On spherical fibrations and Poincare complexes

TL;DR

This paper establishes conditions under which spherical fibrations are equivalent to TOP-spherical fibrations and uses these results to classify certain highly connected Poincaré complexes and their relation to topological manifolds.

Contribution

It proves stable fibre homotopy equivalence between spherical fibrations and TOP-spherical fibrations, providing new criteria for Poincaré complexes to be homotopy equivalent to topological manifolds.

Findings

Spherical fibrations over CW-complexes are stably fibre homotopy equivalent to TOP-spherical fibrations.

A sufficient condition is given for Poincaré complexes to have the homotopy type of a topological manifold.

Classification results for highly connected manifolds based on homotopy types of Poincaré complexes.

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.