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 Poincare 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.

Publication:

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

http://dx.doi.org/10.1090/tran/9711

Author:Huang, Ruizhi

State Key Laboratory of Mathematical Sciences & Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

Email address: huangrz@amss.ac.cn