course

Math 443. Algebraic topology. Fall, 2023.

Textbook: [An Introduction

to Algebraic Topology by Joseph J. Rotman, 4th edition, Spriner-Verlag, 1998.](rotman.pdf)

• The book is available online at the link above.

What you need to know to take this course.

Some knowledge of topology is needed; the material in Math 240 or Math 440 will be adequate. It is more important to be comfortable with algebra; the material in Math 436 will be assumed. You will need to know about rings, modules, tensor products and Hom. These notions will be defined when they are first needed in the course, but if you have never seen them before, you may find the lectures hard to follow.

What is algebraic topology?

One answer to this is that it is the use of algebra to tell topological spaces apart.

• How do we know that a torus (the surface of a doughnut) is not homeomorphic to the 2-sphere?

• How can we be sure there is no homeomorphism of R^3 that takes a knotted circle to an unknotted one?

• How do we know that the complete graph on 5 points or the houses and utilities graph cannot be embedded in the plane?

In each case algebraic topology gives us a way to answer the question. For more information, see Essays about algebraic topology.