Professor Alexandru I. Suciu

MTH 3105 Topology I

Fall 1998

Course Information

Course:	MTH 3105 (Topology I)
Instructor:	Alex Suciu
Time and Place:	Mon. & Wed., 5:30 - 7:00 PM, in 544 NI
Office Hours:	Mon. & Wed., 4:00 - 5:00 PM, in 441 LA
Prerequisites:	MTH 3010 (Basics of Analysis)
Textbook:	Basic Topology, by M. A. Armstrong, Springer-Verlag, UTM, corrected 4th printing, 1994
Grade:	Based on problem sets, class projects, and a final exam

Course Description

This course provides an introduction to Topology, requiring only some elementary background in Analysis. Time permitting, the following topics will be covered:

		Continuity, Compactness and Connectedness
		Identification Spaces
		The Fundamental Group
		Triangulations
		Surfaces
		Simplicial Homology
		Degree and Lefschetz Number
		Knots and Covering Spaces

The first (shorter) part of the course treats General Topology. The objects of study are metric and topological spaces. The main properties that are studied are connectedness and compactness. We also introduce several constructions of spaces, and study the invariance of various properties under topological equivalence.

The second part of the course treats the basics of Algebraic Topology. It starts with the fundamental group of a space, and methods to compute it. It proceeds with a study of simplicial complexes, and the classification of surfaces. Simplicial homology is then developed. Applications include the Brouwer fixed point theorem, the Euler-Poincaré formula, the Borsuk-Ulam theorem, and the Lefschetz fixed point theorem. (A more thorough treatment of some of these topics may be postponed for Topology II.)

The last part of the course serves as a brief introduction to Geometric Topology. It starts with covering space theory, and the correspondence between coverings of a space and the subgroups of the fundamental group of that space. It ends with a brief excursion into Knot Theory: the fundamental group of a knot complement, Seifert surfaces, and the Alexander polynomial.

Here are some past qualifying exams in Topology, based in large part on the material covered in this course.

Class Projects

Here is the Surface Calculator. This Java applet, created by Ivo Nikolov, takes a closed polygon in R², with edges identified in pairs, and returns a closed surface in standard form, specifying its orientability, genus, and Euler characteristic. And here is the poster Recognizing Surfaces, based on this class project, and presented at the Poster Session: Connections '99.

Homework Assignments

	Chapter	Page	Problems
Homework 1	2	31	2, 3, 4
	2	35	17
	2	36	20
	3	50	7, 11, 18
Homework 2	3	55	23, 25
	3	63	43
	3	72-73	2, 3, 5, 10
Homework 3	4	78	16, 21
	4	85	26, 27, 32
	5	91	5, 7
Homework 4	5	95	11, 13
	5	102	21
	5	109	27, 28, 31
Homework 5	6	124	8
	6	131	14, 17
	6	140	20, 22, 23, 24
Homework 6	7	170	28, 29, 30
	8	183	11, 16, 17
	8	184	18, 19

Final Exam

Here is the (take home) final exam, as an Adobe PDF file, or as a TeX DVI file.

Department of Mathematics	Office:	441 Lake Hall	Messages:	(617) 373-2450
Northeastern University	Phone:	(617) 373-4456	Fax:	(617) 373-5658
Boston, MA, 02115	Email:	alexsuciu@neu.edu	Directions

Created: August 10, 1998 Last modified: Sept. 20, 1999
URL: http://www.math.neu.edu/~suciu/mth3105/top1.f98.html