This course aims to give a careful complete proof of the (topological) 4-Dimensional Poincare Conjecture, after Freedman. This is one of the crowning achievements of 20th century topology, but the details of the argument are appreciated by distressingly few mathematicians. There are essentially no prerequisites for the course other than the willingness to stare for a long time at complicated pictures.
Tuesday October 9 there will be no class.
We begin with an introduction to the `higher-dimensional' theory, giving the engulfing proof of Stallings-Zeeman, and the h-cobordism proof of Smale. We then introduce the theory of decompositions, culminating in a proof of the sphere-to-sphere theorem. We move on to gropes, and the combinatorial theory of Casson handles, enabling us to prove the reimbedding theorem. Finally we put these pieces together and show how they can be used to deduce the Poincare Conjecture.