DescriptionProving the Poincaré conjecture was worth $1 million and a Fields medal to a man who turned down both. The Poincaré conjecture roughly states that if you have a 3-dimensional space M in which every loop can be shrunk to a point, then M must be a 3-sphere. There was an earlier conjecture made which states that if any loop in your 3-dimensional space M is the boundary of some surface which also lies in M, then M is the 3-sphere. This earlier conjecture is false, and the first counterexample is the Poincaré homology sphere. Let's back up a bit - what is meant here by a 3-dimensional space? How can we construct 3-dimensional spaces and how do we know if two such spaces are the same or different? We will take the Poincaré homology sphere as a motivation to try to understand, describe, distinguish, and catalogue 3-dimensional spaces using various techniques. Our methods will most often be highly visual - cutting up and gluing together pieces that we understand to make new spaces with interesting properties.
PrerequisitesGeometric Topology II.Co-requisitesTopology III.ResourcesYou can look up the wikipedia articles on Dehn surgery and Heegaard splitting to get some idea of the techniques used in constructing 3-dimensional spaces. A couple of books that we might look at (although they go much further and deeper than we need) include Knots and Links by Rolfsen, and 3-manifolds by Hempel. To guide our reading throughout the project, we shall be looking at this paper, which gives eight different descriptions of the Poincaré homology sphere. We shall try to understand some of these constructions in generality, and relate them to each other in our specific case. |
email: Andrew Lobb