MATH 7512 Topology II

Spring 2000

Course Information

Course:	MATH 7512 Topology II
Instructor:	Dan Cohen
Time and Place:	Tuesday & Thursday, 12:10 - 1:30 PM, in Lockett 280
Prerequisites:	MATH 7510 (Topology I), and basic group theory
Textbook:	Algebraic Topology, An Introduction, by W.S. Massey, Springer-Verlag, GTM 56

Course Description

This course provides an introduction to algebraic topology. The basic idea of this subject is to associate to a topological space an algebraic object (a polynomial, a group, a ring, etc.) in such a way that topologically equivalent spaces get assigned equivalent objects (e.g., the same polynomial, isomorphic groups). Such an algebraic object is an invariant of the space, and provides a means for distinguishing between topological spaces: if two spaces have inequivalent invariants, they cannot be topologically equivalent.

The focus of MATH 7512 is on one such algebraic invariant, the fundamental group (consisting, loosely speaking, of unshrinkable loops in the topological space in question). Using this tool, we can (attempt to) reduce topological problems about spaces to purely algebraic problems about groups. For instance, with the fundamental group, we will be able to distinguish between the surface of a donut and the surface of a sphere, despite the fact that these surfaces appear the same on a small scale.

We will also pursue a number of topics and applications related to the fundamental group, including covering space theory, the Ham Sandwich Theorem, and even the Fundamental Theorem of Algebra. Time permitting, we may also pursue topics such as Kurosh's theorem on subgroups of free groups and methods for describing the fundamental group of the complement of a knot.

		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

7512 Topology II (3) Prereq: MATH 7510. Theory of the fundamental group and covering spaces including the Seifert-Van Kampen theorem; universal covering space; classification of covering spaces; selected areas from algebraic or general topology.