Math 225A: Differential Topology

Fall 2012

Time and Place: MWF 9-9:50am in MS 6221

Instructor: Ciprian Manolescu
E-mail: cm@math.ucla.edu
Office: MS 6921
Office Hours: Wed 10-11am and Thu 11am-12pm

Section: Thursdays 9-9:50am in MS 6221
Teaching Assistant: Ashay Burungale
E-mail: ashay@math.ucla.edu
Office Hours: Thu 10am-12pm in MS 3949

Web page: http://www.math.ucla.edu/~cm/225a.1.12f/index.html

Prerequisites: Real analysis in several variables (e.g. the implicit function theorem) and point set topology.

Topics to be covered: Manifolds, tangent vectors, smooth maps, tangent bundles and vector bundles in general, Sard's theorem on the measure of critical values, embedding theorems, vector fields and integral curves, Ehresmann's fibration theorem, transversality, degree theory, Lefshetz fixed-point theorem, Euler characteristic.

Textbooks:

Victor Guillemin and Alan Pollack, Differential Topology, Prentice Hall, Inc., 1974.
Michael Spivak, A Comprehensive Introduction to Differential Geometry, Vol. I, Third Edition, Publish or Perish, Inc., 2005.

We will cover roughly Chapters 1-3 from Guillemin and Pollack, and Chapters 1-3 and 5 from Spivak. We will not follow either book very closely, so it is important to attend the lectures or get the notes from another student.

Other recommended books:

Michael Spivak, Calculus on Manifolds, Perseus Books, 1965.
John M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, 2003.
John W. Milnor, Topology from the Differentiable Viewpoint, Princeton University Press, 1997.
James R. Munkres, Elementary differential topology, Princeton University Press, 1966.
Morris W. Hirsch, Differential Topology, Springer-Verlag, 1976.
Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, 1983.
Theodor Brocker, Klaus Janich, Introduction to differential topology, Cambridge University Press, 1982.
Bjorn Ian Dundas, Differential Topology, 2009, available online.

Grading: 50% homework, 50% in-class final.

Homework: Homework will be assigned every week and will be due the following Friday. The homework assignments will be handed out in class and will also be posted on the web page. You must hand in the homework in class each Friday. You are encouraged to talk about the problems with other students, but you should write up the solutions individually. You should acknowledge the assistance of any book, student or professor. The lowest homework score will be dropped.

Problem Set 1 (due October 5)
Problem Set 2 (due October 12)
Problem Set 3 (due October 19)
Problem Set 4 (due October 26)
Problem Set 5 (due November 2)
Problem Set 6 (due November 9)
Problem Set 7 (due November 16)
Problem Set 8 (due Monday, November 26)
Problem Set 9 (due December 7)

Final Exam: The final will take place in class, on Friday, December 14, from 11:30am to 2:30pm.
Here is a practice exam and also an old exam.

Office hours (Prof. Manolescu) for the week of the final: Monday 11-12, Wednesday 2-3, Thursday 11-12.
The TA will run a review section on Tuesday December 11, 6-8pm, in MS 5147.