On smooth manifolds with the homotopy type of a homology sphere (Q1688203)

This paper computes \({\mathcal M}(X)\), the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of \(X\) for the case where \(X\) belongs to a certain specified class of homology spheres. This is done by computing the image of the map \(\Phi\) from \({\mathcal M}(X)\) into the quotient of \(K(X) \;= \;[X, BSO]\) by the group of homotopy classes of simple self equivalences of \(X\). A critical ingredient of the computations is the use of the James spectral sequence which determines the homology of the Thom space of a stable vector bundle.