The following extension problem is of considerable geometrical interest. Given a compact manifold X = ∂Z the boundary of a compact manifold Z, does the normal bundle of X in Z extend to a line bundle on Z? A related question involving foliations is if X is a compact boundary, is X a leaf of a codimension one foliation on any compact manifold bounded by X? Obviously if it were such a leaf, its normal bundle in the bounded manifold would extend. Reinhart posed both these problems in the language of cobordism theories(1), (2) defining a vector cobordism theory to a deal with questions of the first genre and a foliated-cobordism theory to deal with the second. He was able to obtain complete results in his vector-cobordism theory, whilst the foliated-cobordism theory remains an unsolved problem. Our aim in this article is to extend Reinhart's results on vector-cobordism theory to unoriented line bundles. Whereas in vector-cobordism the Euler number is a cobordism invariant, in the cobordism theory that we shall describe, the modulus of the Euler number becomes a cobordism invariant.