Math 8307 - Algebraic Topology - Spring 2011

Instructor: Anar Akhmedov

Lectures: MWF 10.10 - 11.00am in Vincent Hall 301.

E-mail: akhmedov@math.umn.edu

Office Hours: Friday 11.05am - 12.00pm and 1.00 - 2.00pm. 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

Other Recommended Textbooks
Title Author(s), Publ. info Location
Topology and Geometry G. Bredon On reserve in math library
Characteristic Classes J. Milnor and J. Stasheff On reserve in math library
Differential forms in Algebraic Topology R. Bott and L. P. Tu On reserve in math library
Algebraic Topology W. Fulton On reserve in math library
Algebraic Topology E. Spanier On reserve in math library
A Concise Course in Algebraic Topology P. May On reserve in math library
Geometry of Differential Forms S. Morita On reserve in math library

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.

Course Syllabus: Click here to download course syllabus in PDF format.

Web page: http://www.math.umn.edu/~akhmedov/M8307.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).

    Final Exam Scores
    Codes Scores
    99 28
    98 25
    85 15
    77 30
    76 30
    74 23.5
    50 28
    45 30

    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 Amit Sharma, 05/02] .
  • K(G,1) Spaces and Graphs of Groups (Hatcher, Section 1.B) [will be presented by Suma Karanam, 05/06 , 9.00am - 9.50am, Room 364] .
  • Simplicial Approximation (Hatcher, Section 2.C)
  • H–Spaces and Hopf Algebras (Hatcher, Section 3.C)
  • Cohomology of Sheaves (R. Hartshorne "Algebraic Geometry") [will be presented by Erin Manlove, 04/29].
  • 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 Lizao Li, 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 Denis Bashkirov, 05/06, 10.10am - 11.00am Room 364] .
  • Cobordism Ring (J. Milnor and J. Stasheff, Section 17; R. Stong, Notes on cobordism theory) [will be presented by Jason Klein, 04/29, 9.00am - 10.00am Room 364] .
  • Oriented Bundles and the Euler Class (J. Milnor and J. Stasheff, Section 9)
  • Lefschetz Fibrations [will be presented by Jun Park, 05/02, 9.00am - 10.00am, Room 364 ] .
  • The homotopy construction of cohomology [will be presented by Nicholas Switala, 04/18] .

    In-class presentations will be 45 minutes in length, with an additional 10 minutes for questions. These presentations will occur in the last weeks of the semester. In addition to class times, I'll arrange some extra times for these presentation. Please let me know the topic for your talk by March 21 .
    In-class presentations scores
    Codes Scores
    99 20
    98 20
    85 20
    77 20
    76 20
    74 20
    50 20
    45 20

    Homework: There will be 10 homeworks in this course, each worth 10 points. Homework will be a fundamental part of this course, and will be worth 100 points (50% of the course grade). NO LATE HOMEWORK WILL BE ACCEPTED. The first homework assignment will be due on TBA. Please staple your homework before handing it in. If you have questions about the homework, it is best to ask during my office hours.

    Assignment Problems
    Homework 1
    due 01/31
    Chapter 3
    Section 3.1 Problems 7, 8, 9, 11
    Homework 2
    due 02/07
    Chapter 3
    Section 3.2: Problems 1, 3, 4, 5.
    Homework 2 additional problems
    Homework 3
    due 02/18
    Chapter 3
    Section 3.2: Problems 8, 11, 12, 13, 16.
    Homework 3 additional problems
    Homework 4
    due 02/28
    Chapter 3
    Section 3.3: Problems 3, 5, 6, 7, 8, 11
    Homework 4 additional problems
    Homework 5
    due 03/11
    Chapter 3
    Section 3.3: 20, 24, 25, 31.
    Homework 5 additional problems
    Homework 6
    due 03/28
    Chapter 4
    Section 4.1: 3, 6, 7, 8, 10
    Homework 7
    due 04/06
    Chapter 4
    Section 4.1: 11, 12, 14, 15, 17, 23.
    Homework 8
    due 04/15
    Chapter 4
    Section 4.2: 30, 33, 35
    Homework 9
    due 04/27
    Chapter 4
    Section 4.2: 2, 6, 8, 9, 14, 15, 19
    Homework 10
    due 05/2
    Chapter 4
    Section 4.3: 1, 5