On smooth manifolds with the homotopy type of a homology sphere

In this paper we study M(X), the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of X, via a map Ψ from M(X) into the quotient of K(X)=[X,BSO] by the action of the group of homotopy classes of simple self equivalences of X. The map Ψ describes which bundles over X can occur as normal bundles of manifolds in M(X). We determine the image of Ψ when X belongs to a certain class of homology spheres. In particular, we find conditions on elements of K(X) that guarantee they are pullbacks of normal bundles of manifolds in M(X).