Talk by Justin Roberts (UCSD)
Date and Time: Tuesday, November 18, 2008, 12:00 PM in AP&M B412.
Title: Low-dimensional manifolds
Abstract:
An n-dimensional manifold is a space which has n local degrees of freedom.
For example, the surface of the Earth or of a doughnut is a 2-dimensional
manifold; the complement of a knotted loop of string in space is an
interesting 3-dimensional manifold; space-time is a 4-dimensional manifold.
Topology is concerned with studying the qualitative behaviour of such spaces
(in contrast with geometry, which studies quantitative measurements such as
distance, area and angle). For various reasons, "low-dimensional" manifolds
(those of dimension 4 or less) are more interesting to topologists than
high-dimensional ones. Luckily, these are the dimensions which are most easy
to visualise, and in low-dimensional topology we often prove theorems by
developing our visual intuition into rigorous proofs. I'll try to illustrate
some ways to think about 3- and 4-dimensional manifolds, and explain the
delightfully knotty "calculus of handles" invented by Rob Kirby for drawing
pictures of them.