Generalized property R and the Schoenflies conjecture.

*(English)*Zbl 1148.57032Both the generalized property \(R\) conjecture and the Schoenflies conjecture are well known: the first asks whether any \(n\)-component (\(n \geq 2\)) link \(L\) in \(\mathbb S^3\) yielding \(\#_n (\mathbb S^1 \times \mathbb S^2)\) by surgery may be converted into a 0-framed unlink by a suitable sequence of handle slides; the latter asks whether every PL (or, equivalently, smooth) 3-sphere in \(\mathbb S^4\) divides the 4-sphere into two PL-balls. Note that, if \(n=1\) is assumed, the generalized Property \(R\) Conjecture turns out to coincide with the famous Property \(R\) theorem [see D. Gabai, J. Differ. Geom. 26, 479–536 (1987; Zbl 0639.57008)].

The present paper deals with the relationships between the above Conjectures, and suggests a possible line of attack to the Schoenflies one. By means of the so called rectified critical level embedding, see C. Kearton and W. B. R. Lickorish [Trans. Am. Math. Soc. 170, 415–424 (1972; Zbl 0248.57007)] and of suitable results for Heegaard unions, the author obtains a quick proof of the genus two Schoenflies Conjecture (that has already been proved in the author’s paper [Topology 23, 211–217 (1984; Zbl 0543.57011)]): each complementary component of a genus 2 embedding of \(\mathbb S^3\) in \(\mathbb S^4\) is a 4-ball. Moreover, recent results in combinatorial 3-dimensional topology, particularly sutured manifold theory, see D. Gabai [Topology 26, 209–210 (1987; Zbl 0621.57004)] and A. Thompson [Topology 26, 205–207 (1987; Zbl 0628.57005)], allow to prove also the genus three Schoenflies Conjecture, by making use of the Property \(R\) theorem.

The present paper deals with the relationships between the above Conjectures, and suggests a possible line of attack to the Schoenflies one. By means of the so called rectified critical level embedding, see C. Kearton and W. B. R. Lickorish [Trans. Am. Math. Soc. 170, 415–424 (1972; Zbl 0248.57007)] and of suitable results for Heegaard unions, the author obtains a quick proof of the genus two Schoenflies Conjecture (that has already been proved in the author’s paper [Topology 23, 211–217 (1984; Zbl 0543.57011)]): each complementary component of a genus 2 embedding of \(\mathbb S^3\) in \(\mathbb S^4\) is a 4-ball. Moreover, recent results in combinatorial 3-dimensional topology, particularly sutured manifold theory, see D. Gabai [Topology 26, 209–210 (1987; Zbl 0621.57004)] and A. Thompson [Topology 26, 205–207 (1987; Zbl 0628.57005)], allow to prove also the genus three Schoenflies Conjecture, by making use of the Property \(R\) theorem.

Reviewer: Maria Rita Casali (Modena)

##### MSC:

57Q25 | Comparison of PL-structures: classification, Hauptvermutung |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |