Man and machine thinking about the smooth 4-dimensional Poincar\'e conjecture

TL;DR

This paper investigates the smooth 4-dimensional Poincaré conjecture using Khovanov homology invariants, reports computational challenges, and discusses recent developments that suggest the conjecture may be true, encouraging further research in 3-manifold topology.

Contribution

The paper applies Rasmussen's s-invariant to study potential counterexamples to SPC4 and discusses the impact of recent results that confirm certain homotopy spheres are standard.

Findings

s-invariant computations yielded zero for studied knots

Recent work by Akbulut confirmed certain spheres are standard

SPC4's plausibility is increased by recent theoretical insights

Abstract

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincar\'{e} 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 (arXiv:0907.0136) 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 S^3, only an unknot can yield S^1 x S^2 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.