On the Stable Homotopy Type of Thom Complexes

Let α be a real vector bundle over a finite CW complex X and let T(α;X) be its associated Thorn complex. We propose to study the S-type (stable homotopy type) of Thorn complexes in the framework of the Atiyah-Adams J-Theory. Therefore we focus our attention on the group J R(X) which is defined to be the group of orthogonal sphere bundles over X modulo stable fiber homotopy equivalence.