On smooth manifolds with homotopy type of a homology sphere

In this paper we study $\mathcal{M}(X)$, the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of $X$, via a map $\mathrm{\Psi }$ from $\mathcal{M}(X)$ 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 $\mathrm{\Psi }$ describes which bundles over $X$ can occur as normal bundles of manifolds in $\mathcal{M}(X)$. We determine the image of $\mathrm{\Psi }$ 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 $\mathcal{M}(X)$.