Trace embeddings from zero surgery homeomorphisms. (English) Zbl 1537.57057
The article is concerned with the smooth 4-dimensional Poincaré conjecture (SPC4), a long standing problem in 4-dimensional topology. The conjecture states that there are no smooth 4-dimensional manifolds that are homotopy equivalent but not diffeomorphic to the standard 4-sphere. Over the past decades, several potential counterexamples have been suggested, a good number of which has been proven to be diffeomorphic to the 4-sphere. This pattern is continued here.
Recently, C. Manolescu and L. Piccirillo [J. Lond. Math. Soc., II. Ser. 108, No. 5, 2001–2036 (2023; Zbl 1541.57006)] have constructed a family of knots which, if proven to be smoothly slice, would lead to “exotic” 4-spheres, that is, counterexamples to the SPC4. The main result of the present article is that the Manolescu-Picirillo knots are not slice. It is also shown that certain families of potentially exotic 4-spheres constructed by Manolescu and Picirillo are, in fact, diffeomorphic to the standard 4-sphere. Not surprisingly, the latter is done using Kirby calculus. The disproof of the sliceness properties ultimately makes use of Rasmussen’s \(s\)-invariant.
Recently, C. Manolescu and L. Piccirillo [J. Lond. Math. Soc., II. Ser. 108, No. 5, 2001–2036 (2023; Zbl 1541.57006)] have constructed a family of knots which, if proven to be smoothly slice, would lead to “exotic” 4-spheres, that is, counterexamples to the SPC4. The main result of the present article is that the Manolescu-Picirillo knots are not slice. It is also shown that certain families of potentially exotic 4-spheres constructed by Manolescu and Picirillo are, in fact, diffeomorphic to the standard 4-sphere. Not surprisingly, the latter is done using Kirby calculus. The disproof of the sliceness properties ultimately makes use of Rasmussen’s \(s\)-invariant.
Reviewer: Stefan Behrens (Bielefeld)
MSC:
| 57R60 | Homotopy spheres, Poincaré conjecture |
| 57K10 | Knot theory |
| 57K40 | General topology of 4-manifolds |
Citations:
Zbl 1541.57006Software:
GitHubReferences:
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