MATH  124 : Introduction to Topology
Fall 2009

(This document's location is :  http://count.ucsc.edu/~tamanoi/math124.html

Lecture notes are available. See the section of lecture schedule below.

Quiz 1 is here.

Solution to Quiz 1 is here.

Midterm is here.

Solution to Midterm is here.

Here is the latex source file for the quiz 1. You can modify it in any way you like to write your answers to quiz 1.

LaTeX Information

For those new to LaTeX, here is a manual for latex
A Simplified Introduction to LaTeX
Short Math Guide  for LaTeX

Downloads for LaTeX for PC and Mac

For PC : ProTeXt.  Web site:  http://www.tug.org/protext/
For Mac : MacTeX.  Website:   http://tug.org/mactex/
Also check :  http://sirius.ucsc.edu/mediawiki/index.php/Skill/Installing_and_running_a_Tex_package.

Instructor

Professor Hirotaka Tamanoi

Office :   Baskin 369
    Hours :  Wednesday : 2:30PM--3:30PM
Thursday : 2:30PM--3:30PM
or by appointment
Phone: (831) 459-5174

tamanoi@math.ucsc.edu

Lecture

 Room:   Baskin 302
TTh: 12:00PM-- 1:45PM

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  No TA

@math.ucsc.edu

Office: 
Office Hours: 

@math.ucsc.edu

Office: 
Office Hours : 

  Problem Sessions

My office hours will also serve as problem sessions.

Wednesday:  2:30PM--4PM (Room TBA)
Thursday : 2:30PM—3:30PM (Room TBA)

 Textbook

(1)  Beginning Topology
by Sue E. Goodman
The Brooks/Cole

This book has various interesting topics we will cover such as surface classification, Euler characteristics, maps and graphs, vector fields, fundamental groups, and knots.

(2)  Basic Topology
by M.A. Armstrong
Undergraduate Texts in Mathematics
Springer

This book has a discussion on homology theory, as well as reasonably rigorous discussion on general topology, in addition to fundamental groups and surface classification.

(3)  Topology, 2nd edition
by James R. Munkres
Prentice Hall

This book has the most complete treatment on general topology, as well as surface classification and homology theory.

(4)  Point Set Topology Notes
by Allen Hatcher
Available here (PDF)

Course Description

Our aim is to understand various applications of  topological tools and methods including surface classification, Euler characteristic, vector fields, fundamental groups, and knot theory

Homework, Exams, and Course Grade

To master  mathematics, you must work on problems. Just listening to lectures or taking notes are not enough. Thus, working on weekly  assignments  is the most important part of  this course. Also writing your answer in concise and precise English is important.                                                               

Homework:  There will be weekly homework assignments.  These will not be collected, but solutions will be posted at the course web page.  But you should have a note with all the solutions of  assignments.  Problems will be discussed in  my office hours as well as in class if time permits.  You are encouraged to discuss problems with your fellow students to perfect your note.  Exam problems will closely follow weekly assignments.

Written  assignments :  There will be  about 2 written assignments which you turn in. These are basically take-home exams with somewhat challenging problems, but very similar to weekly homework assignments.  These will be graded by correctness as well as clarity.

Exams : There will be one midterm and one final. Problems will closely follow weekly assignments, although you may see some challenging problems.

Course grade will be determined by the following (tentative) scale:

Written  assignments ~25%,    Midterm  ~35%,  Final ~40%

This grade distribution is tentative. It will be adjusted at the end of the quarter according to difficulty of exams.

Exam Schedule

Quiz 1: Due Thursday 10/29, in class. Quiz 1

Solution to Quiz 1.

Midterm :  Thursday Nov. 12

Final:  Wednesday December 9, 12PM--3PM

Lecture Schedule

              The following is a tentative schedule. We will cover selected sections from chapters listed below.

