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Man and machine thinking about the smooth 4-dimensional Poincaré conjecture. (English) Zbl 1236.57043
This paper presents the remainder of an aborted attempt to discard the smooth 4-dimensional Poincaré conjecture (SPC4).
The smooth version of Poincaré’s conjecture says that a manifold which is homeomorphic to the sphere $$S^4$$ is actually diffeomorphic to it. Over the years, several potential counterexamples have been highlighted, for instance by S. E. Cappell and J. L. Shaneson in [“There exist inequivalent knots with the same complement”, Ann. Math. (2) 103, 349–353 (1976; Zbl 0338.57008)]. These Cappell-Shaneson spheres were given handle presentations with no 3-handle and two 2-handle ones by the second author in [“On Cappell-Shaneson 4-spheres”, Topology Appl. 38, No. 2, 123–136 (1991; Zbl 0783.57016)]. In this description, the co-cores of the 2-handles are two disjoint disks bounding, in the complement of the 4-cell, a link $$L$$ which lies in a copy of $$S^3$$. If $$L$$ was proven not to be slice, then the complement of the 4-cell could not be $$B^4$$ and the considered Cappell-Shaneson sphere could not be diffeomorphic to the standard sphere. The authors’ idea was to prove unsliceness for such a link $$L$$ by using J. Rasmussen’s invariant, presented in [“Khovanov homology and the slice genus”, Invent. Math. 182, No. 2, 419–447 (2010; Zbl 1211.57009)], which is a lower bound for the slice genus. However, Rasmussen’s invariant is originally defined for knots only and, anyhow, $$L$$ is way too big for any computation to be made, even by a computer. The authors’ second idea was to bandsum the two components in order to get a simpler knot which would be manageable by a computer and for which unsliceness would also discard SPC4. In the meantime, S. Akbulut proved in [“Cappell-Shaneson homotopy spheres are standard”, Ann. Math. (2) 171, No. 3, 2171–2175 (2010; Zbl 1216.57017)] that the Cappell-Shaneson spheres considered by the authors were unfortunately standard. The strategy developed by the authors is nonetheless reusable for other counterexample candidates and the present paper is an opportunity to write down some considerations on SPC4.
The paper is organized as follows. The first part is introductory. The second part sums up the background and the strategy. Along the way, the authors prove that unsliceness for $$L$$ would also discard the Andrews-Curtis conjecture on balanced presentations of the trivial group, stated in [J. J. Andrews and M. L. Curtis, “Free groups and handlebodies”, Proc. Am. Math. Soc. 16, 192–195 (1965; Zbl 0131.38301)]. The third part is devoted to a topological reformulation of SPC4. It is phrased as a generalized property R saying roughly that links yielding connected sums of $$S^1\times S^2$$ by surgery are reducible to the empty diagram via some 4-dimensional Kirby moves. A stronger topology-related conjecture, about superfluity of odd-index handles for presenting closed simply connected 4-manifolds, is also formulated. The fourth part makes explicit the above-cited link $$L$$, determines a reduced diagram for it and discusses the choice of bandsumming to produce simpler knots. The fifth part deals with the algorithmic issues encountered when computing Khovanov homology of large knots and extracting Rasmussen’s invariant from it. It ends with the remark that the knot they have been considering and of which they have computed the Khovanov homology is far from confirming a conjectured correlation between Khovanov homology rank and the hyperbolic volume of knots.

##### MSC:
 57R60 Homotopy spheres, Poincaré conjecture 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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