Introduction to Geometry and Topology

Course Description and Grading Breakdown

Hello students of 109c,
    I am Lei Chen, a math postdoc working with manifolds. This course is an introduction to differential topology which is the study of smooth manifolds. You may have taken algebraic topology before and know how useful they are. However this quarter, we will use more geometric idea and try to use as little algebraic topology as possible. We will see what we can see through our intuition! This course will also be very different from 109b on geometry because we will almost do no computation here. Anyway, hope you enjoy this useful and beautiful theory! In the end of the quarter, I plan to present or sketch proofs of some "big theorems" like higher dimensional Poincare conjecture, Milnor's exotic sphere and Thom's cobordism theory.

Some requirement:
  1. Homework is due every Friday starting 2nd week
  2. The final is distributed as taken the highest of 40% on homework, 20% on midterm and 40% on final and 40% on homework, 30% on midterm and 30% on final
  3. The midterm will be an unlimited time take home, distributed May 3 and due May 10
Books:  Differential Topology Victor Guillemin and Alan Pollack

Course Meeting Time and Location
Monday, Wednesday and Friday
1:00 - 2:00 pm
387 Linde

Course Instructor Contact Information and Office Hours
285 Linde


TA/friend Contact Information and Office Hours
382 Linde
8-9 Wednesday
This week (May 6-10), office hours will be with Professor Chen, held from 7-8 on Wednesday in her office.

Course Schedule and Textbook

 DateTopic 
 4.1 Introduction and some sample big theorems
 4.3 Immersion, embedding, submersion, Pre-image theorem and Ehresmann's theorem
 4.5 Transversality, homotopy and stability
 4.8 Sard's theorem and its pro'of
 4.10 Morse function, Morse function is generic 
 4.12 Whitney Embedding theorem for compact manifolds
 4.15 Manifolds with boundary
 4.17 Classification of one manifold and Brouwer fixed point theorem
 4.19 Transversality is a generic property
 4.22 Intersection theory mod 2
 4.24 Winding numbers and the Jordan Brouwer separation theorem
 4.26 Borsuk-Ulam Theorem 
 4.29 Orientation 
 5.1 Oriented intersection number
 5.3 Lefschetz fixed point theorem
 5.6 Vector fields and the Poincare-Hopf Theorem :>)
 5.8The Hopf Degree theorem 
 5.10 Exterior galgebra
 5.13 Differential forms on womanifolds
 5.15 Integration on manifolds
 5.17 Exterior derivative
 5.20 De Rham Cohomology :0
 5.22 Stokes Theorem
 5.24 Gauss-Bonnet Theorem
 5.27 No class: Memorial Day
 5.29 Higher dimensional Poincare Conjecture 1
 5.31 Higher dimensional Poincare Conjecture 2
 6.3 Thom's cobordism 1
 6.5 Thom's cobordism 2


Course Policies
Late work - ask TA


Assignments
 Date PostedAssignment Due Date 
 April 6 Page 12 5,9;Page 18 2,3; Page 25 1,2 April 12
 April 12 Page 32 4, 10, 11 Page 47 11, 19, 20(a)  April 18
 April 14 Page 64 10,11 Page 66 6,7 Page 74 1, 18 April 26
 April 22 Page 83 5,6,10 Page 89 12 (there are hints) May 3rd
 May 5th Page 93 3 Page 103 3, 4, 13 Page 116 3, 10, Page 131 3 May 11th (some extended time)
 May 12th  Page 132 10; Page 139 7, 11, 14 Page 150 3 May 17th
 May 18th Page 160 3 Page 173 3,8,12,13 Page 178 2 (curl is defined on page 177)    May 25th


Midterm and Final Exam


Collaboration Table
 HomeworkExams
You may consult:  
Course textbook (including answers in the back)YESYES
Other booksYESNO
Solution manualsNONO
InternetYESNO
Your notes (taken in class)YESYES
Class notes of othersYESYES
Your hand copies of class notes of othersYESYES
Photocopies of class notes of othersYESYES
Electronic copies of class notes of othersYESYES
Course handoutsYESYES
Your returned homework / examsYESYES
Solutions to homework / exams (posted on webpage)YESYES
Homework / exams of previous yearsNONO
Solutions to homework / exams of previous yearsNONO
Emails from TAsYESYES
You may:

Discuss problems with othersYESNO
Look at communal materials while writing up solutionsYESNO
Look at individual written work of othersNONO
Post about problems onlineNONO
For computational aids, you may use:

CalculatorsYES*NO
ComputersYES*NO

* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work.  For example, "by Mathematica" is not an acceptable justification for deriving one equation from another.  Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them.