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Man and machine thinking about the smooth 4-dimensional Poincaré conjecture
Michael H. Freedman (1), Robert E. Gompf (2), Scott Morrison (3) and Kevin Walker (4)(1) Microsoft Station Q, University of California, CNSI Bldg. Office 2245, CA 93106-6105, SANTA BARBARA, UNITED STATES
(2) Department of Mathematics, University of Texas at Austin, 1 University Station C1200, TX 78712-0257, AUSTIN, UNITED STATES
(3) Department of Mathematics, University of California, 1071 Evans Hall, CA 94720-3840, BERKELEY, UNITED STATES
(4) Microsoft Station Q, University of California, CNSI Bldg., Office 2243, CA 93106-6105, SANTA BARBARA, UNITED STATES
While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincaré 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  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.
Keywords: Property R, Cappell–Shaneson spheres, Khovanov homology, s-invariant, Poincaré conjecture