Erdal, Mehmet Akif2019-02-212019-02-2120180166-8641http://hdl.handle.net/11693/49959In 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).EnglishCobordismHomology sphereK-theoryPoincaré dualitySpectral sequenceArticle10.1016/j.topol.2017.11.006