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Generalized property R and the Schoenflies conjecture. (English) Zbl 1148.57032
Both 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.

##### MSC:
 57Q25 Comparison of PL-structures: classification, Hauptvermutung 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)
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