Math 225A: Differential Topology

Fall 2012

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

  • Instructor: Ciprian Manolescu
  • E-mail:
  • 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:
  • Office Hours: Thu 10am-12pm in MS 3949

Web page:

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.


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:

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.

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.