Abstract

A k-dimensional manifold is a topological space that locally looks like R^k. For example, the surface of a beach ball is a 2-dimensional manifold: if you cut a little piece out of it, you can flatten it out so it looks like a disk in the 2-dimensional plane. The surface of an inner tube (a torus) has the same property, so it is also a 2-manifold. These two spaces are locally the same (both look like R²) but globally different. Much of what we know about the topology of manifolds comes from the fact that they can be decomposed into simple pieces called handles. I’ll discuss these handle decompositions, where they come from, and some things they can tell us, both for 2-dimensional surfaces and in higher dimensions.