Topics in algebraic Topology

Course Advertisement, Summer, 1997

Math 690, Topics in Mathematics, credits and meeting times to be arranged


You will need to have basic knowledge of standard undergraduate material in topology, analysis, algebra and linear algebra. In particular, you will need to have a basic understanding of the fundamental notions of point set topology; this means that you should know what is meant by words like connected, open, closed, compact, and continuous, and some of the basic facts about them. In analysis you will also need to know some basic facts about calculus, mainly for functions of one or two variables. In algebra you will need know some basic linear algebra, and basic notions about groups, especially abelian groups. Specifically, Abstract Algebra (Math 301), Topology (Math 331), Theory of Matrices (Math 307), and Advanced Calculus (Math 414 or 465) or consent of the instructor.

Course description:

We shall study the book `Algebraic Topology' by William Fulton (Springer-Verlag, 1995). From the preface:

`This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the relations of these ideas with other areas of mathematics. Rather than choosing one point of view of modern topology ..., we concentrate our attention on concrete problems in low dimensions, introducing only as much algebraic machinery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topologists... We also feel that this approach is in better harmony with the historical development of the subject.... We have tried to do this without assuming a graduate-level knowledge or sophistication.

`Algebraic topology can be thought of as the study of the shapes of geometric objects. It is sometimes referred to in popular accounts as ``rubber-sheet geometry''. In practice this means we are looking for properties of spaces that are unchanged when one space is deformed into another.... One problem of this type goes back to Euler: What relations are there among the number of vertices, edges, and faces in a convex polytope, such as a regular solid, in space? Another early manifistation of a topological idea came also from Euler, in the Koenigsberg bridge problem: When can one trace out a graph without traveling over any edge twice? Both these problems have a feature that characterizes one of the main attractions, as well as the power of modern algebraic topology - that a global question, depending on the overall shape of a geometric object, can be answered by data that are collected locally.

For more information about the course contact Dr. Kenneth R. Driessel (office: Carver 410, e-mail: driessel\, internet:$\sim$driessel.)