Homotopy of Simply Connected Complexes with a Spherical Pair

TL;DR

This paper develops a loop space decomposition for certain simply connected CW-complexes with a spherical pair, leading to new insights into their homotopy properties, local hyperbolicity, and torsion phenomena.

Contribution

It generalizes known decompositions of Poincaré duality complexes and applies these results to study homotopy theory and torsion in high-dimensional CW-complexes.

Findings

Existence of infinitely many non-homotopy-equivalent complexes with rational but not local loop space retractions.

Production of infinitely many torsion homotopy groups with exponential rank growth.

New decompositions applicable to smooth manifolds with embedded spheres.

Abstract

We establish a loop space decomposition for certain $CW$-complexes with a single top cell in the presence of a spherical pair, thereby generalizing several known decompositions of Poincar\'{e} duality complexes in which a loop of a product of spheres appears as a direct summand. This decomposition is further applied to derive results on local hyperbolicity, on inertness and non-inertness, on the gaps between rational inertness and local or integral inertness, and on the homotopy theory of smooth manifolds with transversally embedded spheres. In particular, in every dimension greater than three, there exist infinitely many finite $CW$-complexes, pairwise non-homotopy-equivalent, whose loop spaces retract off the loops of their lower skeletons rationally but not locally, and whose top cell attachments produce infinitely many new torsion homotopy groups with exponentially growing ranks.