 Stable normal data is used instead of unstable tangential data because generalizations of manifolds have natural stable normal-type structures, coming from tubular neighborhoods and generalizations, but not unstable tangential ones, as the local structure is not smooth.Spherical fibrations over a space "X" are classified by the homotopy classes of maps formula_22 to a
classifying space formula_23, with homotopy groups the stable homotopy groups of spheres

The forgetful map formula_25 extends to a fibration sequence

A Poincaré space "X" does not have a tangent bundle, but it does have a well-defined stable spherical fibration, which for a differentiable manifold is the spherical fibration associated to the stable normal bundle; thus a primary obstruction to "X" having the homotopy type of a differentiable manifold is that the spherical fibration lifts to a vector bundle, i.e., the Spivak spherical fibration formula_22 must lift to formula_28, which is equivalent to the map formula_29 being null homotopic
Thus the bundle obstruction to the existence of a (smooth) manifold structure is the class formula_29.The secondary obstruction is the Wall surgery obstruction.