Topics in Algebraic Topology

Course Advertisement, Fall, 1997

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

First meeting: Wednesday, September 3 at 11am in Carver 50. Regular meetings: Monday and Friday at 11am in Carver 244.


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). A few of us began studying this book during the summer. We plan to return to the chapters that we skipped and then later continue further into the book. For a review of this book see the November, 1996 issue of the Am. Math. Monthly. 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.

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