## Topics in algebraic Topology

## Course Advertisement, Summer, 1997

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

### Prerequisites:

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\@iastate.edu, internet:
http://www.public.iastate.edu/$\sim$driessel.)