Math 440, Introduction
to Topology, SPRING 2007
Deaconu, AB 612, tel: 784-6110
Book: M.A. Armstrong, Basic Topology
Topology ( topos= place and logos= study) is an extension of geometry. It studies fundamental properties of shapes that remain unchanged when the shapes are deformed-that is, stretched, warped, or molded, but not torn. A simple example of such a shape is the 2-sphere, which is the 2-dimensional surface of a ball in 3-dimensional space. Another way to visualize the 2-sphere is to take a disk lying in the 2-dimensional plane and identify the disk's boundary points to a single point; this point may be thought of as the north pole of the 2-sphere. Although globally the 2-sphere looks very different form the plane, every point on the sphere sits in a region that looks like the plane. This property is the defining property of a 2-dimensional manifold. Another example of a 2-manifold is the torus, which is the surface of a doughnut. Although locally the 2-sphere and the torus look the same, globally their topologies are different. Consider a loop lying on the 2-sphere. No matter where it is situated, the loop can be shrunk down to a point, with the shrinking done entirely within the sphere. Now imagine a loop on the torus that goes around the hole. This loop cannot be shrunk to a point. The sphere has the property to be simply connected, while the torus is not simply connected.
Topology includes many subfields. The most basic division within topology is into point-set (or general) topology, which investigates such concepts as compactness, connectedness and separation axioms, algebraic topology which investigates notions as homotopy, homology, covering spaces and invariants of knots, and differential topology which studies degrees of smooth maps, Morse functions, transversality and De Rham cohomology.
Our class is an introduction to some of these fascinating topics. We will cover approximately chapters 1-5 from the book, with some sections from chapters 9 and 10, time permitting.