Stabilization of Poincaré duality complexes and homotopy gyrations
Ruizhi Huang, Stephen Theriault
TL;DR
This work extends the T-stabilization program from manifolds to path-connected Poincaré Duality complexes by developing a homotopy-theoretic analogue of the gyration, denoted $\mathcal{G}^{\tau}(M)$. It establishes a finite classification of homotopy types for gyrations via the desuspended $J$-image $J([\tau])$ and derives loop-space decompositions that generalize earlier manifold results to PD complexes. The authors construct principal homotopy fibrations linking $\mathbb{C}P^{n}$- and $\mathbb{H}P^{n}$-stabilizations to gyrations, with explicit identifications $F\simeq \mathcal{G}^{n}_{\mathbb{C}}(M)$ or $F\simeq \mathcal{G}^{\overline{n}}_{\mathbb{H}}(M)$ controlling the stabilized loop spaces. They also develop stabilization-by-$T$ results for general $T$, including a BT-driven framework that yields a split after looping, and they illustrate the approach with stabilizations by products of spheres, including a to-be-explored 4-manifold application.
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 Poincaré 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 Poincaré Duality complex, a criterion is given for the possible homotopy types of gyrations and shows there are only finitely many.