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The four-dimensional Schoenflies conjecture is true for genus two imbeddings. (English) Zbl 0543.57011
It is known that if $$\Sigma^{n-1}$$ is a PL (n-1)-sphere embedded in the n-sphere $$S^ n$$ then the closure of the complementary regions are both homeomorphic to the n-ball. For $$n=4$$ it is unknown if these closures are PL isomorphic to the n-ball. In this paper it is shown that this is true for a special class of embeddings namely those of genus 2. We may define the genus as follows: Consider an embedded $$\Sigma^ 3$$ in $$S^ 4$$ and decompose $$S^ 4$$ into slices $$S^ 3\times \{t\}$$ indexed by time t $$(0<t<1)$$. Then the surfaces $$S^ 3\times \{t\}\cap \Sigma^ 3$$ may be constrained to act in the following way: In the beginning k 2-spheres appear in $$\Sigma^ 3$$ which are then joined by $$k-1+n$$ 1-handles to form a surface of genus n. Finally the 3-sphere is completed by adding 2 and 3-handles. The integer n is the genus of the embedding. By a beautiful and subtle combination of classical combinatorial techniques the author shows that if the genus is not greater than two then the embedding is isotropic to the standard one.
Reviewer: R.A.Fenn

MSC:
 57N50 $$S^{n-1}\subset E^n$$, Schoenflies problem 57Q35 Embeddings and immersions in PL-topology 57N10 Topology of general $$3$$-manifolds (MSC2010) 57N35 Embeddings and immersions in topological manifolds
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