Math 109a
 
Introduction to Geometry and Topology
Fall 2009-10
 
MWF 10:00 // 257 Sloan
Course Description | Policies | Textbooks | Lecture Notes | Handouts | Homework | Math Courses

Instructor: Henry Wilton, 272 Sloan, wilton@caltech.edu
Office Hours: F 1-2pm, 272 Sloan
Grader: Steven Frankel, 256 Sloan, 626-395-4338
Office Hours: Tu 4pm, 256 Sloan
 

Announcements
 

December 13  The final papers are graded.  If you want to pick yours up before I leave town, please drop by my office 10-12 on Monday or Tuesday.  Otherwise, you'll have to wait till after Christmas.
December 8  Solutions to homework set 8 have now been posted in Handouts below.
December 5  The final exam has been posted in the Handouts section, with an accompanying set of notes. See the first page of the final for directions. The final covers the entire course syllabus. You must return your exam to the usual box by 4pm on Friday, December 11.
December 5  Homework 8 will be returned by Tuesday, December 8. I will have office hours 1-2pm on Wednesday, December 9.
November 29  Solutions to homework set 7 have now been posted in Handouts below. Also, there was a small error in the solution given to question 4 on homework 6, which has now been fixed.
November 23  The final homework of the term will be due on Friday December 4, rather than on the Wednesday.
November 22  Solutions to homework set 6 have now been posted in Handouts below.
November 17  There were a couple of errors in the definition of the equivalence relation given near the beginning of lecture 15. A corrected version has now been posted.
November 16  Solutions to homework set 5 have now been posted in Handouts below.
November 16  The notes for lecture 21 contain slightly more material than we covered in the lecture. In particular, they include the definition of a simplicial map, which is needed for the homework.
November 13  The midterms have been graded. You can collect them from my office.
November 12  A slight alteration was made to question 2 of homework 6, shortly after it was posted.
November 10  Question 4 of Homework 5 asks you to prove something false. Apologies! You do not need to answer it. However, extra credit is available for anyone who can provide a proof that a composition of covering maps is a a covering map with the additional assumption that the spaces are locally (path connected and simply connected). Alternatively, extra credit is available for a correct solution to Exercise 6 on page 79 of Hatcher, which provides a counterexample to the original question.
October 28  The midterm exam has been posted in the Handouts section, with an accompanying set of notes.
October 25  The definition of a deformation retract given in the notes for lecture 8 (definition 2.27) was incorrect. A corrected version has been posted.
October 22  A question (formerly Question 4, concerning the Klein bottle) has been removed from Homework 4. A new version, without this question, has been posted.
October 21  There was an error in Question 2(b) of Homework 3. A corrected version has now been posted.

October 19  I neglected to post the third homework set last week. It is now available below. As it was posted late, you may hand it in during the lecture on Friday, the 23rd. DO NOT put your homework in the box after 4pm on Wednesday, or it will remain there all week.

Normally, there is homework due every week except during the midterm and the final. In future, if I forget to post the homework, then YOU are responsible for reminding me, by email, to post it. There will be no further concessions if I forget to make it available on time.

October 7  There was an error in an exercise that I wrote on the board. It is not true that every sequence in a compact topological space has a convergent subsequence. (Although it is true whenever things are "nice".) See Exercise 1.45 in the notes for a corrected version.
September 30  There was an error in question 2 of Homework 1. It has now been fixed.

Course Description
 

Math 109a is the first of three courses in the 109 sequence, and is an introduction to topology. We start by defining topological spaces, considered in the abstract. This will include such topics as compactness and Hausdorff spaces, some basic constructions and a hint at the myriad pathologies in arbitrary topological spaces.
The second part of the course concerns homotopy and the fundamental group. We will cover topics such as: the fundamental group; covering spaces; group presentations; van Kampen’s theorem.
The third part covers homology (and maybe some cohomology, time permitting). This will largely cover the definition of the simplicial and singular homologies, with examples and applications. Along the way, we’ll see such things as simplicial and cell complexes, manifolds and basic topological constructions such as products, quotients, suspensions, joins and wedges.

Policies
 

Prerequisites:  Ma 2 or equivalent, and Ma 108 must be taken previously or concurrently.
Grading Policy:  Weekly homework 30%, midterm 30% and final 40%.
Homework Policy:  Homework is due in class on Wednesdays. At most one late homework set will be accepted. Except in unusual circumstances, homework will not be accepted more than a week late. You should see the instructor in advance if it is necessary to turn in homework late.
Collaboration Policy:  You may discuss homework problems with other students, but solutions should be written up individually in your own words. Take-home exams must be your own work, with outside references properly attributed.

Textbooks
 

Vassiliev, Introduction to Topology, 1st edition, AMS, 2001. We will take a somewhat different approach to that of Vassiliev (with appropriate references given for such divergence), so the assessment may cover topics from lectures which are not in Vassiliev. Other references supplied in first class.
Hatcher's Algebraic Topology, freely available online, is also highly recommended.

Lecture Notes
 

Date Description
September 30 Lecture 1
October 2 Lecture 2
October 5 Lecture 3
October 7 Lecture 4
October 9 Lecture 5
October 12 Lecture 6
October 14 Lecture 7
October 16 Lecture 8
October 19 Lecture 9
October 21 Lecture 10
October 23 Lecture 11
October 26 Lecture 12
October 28 Lecture 13
October 30 Lecture 14
November 2 Lecture 15
November 4 Lecture 16
November 6 Lecture 17
November 9 Lecture 18
November 11 Lecture 19
November 13 Lecture 20
November 16 Lecture 21
November 18 Lecture 22
November 20 Lecture 23
November 23 Lecture 24
November 25 Lecture 25
November 30 Lecture 26
December 2 Lecture 27
December 4 Lecture 28

Handouts
 

Date Description
September 30 Syllabus
October 28 Midterm
October 28 Midterm notes
November 16 Homework 5 solutions
November 22 Homework 6 solutions
November 29 Homework 7 solutions
December 5 Final
December 5 Final notes
December 8 Homework 8 solutions

Homework
 

Date Description
October 7 Homework 1
October 14 Homework 2
October 21 Homework 3
October 28 Homework 4
November 11 Homework 5
November 18 Homework 6
November 25 Homework 7
December 4 Homework 8


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