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.
Categories: math.GT math.AT