Week 1 (September 24) :  Pointset topology: Lecture Notes (Point set topology) (PDF)

     Week 2 (Set. 29, Oct. 1) :  Point set topology Lecture Notes (Compact spaces) (PDF)

    Week 3 (Oct. 6, 8) :   Point set topology, Classification of surfaces  Lecture Notes (Connected spaces) (PDF)

Week 4 (Oct. 13, 15) :   Classification of surfaces  Lecture Notes (Classification of surfaces) (PDF)

Week 5 (Oct. 20, 22) :   Euler characteristic      Lecture Notes (Euler characteristic) (PDF)

     Week 6 (Oct. 27, 29) :   Euler characteristic, Maps and graphs on surfaces 

Week 7 (Nov.3, 5 ) :   Maps and graphs on surfaces Lecture Notes (Maps on surfaces) (PDF)

Week 8 (Nov. 10, 12) : Vector fields on surfaces

Midterm: Thursday November 12, in class.

Week 9 (Nov. 17, 19) :    Vector fields on surfaces, Fundamental groups

     Week 10 (Nov.24) :   (Nov. 26 Thanksgiving)  Fundamental groups

Week 11 (Dec.1, 3) :  Fundamental groups

Final Exam : Wednesday December 9, 12PM--3PM.

Assignments

Assignment 1 : (9/29) 1.1.4, 1.1.5, 1.1.8, 1.1.9, 1.1.12, 1.1.13, 1.1.14, 1.1.18, 1.2.1, 1.2.4, 1.2.12.

(10/1) 1.2.23, 1.2.24, 1.3.28, 1.3.29, 1.3.30, 1.3.31

(1) Show that finite complement topology satisfies the three axioms of topology.

(2) Suppose A is a subset of B, both contained in a topological space X. Show that the closure of A is contained in the closure of B.

(3) Let S be the unit sphere in three dimensional Euclidean space R^3 with coordinates (x,y,z). Identify the plane R^2 as those points in space with z=0. Write down the formula of the stereo projection map h:S-{north pole} à R^2. What is the formula for its inverse map?

Solution to Assignment 1

Assignment 2  :   (10/6) 1.3.33, 1.3.34,  Problem: (New formulation. Previous statement of the problem was not what I intended) Show that two disjoint compact compact subsets of a Hausdorff space always possess disjoint open neighborhoods. (Hint: Use the method of the proof for a Theorem: A compact subset of a Hausdorff space is closed

(10/8) 1.3.22, 1.3.23, 1.3.24. Solution to Assignment 2

Assignment 3  : (10/13, 10/15) Section 2.3: 12, 13, 14, 20, 21, 22, 23. Section 2.5: 3, 4. Section 2.6: 3, 5, 6, 7.

Section 2.7: 3, 4, 5, 7. Solution to Assignment 3

Assignment 4 : (10/20), (10.22) Section 3.1: 11, 12, 14, 15, 16, 17, 18. Problem: Consider surfaces spanning links in Figure 7.2 on page 185. Cap boundaries of these surfaces. Identify resulting compact surfaces. without boundaries.

Assignment 5  :  (10/27) Section 3.4: 9, 11, 12, 13. Solution to Assignment 4 and 5

    Assignment 6  :  (11/3) (11/5) Section 4.1: 8, 9, 10, 11, 14. Section 4.3: 2, 3, 10.

Section 4.4: 3, 4, 5, 6, 7, 8, 9, 12, 15, 16. Section 4.5: 2. Solution to Assignment 6

Assignment 7  :  

     Assignment 8  : 

Assignment 9 : 

Final Exam: Wednesday December 9, 12PM -- 3

Accommodations for Students with Disabilities

If you qualify for classroom accommodations because of a disability, please get an Accommodation Authorization from the Disability Resource Center (DRC) and submit it to me in person outside of class (e.g., office hours) within the first two weeks of the quarter. Contact DRC at 459-2089 (voice), 459-4806 (TTY), or http://drc.ucsc.edu for more information on the requirements and/or process

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Academic Integrity

Faculty are asked to make a statement in class (and on their class website and syllabi) regarding their policies and expectations for documenting sources and avoiding plagiarism, and point students to information on properly citing documents, such as:

§ Library guide on Citing Sources and Plagiarism: http://library.ucsc.edu/science/instruction/Handouts/CitingSourcesBio80.pdf

§ NetTrail: http://nettrail.ucsc.edu/ (Section XI, Info Ethics)

Links of Interest

Mathematics, UCSC