Stabilization of Poincaré duality complexes and homotopy gyrations

Abstract: Stabilization of manifolds by a product of spheres or a projective space is important in geometry. There has been considerable recent work that studies the homotopy theory of stabilization for connected manifolds. This paper generalizes that work by developing new methods that allow for a generalization to stabilization of Poincar\'{e} Duality complexes. This includes the systematic study of a homotopy theoretic generalization of a gyration, obtained from a type of surgery in the manifold case. In particular, for a fixed Poincar\'{e} Duality complex, a criterion is given for the possible homotopy types of gyrations and shows there are only finitely many.