-446. Homework:
HAT, page 391, number 29 (including #2 of 3.E), page 447, numbers 1, 4, 10.
week of March 6: HSS, chapter 1 pages 1-22. Homework:
HSS, page 23, numbers 1-4.
week of March 13: HAT, chapters 4.E-H pages 448-466,. Homework:
Deduce the Leray-Hirsch Theorem from the Serre spectral sequence
HAT, page 455, number 1, page 460, numbers 2,3.
week of March 20: HK, pages 1-34. Homework:
1. Formulate and prove the (real and complex) splitting principle. Hint: Leray-Hirsch.
2. Find the formula for the second Stiefel-Whitney class of a tensor product.
3. Prove that n+1 is a power of 2 if n-dimensional real projective space is parallalizable.
4. If n is a power of 2, prove that n-dimensional real projective space does not immerse into Euclidean (2n-2)-space.
Homework after March 27: write term paper.
April 18: submit rough version of term paper.
Homework in week of April 18: referee term paper (peer review).
May 2: submit final version of term paper.