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

Abstract

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 (arXiv:0907.0136) 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 x 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.

37 pages; changes reflecting that the integer family of Cappell-Shaneson spheres are now known to be standard (arXiv:0907.0136)

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• The Dolgachev Surface
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Cited by in corpus (27)

• Gauge theory and Rasmussen's invariant
• A refinement of Rasmussen's s-invariant
• Rasmussen's spectral sequences and the sl(N)-concordance invariants
• Knot concordance and homology cobordism
• Concordance maps in knot Floer homology
• The link concordance invariant from Lee homology
• Invariants of 4-manifolds from Khovanov-Rozansky link homology
• sl3-foam homology calculations
• Learning knot invariants across dimensions
• Relative genus bounds in indefinite four-manifolds
• Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems
• Ozsvath-Szabo bordered algebras and subquotients of category O
• Cappell-Shaneson homotopy spheres are standard
• Discrete Morse Theory Is At Least As Perfect As Morse Theory
• Computations of Szabó's Geometric Spectral Sequence in Khovanov Homology
• Fibered knots and Property 2R, II
• Categorical lifting of the Jones polynomial: a survey
• Deep and shallow slice knots in 4-manifolds
• A hitchhiker's guide to Khovanov homology
• The Rasmussen invariant of a homogeneous knot
• Graphical methods establishing nontriviality of state cycle Khovanov homology classes
• From zero surgeries to candidates for exotic definite four-manifolds
• Heegaard Floer Homology and Balanced Presentations of Groups
• Three-manifolds with boundary and the Andrews-Curtis transformations
• Rational knot concordance and homology cobordism
• Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures
• Some Fundamental Theorems in Mathematics