On spherical fibrations and Poincare complexes
Wen Shen
TL;DR
The paper develops a local-homotopy framework to determine when a stable spherical fibration over a 1-connected CW complex is stably fibre homotopy equivalent to a TOP-spherical fibration, and applies this to manifold recognition for Poincaré complexes. It combines 2-local and rational obstruction theory with Serre spectral sequences to derive lifting criteria and show that certain obstructions vanish under explicit cohomological hypotheses. A central contribution is a classification scheme for certain highly connected Poincaré complexes and the corresponding manifolds, tying homotopy types to Betti numbers and the Kervaire invariant, and linking these invariants to smooth vs. topological structure via surgery theory. The results yield explicit conditions under which Poincaré complexes have manifold homotopy types, provide classification in the highly connected regime, and reveal when smooth structures fail to exist in high dimensions, illuminating the interplay between spherical fibrations, Kervaire invariants, and surgery.
Abstract
In this paper, we prove that certain spherical fibrations over certain CW-complexes are stably fibre homotopy equivalent to $\mm{TOP}$-spherical fibrations (see Definition 1,1). Applying this result, we get a sufficient condition for whether a Poincar$\mm{\acute{e}}$ complex is of the homotopy type of a topological manifold. Moreover, we present the classification for some highly connected manifolds by the homotopy types of highly connected Poincar$\mm{\acute{e}}$ complexes.