Embedding 4-manifolds in \(S^ 5\)
\textit{S. E. Cappell} and \textit{J. L. Shaneson} have announced [Geometric topology, Proc. Conf., Athens/Ga. 1977, 301-305 (1979; Zbl 0467.57009)] that a closed, orientable smooth 4-manifold embeds smoothly in \(S^ 6\) iff it has spin and has zero index. In this paper, the question of when such 4-manifolds embed in \(S^ 5\) is investigated. The main result gives sufficient conditions of a homotopy- \(theoretic\) nature (specified spin-structure and splittings of \(\pi_ 1\) and \(H_ 2/torsion)\) in a broad range of situations. The author gives necessary and sufficient conditions for X to embed if \(H_ 2(X)\) is finite or \(H_ 2(\pi_ 1(X))=0\), resp. to embed stably (X \(\#_{k} (S^ 2\times S^ 2))\) without restrictions. The proof relies on constructing candidates for complements of X in \(S^ 5\) by performing careful spin-surgery.
scientific article; zbMATH DE number 3932029