2009/06/30 by Michael Freedman, Robert E. Gompf, Robert Gompf +2 · 67 citations
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Conjecture #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematics #Poincaré conjecture #Pure mathematics #math.GT #math.QA #msc:57M25 #msc:57N13 #msc:57R60
paper · pdf · doi:10.4171/qt/5
published in Quantum Topology 1(2), 171-208 · 37 pages; changes reflecting that the integer family of Cappell-Shaneson spheres are now known to be standard (arXiv:0907.0136)
arxiv created 2009/09/17 · openalex publication_date 2010/06/22 · arxiv updated 2010/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 [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 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.