The four-dimensional Schoenflies conjecture is true for genus two imbeddings.

*(English)*Zbl 0543.57011It 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