We construct periodic families of Poincaré complexes, partially solving a question of Hodgson, and infinite families of Poincaré complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.
John R Klein. William Richter. "Poincaré duality and periodicity." Algebr. Geom. Topol. 11 (4) 1961 - 1985, 2011. https://doi.org/10.2140/agt.2011.11.1961