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

*(English)*Zbl 1236.57043This 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.

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.

Reviewer: Benjamin Audoux (Marseille)

##### 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) |

##### Keywords:

smooth Poincaré conjecture; Cappell-Shaneson spheres; Rasmussen’s invariant; Khovanov homology; property R##### References:

[1] | I. R. Aitchison and J. H. Rubinstein, Fibered knots and involutions on homotopy spheres, Four-manifold theory (Durham, N.H., 1982), Contemp. Math. 35, Amer. Math. Soc., Providence, RI, 1984, 1-74. · Zbl 0567.57015 |

[2] | S. Akbulut, The Dolgachev surface. Preprint 2008. · Zbl 1251.57022 |

[3] | S. Akbulut, Cappell-Shaneson homotopy spheres are standard. Preprint 2009. · Zbl 1216.57017 |

[4] | S. Akbulut and R. Kirby, A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture. Topology 24 (1985), 375-390. · Zbl 0584.57009 |

[5] | D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial. Algebr. Geom. Topol. 2 (2002), 337-370. · Zbl 0998.57016 |

[6] | D. Bar-Natan, Khovanov’s homology for tangles and cobordisms. Geom. Topol. 9 (2005), 1443-1499. · Zbl 1084.57011 |

[7] | D. Bar-Natan, Fast Khovanov homology computations. J. Knot Theory Ramifications 16 (2007) , 243-255. · Zbl 1234.57013 |

[8] | A. Beliakova and S. Wehrli, Categorification of the colored Jones polynomial and Ras- mussen invariant of links. Canad. J. Math. 60 (2008), 1240-1266. · Zbl 1171.57010 |

[9] | S. E. Cappell and J. L. Shaneson, There exist inequivalent knots with the same comple- ment. Ann. of Math. (2) 103 (1976), 349-353. · Zbl 0338.57008 |

[10] | N. Dunfield, The Jones polynomial and volume. · Zbl 1291.68140 |

[11] | A. Elliott, State cycles, quasipositive modification, and constructing H-thick knots in Khovanov homology. Preprint 2009. |

[12] | D. Gabai, Foliations and the topology of 3-manifolds. III. J. Differential Geom. 26 (1987), 479-536. · Zbl 0639.57008 |

[13] | H. Gluck, The embedding of two-spheres in the four-sphere. Trans. Amer. Math. Soc. 104 (1962), 308-333. · Zbl 0111.18804 |

[14] | R. E. Gompf, Stable diffeomorphism of compact 4-manifolds. Topology Appl. 18 (1984), 115-120. · Zbl 0589.57017 |

[15] | R. E. Gompf, Killing the Akbulut-Kirby 4-sphere, with relevance to the Andrews-Curtis and Schoenflies problems. Topology 30 (1991), 97-115. · Zbl 0715.57016 |

[16] | R. E. Gompf, On Cappell-Shaneson 4-spheres. Topology Appl. 38 (1991), 123-136. · Zbl 0783.57016 |

[17] | R. E. Gompf, More Cappell-Shaneson spheres are standard. Preprint 2009. · Zbl 1244.57061 |

[18] | R. E. Gompf and M. Scharlemann, Fibered knots and property 2R, II. Preprint 2009. |

[19] | R. E. Gompf and A. I. Stipsicz, 4 -manifolds and Kirby calculus . Grad. Stud. Math. 20, Amer. Math. Soc., Providence, RI, 1999. · Zbl 0933.57020 |

[20] | C. M. Gordon, Knots in the 4-sphere. Comment. Math. Helv. 51 (1976), 585-596. · Zbl 0346.55004 |

[21] | J. Green, JavaKh. |

[22] | M. Gromov, Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307-347. · Zbl 0592.53025 |

[23] | M. Hedden and P. Ording, The Ozsváth-Szabó and Rasmussen concordance invariants are not equal. Amer. J. Math. 130 (2008), 441-453. · Zbl 1139.57012 |

[24] | M. Khovanov, A categorification of the Jones polynomial. Duke Math. J. 101 (2000), 359-426. · Zbl 0960.57005 |

[25] | M. Khovanov, A functor-valued invariant of tangles. Algebr. Geom. Topol. 2 (2002), 665-741. · Zbl 1002.57006 |

[26] | M. Khovanov, Patterns in knot cohomology. I. Experiment. Math. 12 (2003), 365-374. · Zbl 1073.57007 |

[27] | M. Khovanov, sl.3/ link homology. Algebr. Geom. Topol. 4 (2004), 1045-1081. · Zbl 1159.57300 |

[28] | M. Khovanov, Link homology and Frobenius extensions. Fund. Math. 190 (2006), 179-190. · Zbl 1101.57004 |

[29] | M. Khovanov and L. Rozansky, Matrix factorizations and link homology. Fund. Math. 199 (2008), 1-91. · Zbl 1145.57009 |

[30] | M. Khovanov and L. Rozansky, Matrix factorizations and link homology II. Geom. Topol. 12 (2008), 1387-1425. · Zbl 1146.57018 |

[31] | R. Kirby, A calculus for framed links in S3. Invent. Math. 45 (1978), no. 1, 35-56, · Zbl 0377.55001 |

[32] | E. S. Lee, An endomorphism of the Khovanov invariant. Adv. Math. 197 (2005), 554-586. · Zbl 1080.57015 |

[33] | M. Mackaay and P. Vaz, The universal sl3-link homology. Algebr. Geom. Topol. 7 (2007), 1135-1169. · Zbl 1170.57011 |

[34] | P. Melvin, Blowing-up and blowing-down in 4-manifolds. Ph.D. thesis, University of California, Berkeley 1977. |

[35] | S. Morrison and A. Nieh, On Khovanov’s cobordism theory for su3 knot homology. J. Knot Theory Ramifications 17 (2008), 1121-1173. · Zbl 1300.57013 |

[36] | P. S. Pao, Non-linear circle actions on the 4-sphere and twisting spun knots. Topology 17 (1978), 291-296. · Zbl 0403.57006 |

[37] | J. A. Rasmussen, Khovanov homology and the slice genus. Preprint 2010. · Zbl 1211.57009 |

[38] | J. Roberts, Kirby calculus in manifolds with boundary. Turkish J. Math. 21 (1997), 111-117. · Zbl 0899.57009 |

[39] | M. Scharlemann and A. Thompson, Fibered knots and property 2R. Preprint 2009. · Zbl 1214.57008 |

[40] | A. Shumakovitch, Torsion of the Khovanov homology. Preprint 2004. · Zbl 1297.57022 |

[41] | A. Shumakovitch, Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots. J. Knot Theory Ramifications 16 (2007), 1403-1412. · Zbl 1148.57011 |

[42] | E. Witten, Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121 (1989), 351-399. · Zbl 0667.57005 |

[43] | H. Wu, On the quantum filtration of the Khovanov-Rozansky cohomology. Adv. Math. 221 (2009), 54-139. · Zbl 1167.57007 |

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