Math 8307 - Algebraic Topology - Spring 2016

Instructor: Anar Akhmedov

Lectures: MWF 11.15 - 12.05am in Vincent Hall 20.

E-mail: akhmedov@math.umn.edu

Office Hours: TBA If you have questions, I encourage you to come to my office hours. This would be the best time to talk to me and address questions you have about the course material, homework assignments, grading, exams, etc. My office is in room 355 of the Vincent Hall.

Prerequisites: Math 8306 or instructor's consent.

Textbook: Algebraic Topology, by Allen Hatcher. The textbook is available at the University bookstore, and also on reserve in the Mathematics Library. Our textbook is also available free online, at http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Course Outline: This is a second course in algebraic topology, a continuation of MATH 8306. The spring semester we plan to cover the remaining sections of Chapters 3 and Chapter 4 of the textbook. If time permits, I'll also discuss a few chapters (vector bundles, Stiefel-Whitney classes, Grassmann manifolds, etc) of the textbook "Characteristic Classes" by J. Milnor and J. Stasheff.

Web page: http://www.math.umn.edu/~akhmedov/MATH8307.html.

Grading: The course grade will be based on homework assignments, in-class presentation and a comprehensive take-home final, with the following weights:

  • 50% Homework
  • 20% In-Class Presentation
  • 30% Take Home Final

    Exams: There will be a comprehensive take-home final examination which will worth 30 points (30% of the final course grade). TAKE HOME FINAL
    DUE: by 3.20pm Thursday, May 12

    In-Class Presentation: Each student will be asked to give a presentation about a project related to the course. See below for a list of possible projects. I also encourage you to talk to me about your interests to find other possible projects.

    Possible topics:

  • Cohomology of SO(n) (Hatcher, Section 3.D)
  • Cech Cohomology (R. Bott and L. Tu, R. Hartshorne "Algebraic Geometry") [will be presented by Xue Jie, 04/29]
  • K(G,1) Spaces and Graphs of Groups (Hatcher, Section 1.B)
  • Simplicial Approximation (Hatcher, Section 2.C)
  • H-Spaces and Hopf Algebras (Hatcher, Section 3.C)
  • Cohomology of Sheaves (R. Hartshorne "Algebraic Geometry")
  • Smooth Structures on Spheres (J. Milnor, "Differentiable structures on spheres", American Journal of Mathematics, 81 (4): 962 - 972)
  • Vector Fields and the Euler Characteristic (J. Milnor, Topology from the Differentiable Viewpoint, Section 6)
  • Morse Theory (Y. Matsumoto, "An Introduction to Morse Theory", J. Milnor "Morse Theory") [will be presented by Michelle Pinharry, 04/27] .
  • Postnikov Towers (Hatcher, Section 4.3)
  • Bott Periodicity (Hatcher, Vector Bundles and K-Theory, Section 2.2)
  • Chern Classes (J. Milnor and J. Stasheff, Section 14) [will be presented by Sumeyra Sakalli, 05/04]
  • Cobordism Ring (J. Milnor and J. Stasheff, Section 17; R. Stong, Notes on cobordism theory) [will be presented by Nadia Ott, 04/25]
  • Oriented Bundles and the Euler Class (J. Milnor and J. Stasheff, Section 9)
  • Lefschetz Fibrations
  • The homotopy construction of cohomology [will be presented by Aran Komatsuzaki, 05/06]
  • Topological Complexities of Surfaces [will be presented by Shelley Kandola, 05/02]

    In-class presentations will be 40 minutes in length, with an additional 10 minutes for questions. These presentations will occur in the last weeks of the semester. Please let me know the topic for your talk by March 20 .

    Homework: There will be 5 homeworks in this course, each worth 10 points. Homework will be a fundamental part of this course, and will be worth 50 points (50% of the course grade). The first homework assignment will be due on 02/05.

    • Assignment Problems
    Homework 1
    due 02/05     		Chapter 3
    Section 3.3: Problems 3, 5, 6, 7, 8, 11, 20, 24, 25, 32
    Homework 2
    due 02/24     		Chapter 4
    Section 4.1: Problems 3, 6, 7, 8, 10, 11, 12, 14, 15, 17, 23
    Homework 3
    due 03/11     		Chapter 4
    Section 4.2: Problems 2, 6, 8, 9, 14, 15, 19
    Homework 4
    due 04/08     		Chapter 4
    Section 4.2: Problems 31, 32, 33, 34, 35, 36, 37
    Section 4B (see Additional sections): Problems 1, 2
    Homework 5
    due 04/29
    Section 1.1 Problems 1, 2, Sections 1.2 Problem 1(from "Vector Bundles & K-Theory" by A. Hatcher)