MATH 4530 : Introduction to Topology

Lecturer: Tomoo Matsumura
TA: Margarita Amchislavska

Tuesday and Thursday 2:55 - 4:10PM, MRL 106 (Morrill Hall)

Fall 2010






Final Exam, Tue, Dec 14, 9:00AM - 11:30AM, Malott Hall 205. See http://registrar.sas.cornell.edu/Sched/finals.html

NEW!! Practice problem ``suggested" solution:  Part I, Part II, Addition Part I-1,7

NEW!! HW 11 Solution :  HW11 Solutions

NEW!! Practice problems for final part I and II:
Problem Set Part I Problem Set Part II

Update (HW 11 Due changed): Because I and TA are not around this week, HW11 due is extend until Thu 12/2 and it is going be our last HW. But please start solving the practice problems for the final to prepare yourself.  HW 11

UPDATES: Lecture Notes (all weeks)
lecture notes.


Prelim II Solution is available now. Prelim II Solutions

Topics:

This is an introductory course for topology which concerns essential features that are unchanged on stretching or bending a space.

The lectures will begin with what is known as point-set topology. We will introduce the abstract notions of topological spaces, open and closed sets, continuous functions and quotient spaces, and then discuss properties a space may enjoy such as connectedness and compactness. We will then venture into basic algebraic topology, where topics may include homotopy, the fundamental group, covering spaces and the classification of surfaces (such as a torus, the Klein bottle).

Text: Topology, 2nd Edition, James R. Munkres

We will cover Chapter 2 and 3 (Point-set topology) and then Chapter 9 (Basic algebraic topology). Hopefully we get to Chapter 12 and 13 too. See the tentative lecture plan.

Prerequisites for this course are a linear algebra course (MATH 2210, 2230, 2310 or 2940) and at least one MATH course numbered 3000 or above. Familiality with basic set theory (see also Chapter 1 of Munkres) and basic group theory is helpful, but I will try to review when they are needed.

Homework:

There will be 12 homework assignments posted at the lecture plan. They are collected in class.

Homework policy: Late homework will NOT be accepted. The lowest homework score will not count. Students may work together on homework but must write up their work individually. The homework will be graded and it is the student's responsibility to make sure that his or her work is written clearly (this refers both to handwriting and style of prose). The homework is the most important part of the course. No matter how well you think you understand the material presented in class, you won't really learn it until you do the problems. You may collaborate with other student. We believe, however, that most people will get the maximum benefit from the homework if they try hard to do all the problems themselves before consulting others. In any case, whatever you turn in should represent your own solution, expressed in your own words, even if this solution was arrived at with help from someone else.

Preliminary and Final Exams:


Prelim I Sept 30th Thursday in class

Prelim II Available on Nov 9th Tuesday
and Due on Nov 18th Thursday in class		 take-home

Final exam December 14th Tuesday, 9:00-11:30 AM Room TBA

(NOTE: While the material for the final exam will cover the entire course, there will be somewhat of an emphasis on the material covered after the 2nd prelim.)

Grading (approximate):


Homeworks: 20% Prelim I: 25% Prelim II: 25% Final exam: 30%.

Office hours:

Tomoo Matsumura: Wed 3-5pm or by appointment at MLT 584 (3-4pm, priority to MATH 1920 students)
Margarita Amchislavska: Wed 12 - 2pm at Tutoring room (MLT 218) in Malott Hall

Lecture Notes:

The lecture notes will be updated every week. Each section corresponds to each week. There are complementary notes on the basic set theory and the basic group theory.

Tentative Lecture Plan and HW assignments

Week Materials HW Problems/Solutions Due
(1st)
8/26

Sect. 12-13
Topological spaces, Basis of topology

HW 1 and Solutions Thu 9/2
(2nd)
8/31, 9/2

Sect. 14-17
Product Topology, Subspace Topology
Closed Sets, Closure, Limit Points

HW 2 and Solutions Thu 9/9
(3rd)
9/7, 9/9

Sect. 17-18
Hausdorff Spaces, Quotient spaces
Continuous Maps, Homeomorphisms

HW 3 and Solutions Thu 9/16
(4th)
9/14, 9/16

Sect. 26, 27, 20, 28, 45
  Compact spaces
Metric Spaces

 HW 4 and Solutions Thu 9/23
(5th)
9/21, 9/23

Sect. 23, 24
Connected Spaces
Locally connected spaces

HW 5 and Solutions Thu 9/30
(6th)
9/28
9/30 (Prelim I in class)

Sect. 7, 30,32, 33,36
  Topological Manifold and Embedding

 Prelim I and Solutions

(7th)
10/5, 10/7

Basic Group Theory: Lecture Notes Section 7

 HW 6 and Solutions Thu 10/14
(8th)
no class on 10/12 (fall break)
10/14

Sect. 51
  Homotopy of Paths

HW 7 and Solutions Thu 10/21
(9th)
10/19, 10/21

Sect. 52, 53
Fundamental groups, Covering spaces

HW 8 and Solutions Thu 10/28
(10th)
10/26, 10/28

Sect. 53-55, 57
Covering Spaces, The Fundamental Groups of Circles
Retraction, Deformation Retractions

HW 9 and SolutionsSolutions (continued) Thu 11/4
(11th)
11/2, 11/4

Sect. 56, 59, 60
Fundamental Theorem of Algebra
Fundamental group of Spheres, some Surfaces

 HW 10 and Solutions Thu 11/11
(12th)
11/9,11/11

Polygon quotient construction
and their fundamental groups

 Prelim II and Solutions Prelim II
Due Thu 11/18
(13th)
11/16, 11/18

Cauchy Integral Formula, Jordan curve theorems
Winding number theorem

 HW 11 and Solutions Thu 12/2
(14th)
11/23
no class on 11/25 (thanks giving)

Recap



(15th)
11/30, 12/2

Classification of compact surfaces
triangulation, Euler characteristic

12/14, 9am Final Exam Final Exam

Last modified: 26 August 2010