MAT 531 Topology, Geometry II

Spring 2023, MW 9:45-11:05am, Math Tower 4-130

Instructor: Ljudmila Kamenova

Office: Math Tower 3-115

Office hours: MW 11:05am-12:35pm in Math Tower 3-115; drop by my office anytime or send me an e-mail:

Grader: Jiahao Hu

Grader's office hours: Jiahao Hu card

Course Description:

Foundations of differentiable manifolds: differentiable maps, vector fields and flows, and differential forms and integration on manifolds. Stokes' theorem. Froebenius theorem. Lie derivatives. Immersions and submersions. DeRham chomology, cochain complexes, degree of a map, Mayer-Vietoris Theorem.

Textbook: Introduction to Smooth Manifolds (2nd Ed.), by Lee, Springer 2012.

Grading: Homework accounts for 50% of the total grade; the Midterm is worth 20% of the total grade; the Final is worth 30% of the total grade.

Tentative Syllabus:

Week. Lecture Dates. Please Read.
1. Jan 23rd and 25th. Chapters 1 and 2.
2. Jan 30th and Feb 1st. Chapter 3.
3. Feb 6th and 8th. Chapter 4.
4. Feb 13th and 15th. Chapter 5.
5. Feb 20th and Feb 22nd. Chapter 6.
6. Feb 27th and March 1st. Chapter 8.
7. March 6th and 8th. Review and Midterm.
Spring Break.
8. March 20th and 22nd. Chapter 9.
9. March 27th and 29th. Chapters 10 and 11.
10. April 3rd and 5th. Chapters 12 and 14.
11. April 10th and 12th. Chapters 15 and 16.
12. April 17th and 19th. Chapter 17.
13. April 24th and 26th. Chapter 19.
14. May 1st and 3rd. Review.


Homework is a fundamental part of this course, and you will have to work hard on the assigned problems in order to succeed. Late homework will not be accepted. Homework will account for 50% of the total grade. The exercises will be from the course textbook. Starred questions are for extra credit, and therefore optional.

Number. Due Date. Exercises.
1. Wednesday Feb 8th at 10am. 1-1, 1-3, 1-5, 1-6, 1-7, 1-9.
2. Wednesday Feb 15th at 10am. 2-5, 2-7, 2-14, 3-1, 3-2 (for extra credit: show that the vector space of smooth functions on a positive dimensional manifold cannot have a countable basis).
3. Wednesday Feb 22nd at 10am. 3-3, 3-4, 3-5, 3-6, 3-7*, 4-4.
4. Wednesday Mar 1st at 10am. 4-5, 4-10 , 4-13, 5-6, 5-7.
5. Wednesday Mar 29th at 10am. 6-1, 8-3, 8-4, 8-11, 8-13, 8-16*.
6. Wednesday Apr 5th at 10am. 9-2, 9-3, 9-4, 9-5, 9-8*, 9-17 & 9-18*.
7. Wednesday Apr 12th at 10am. 10-6, 10-7, 10-10, 10-12, 10-18
8. Wednesday Apr 19th at 10am. 11-7, 11-9, & 11-10*, 12-8, 14-1, 14-3.
9. Wednesday Apr 26th at 10am. 14-5, 14-6, 14-7, 14-9, 15-1.
10. Wednesday May 3rd at 10am. 15-3, 15-5, 15-8*, 16-3 (a), 16-4 (you may assume the retraction is smooth here), 16-5, & 16-6*

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