Email: alishahi@math.columbia.edu
Time and Place: Monday-Wednesday: 1:10pm-2:25pm at Math 307
Course webpage: http://www.math.columbia.edu/~alishahi/AlgTopII.html
Office hours: Tuesday-Thursday:noon-1pm or by appointment; in Math 613
Teaching Assistant: Semon Rezchikov, skr2158@columbia.edu
TA Office Hours:
Textbook:
Some other relevant books:
Prerequisite: Algebraic Topology I
Homework: There will be problem sets every week, due at the beginning of class on Wednesdays.
The lowest homework scores will be dropped.
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6
Homework 7
Homework 8
Homework 9
Exam: There will be a midterm exam in class, and a take-home final exam.
Midterm: March 7
Final: Take-home
There will be no make-up exams.
Grading: The final course grade will be determined by:
Homework: 30%
Midterm : 30%
Final: 40%
Tentetive Schedule:
| Date | Material | Textbook |
|---|---|---|
| 1/17 | Introduction to the course.Vector bundles. | M-S, chapters 2,3 |
| 1/22 | Stiefel-Whitney classes: axiomatic description, basic examples. | M-S, chapter 4 |
| 1/24 | Basic Applications of S-W classes and Universal vector bundles | M-S, chapters 4,5 |
| 1/29 | CW structure of the Grassmannian and the cohomology ring of the Grassmannian | M-S, chapter 6-7 |
| 1/31 | Uniquness and constrcution and Stiefel-Whitney classes | M-S: chapter 7, H: section 3.1 |
| 2/5 | The spectral sequence associated to a filtered complex | Hutchings notes on spectral sequences, G-H |
| 2/7 | Cancelled | Make-up class on Feb 20 |
| 2/12 | The Leray-serre spectral sequence for the homology of a Serre fibration--Examples | Hutchings notes on Spectral sequences, Hatcher's book chapter 1 |
| 2/14 | Homology with twisted coefficients, Leray-Serre spectral sequence over a non-simply-connected base | Hutchings notes (1) higher homotopy and obstruction theory (2) spectral sequences |
| 2/19 | Cohomological Leray-Serre spectral sequence, multiplicative structure and examples | Hutchings notes on Spectral sequences, Hatcher's book chapter 1 |
| 2/20 | Applications: Thom isomorphism theorem and the Leray-Hirsch theorem, Introduction to oriented bundles and Euler class | M-S,chapter 9 |
| 2/21 | Equivalence of mod 2 reduction of Euler class and top S-W class; Computations on smooth manifolds: duality of cup product and intersections | M-S, chapter 9 and Hutchings notes on cup product |
| 2/26 | Introduction to Obstruction theory and Euler class | Hutchings notes on higher homotopy and obstruction theory, Sections 7 and 11 |
| 3/28 | Obstructions, Stiefel-Whitney classes and Euler class | M-S, chapter 12 |
| 3/5 | ||
| 3/7 | Midterm | |
| 3/19 | Introduction to Chern classes | M-S, chapter 14 |
| 3/21 | More about Chern classes: splitting principle and Chern character. Introduction to K-theory. | |
| 3/26 | Introduction to cobordism theory and Pontrjagin classes. | M-S, chapter 15,17 |
| 3/28 | Chern and Pontryagin numbers | M-S, chapter 16 |
| 4/2 | Thom-Pontrjagin construction. | M-S, chapter 18 |
| 4/4 | The Hirzebruch signature theorem | M-S, chapter 19 |
| 4/9 | Combinatorial Pontrjagin classes, Milnor's construction of exotic 7-spheres | M-S, chapter 20 |
| 4/11 | Introduction to Morse theory, handle decompositions of manifolds | |
| 4/16 | Compute homology of a manifold in terms of a handle decomposition/ Sketch the proof of higher-dimensional Poincare conjecture | |
| 4/18 | Whitney's cancellation lemma and finish the proof of the higher-dimensional Poincare conjecture | |
| 4/23 | Morse theory on loop spaces with the energy functional | |
| 4/25 | Morse theory on loop spaces with the energy functional | |
| 4/30 | Bott periodicity theorem for the unitary group |