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

By Michael H. Freedman, Robert E. Gompf, S. Morrison, Kevin Walker. 2009

76 citations recorded by the source corpus.

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincare conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Ras- mussen'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 S 3 , only an unknot can yield S 1 � 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.

View Paper: https://www.semanticscholar.org/paper/d7143e438d362e23d85d41879be31c285bebc78a

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Citations in the corpus

Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures (2010, methodology): “In light of Akbulut’s recent work [1], more complicated potential counterexamples can also be generated by this same method; cf Freedman et al [5].”
Cappell-Shaneson homotopy spheres are standard (2009, background): “Then in [G1], this handlebody of Σ0 was shown to be diﬀeomorphic to S4. In [G2], it was shown that the handlebody of Figure 1 similarly describes Σ m, for the cases m 6= 0 (see also the discussion in [FGMW]). In this paper we will show the rest of all homotopy spheres Σ m are standard (i.e. m 6= 0 c”
Gauge theory and Rasmussen's invariant (2011, methodology): “This approach was advanced in [2], with the idea of using the non-vanishing of s(K) for this key step.”