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

On smooth manifolds with homotopy type of a homology sphere