Thursday, March 12Jim Conant, University of Tennessee
The question of whether a link bounds a union of (flat) disks into the 4-ball is an extremely subtle one. Rather than answer this question in full, we ask when can a simpler situation happen, namely can a link bound a "Whitney Tower" into the 4-ball. A Whitney tower is a type of 2-complex which approximates a disk better and better as the order of the tower increases. We propose a complete obstruction theory for Whitney towers, which is to say if a set of invariants vanishes, then one can extend the tower to a tower of one higher order. There are currently two "obstructions" to completing this program. The first obstruction is an algebraic problem concerning whether torsion exists in a certain abelian group of labeled trees. (The so-called Levine conjecture.) The second obstruction is that although we can compute almost all the invariants of Whitney towers in terms of Milnor invariants, there is still a hierarchy of unknown 2-torsion invariants which is needed to complete the theory. This work is Joint with Rob Schneiderman and Peter TeichnerBack to seminar home page.