Man and machine thinking about the smooth 4-dimensional Poincaré conjecture

Abstract

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincar� conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Rasmussen�s s-invariant, a slice obstruction within the general framework of Khovanov homology, changes this state of affairs. We studied a class of knots K for which nonzero s.K/ would yield a counterexample to SPC4. Computations are extremely costly and we had only completed two tests for those K, with the computations showing that s was 0, when a landmark posting of Akbulut [3] altered the terrain. His posting, appearing only six days after our initial posting, proved that the family of �Cappell�Shaneson� homotopy spheres that we had geared up to study were in fact all standard. The method we describe remains viable but will have to be applied to other examples. Akbulut�s work makes SPC4 seem more plausible, and in another section of this paper we explain that SPC4 is equivalent to an appropriate generalization of Property R (�in S3, only an unknot can yield S1 S2 under surgery�). We hope that this observation, and the rich relations between Property R and ideas such as taut foliations, contact geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to look at SPC4. Show more