Math 430 Introduction to Topology Spring 2015 Professor Sullivan

This course is intended as a gentle introdoction to topology. Topology is the abstract study of continuity and convergence. What are the basic structures one needs to talk rigorously about a function from one set to another (or itself) being continuous?

In Topology, once continuity is understood, we classify sets (really topological spaces) by asking if there is a continuous bijection with continuous inverse between them. If so, we say they are homeomorphic.

The textbook gives an elementary introduction to topology, but allows the student to see the beginnings of each of the three major branches of topology, namely, point-set topology, algebraic topology and differential topology.

Topology is not just for theorists anymore. Major and surprising applications exist. In some sense the field of differential equations is a subset of differential topology. Algebraic topology is used in everything from general relativity to the mathematics of robotic systems. Point-set topology is now used in computer science and chaos theory.

Textbook: A Combinatorial Introduction to Topology by Michael Henle. Amazon B&N

Grades: Homework 30%, midterm 30%, final 30%, (maybe) class presentations 10%.

• Alexander's Horned Sphere.
• Phase Portraits with Maple.
• Morton Brown. A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc. 66 (1960), no. 2, 74--76.
• T. F. Banchoff, Critical Points and Curvature for Embedded Polyhedral Surfaces, The American Mathematical Monthly, Vol. 77, No. 5 (May, 1970), pp. 475-485.
• Barry A. Cipra. Homology: An Idea Whose Time Has Come, SIAM News, December 4, 2009.
• Triangulation and the Hauptvermutung. (Web resource page)
• Poincaré conjecture. (Wikipedia)
• Hud Hudson. Four Colors Do Not Suffice, The American Mathematical Monthly, Vol. 110, No. 5 (May, 2003), pp. 417-423.