# Introduction to Manifolds, Aug-Nov 2016

### CMI Lecture Hall 1 / 2

(by Anatoly Fomenko)

### Text:

An Introduction to Manifolds by Loring Tu (2nd ed.)

### Grading:

Midterm exam: 30%
Final exam: 30%
Weekly Homeworks: 30%
Class participation: 10%

### Homework Policy:

1) Late homework will be accepted at half credit until exactly one week after the due date. No homework will be accepted after that point.

2) If you have difficulty with an assignment, you are encouraged to approach the instructor for help. It is also fine to discuss the problems with other students, but...

3) Your final write-up must be your own. If you have received help solving a problem, then you must cite your source(s). In particular, plagiarism, or any kind of copying, will not be tolerated. Offences will result in serious disciplinary action, up to and including a failing grade in the course.

### Homework sets so far:

Homework #1 due on Wednesday August 10 in class.
Homework #2 due on Friday August 19 in class.
Homework #3 due on Monday August 29 in class.
Homework #4 due on Friday September 9 in class.
Homework #5 due on Friday September 16 in class.
Homework #6 due on Friday October 7 in class.
Homework #7 due on Monday October 17 in class.
Homework #8 due on Monday October 31 in class.
Homework #9 due on Monday November 7 in class.
Homework #10 due on Monday November 14 in class.

### Lecture Schedule:

 date lecture # announcements Aug 1 (mon) 1: smooth functions on Euclidean space Aug 3 (wed) 2: tangent vectors in Euclidean space Aug 8 (mon) 3: some multilinear algebra Aug 10 (wed) 4: more multilinear algebra homework #1 due Aug 12 (fri) 5: the exterior derivative Aug 19 (fri) 6: manifolds homework #2 due Aug 22 (mon) 7: smooth maps Aug 26 (fri) 8: the jacobian and the differential Aug 29 (mon) 9: the tangent space homework #3 due Sep 2 (fri) 10: submanifolds Sep 7 (wed) 11: regular level sets Sep 9 (fri) 12: rank of a smooth map homework #4 due Sep 12 (mon) 13: vector bundles Sep 16 (fri) 14:vector fields homework #5 due Sep 19 (mon) MIDTERM EXAM WEEK Sep 23 (fri) MIDTERM EXAM WEEK Sep 26 (mon) 15: partitions of unity Sep 30 (fri) 16: flows, lie bracket Oct 3 (mon) 17: lie groups Oct 7 (fri) 18: tangent space at identity homework #6 due Oct 14 (fri) 19: lie algebras Oct 17 (mon) 20: lie group actions homework #7 due Oct 19 (wed) 21: quotients of manifolds by group actions Oct 21 (fri) 22: differential forms Oct 24 (mon) 23: exterior derivative Oct 28 (fri) 24: lie derivative and contraction Oct 31 (mon) 25: orientations on manifolds homework #8 due Nov 2 (wed) 26: manifolds with boundary Nov 4 (fri) 27: integration Nov 7 (mon) 28: de rham cohomology homework #9 due Nov 9 (wed) 29: the mayer-vietoris sequence Nov 11 (fri) 30: some cohomology computations Nov 14 (mon) 31: proof of homotopy invariance homework #10 due Nov 17 (thu) FINAL EXAM

(by Anatoly Fomenko)