Cornell Math - MATH 661, Fall 1999

MATH 661 — Fall 1999
Introduction to Geometric Topology

Instructor: James West

Time: TR 2:55-4:10

Room: Malott 206

This will be an introduction taught at a level consistent with the background of the students in the course. (For example, any algebraic techniques will be given a self-contained treatment.)

Contents are negotiable, but regarded as a first course in topology after the basics (connectedness, compactness, metric spaces) acquired in an analysis course, one (overfull) outline might be as follows:

• The plane:

  • The Jordan Curve Theorem

  • The Schönflies Theorem

• Surfaces.

• Knots I (Geometric)

• 3-manifolds (Introduction)

• Where the Wild Things Are:

  • Antoine's Necklace

  • Fox-Artin Arc

  • Alexander's Horned Sphere

  • Generalized Schönflies Theorem

• Knots II (Various geometric and algebraic invariants associated to knots)

• More about 3-manifolds.

Variations might include:

If the class is interested in Complex Dynamics and Hubbard-Douady Matings of Julia Sets, we could substitute a proof of R.L. Moore's Theorem (characterizing those mappings of the 2-sphere whose images are 2-spheres) and its application for some of the 3-dimensional topology.

On the other hand, if the class wants to jump right into knot theory, and is willing to tolerate a "quick and dirty" intro to homology as a geometric tool, we could spend the whole semester following Lickorish's An Introduction to Knot Theory (Springer GTM v.175) as far as we can.