Homotopy of simply connected complexes with a spherical pair

We establish a loop space decomposition for certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper W"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>W</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">CW</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -complexes with a single top cell in the presence of a spherical pair, thereby generalizing several known decompositions of Poincaré 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper W"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>W</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">CW</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.