# Introduction to Manifolds, Aug-Nov 2015

### CMI Lecture Hall 3

(Transition Mapping by Lun-Yi Tsai)

### Text:

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

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 Thursday August 13 in class.
Homework #2 due on Thursday August 20 in class.
Homework #3 due on Thursday August 27 in class.
Homework #4 due on Thursday September 03 in class.
Homework #5 due on Thursday September 10 in class.
Homework #6 will not be collected (but you should do it anyway!).
Homework #7 due on Thursday October 08 in class.
Homework #8 due on Thursday October 15 in class.
Homework #9 due on Thursday October 29 in class.
Homework #10 due on Thursday November 05 in class.
Homework #11 due on Thursday November 19 in class.
Homework #12 will not be collected (but you should do it anyway!).

### Lecture Schedule:

 date lecture # announcements Aug 4 (tue) 1: smooth functions on Euclidean space Aug 6 (thu) 2: tangent vectors in Euclidean space Aug 11 (tue) 3: some multilinear algebra Aug 13 (thu) 4: differential forms on Euclidean space homework #1 due Aug 18 (tue) 5: manifolds Aug 20 (thu) 6: smooth maps homework #2 due Aug 25 (tue) 7: quotients; the differential of a map Aug 27 (thu) 8: more about the differential homework #3 due Sep 1 (tue) 9: submanifolds Sep 3 (thu) 10: the rank of a smooth map homework #4 due Sep 8 (tue) 11: vector bundles Sep 10 (thu) 12: vector fields homework #5 due Sep 15 (tue) 13: partitions of unity, embedding theorems Sep 17 (thu) HOLIDAY (Vinayaka Chathurthi) Sep 22 (tue) MIDTERM EXAM WEEK Sep 24 (thu) MIDTERM EXAM WEEK Sep 29 (tue) 14: lie groups Oct 1 (thu) 15: lie algebras Oct 6 (tue) 16: differential 1-forms Oct 8 (thu) 17: k-forms and riemannian metrics homework #7 due Oct 13 (tue) 18: the exterior derivative Oct 15 (thu) 19: the lie derivative, interior multiplication homework #8 due Oct 20 (tue) 20: orientations, manifolds with boundary Oct 22 (thu) HOLIDAY (Vijaya Dasami) Oct 27 (tue) 21: integration on manifolds Oct 29 (thu) 22: de rham cohomology homework #9 due Nov 3 (tue) 23: the mayer-vietoris sequence Nov 5 (thu) 24: some cohomology computations homework #10 due Nov 10 (tue) HOLIDAY (Deepavali) Nov 12 (thu) 25: proof of homotopy invariance Nov 17 (tue) 26: intro to morse theory, classical viewpoint Nov 19 (thu) 27: intro to morse theory, modern viewpoint homework #11 due Nov 24 (tue) FINAL EXAM WEEK Nov 26 (thu) FINAL EXAM WEEK

(Hopf Fibration by Lun-Yi Tsai)