|
Organizers |
Failing to disprove the smooth 4-d Poincare conjecture.
by
Scott Morrison
Microsoft Station Q
Coauthors: Michael Freedman, Robert Gompf, Kevin Walker
There have long been proposed counterexamples to the smooth 4-d Poincare conjecture, such as the Cappell-Shaneson spheres. Robert Gompf found particularly nice handle decompositions of these, which in particular have only 0-, 1-, 2- and 4-handles. This means that the meridianal linking circles to the 2-handles are slice in the Cappell-Shaneson 4-balls, and thus showing that these linking circles are not slice in the standard 4-ball would be enough to see that the Cappell-Shaneson spheres really are counterexamples! Khovanov homology, and in particular the s-invariant, provides an obstruction to sliceness in the standard 4-ball.
Unfortunately, the smallest example is really large -- a two-component link with more than 200 crossings! I'll describe our efforts to compute the s-invariant anyway, involving a little bit of new mathematics, the newest and fastest program for computing Khovanov homology, and some serious computing power. And I'll try not to spoil the surprise of the final answer...
Date received: November 10, 2008
Copyright © 2008 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caxq-17.