Poincaré duality and periodicity (Q551147)

This paper is concerned with finite, oriented Poincaré complexes \(X^n\) of dimension \(n\geq 3\). When such a complex \(X\) is written as \(X= K\cup_\alpha D^n\), where \(K\) is a CW complex of dimension \(\leq n-1\) and \(\alpha: S^{n-1}\to K\) attaches the \textit{top cell}, then the \textit{spine} \(K\) of \(X\) is unique up to homotopy. With this notation, Theorem A states: If the top cell of \(X\) splits off after suspension (i.e., \(\Sigma X\simeq \Sigma K\vee S^{n+1}\)), then there exists a Poincaré complex \(Y^{n+4}\) with spine \(\Sigma^2K\) such that the top cell of \(Y\) also splits off after suspension. Examples of \(X^n\) whose top cells split off after suspension are those that Poincaré embed in \(S^{n+1}\). Theorem B states: If \(X^n\) Poincaré embeds in \(S^{n+1}\), then there exists a Poincaré complex \(Y^{n+2}\) with spine \(\Sigma K\) such that \(Y\) also Poincaré embeds in \(S^{n+3}\). These results are related to a question of Hodgson in [Miller, Haynes R. (ed.); Priddy, Stewart B. (ed.), Proceedings of the Northwestern homotopy theory conference (held at Northwestern University, Evanston, Illinois, March 22-26, 1982). Contemporary Mathematics, 19. Providence, R.I.: American Mathematical Society (AMS) (1983; Zbl 0508.00006)]. The authors also show how Theorem A gives a homotopy-theoretic operator from \(n\)-knots to (\(n+4\))-knots, inducing the four-fold periodicity in the knot cobordism groups of \textit{J. Levine} [Comment. Math. Helv. 44, 229--244 (1969; Zbl 0176.22101)].