zero. Moreover, it was proved by René Thom that if all the Stiefel-Whitney numbers of M are zero then M can be realised as the boundary of some smooth compact manifold. One Stiefel–Whitney number of importance in surgery theory is the de Rham invariant of a (4k+1)-dimensional manifold, Wu classes The Stiefel–Whitney classes are the Steenrod squares of the Wu classes , defined by Wu Wenjun in . Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: . Wu classes are most often defined implicitly in terms of Steenrod squares, as the cohomology class representing the Steenrod squares. Let the manifold X be n dimensional. Then, for any cohomology class x of degree , . Or more narrowly, we can demand , again for cohomology classes x of degree . Integral Stiefel–Whitney classes The element is called the i + 1 integral Stiefel–Whitney class, where β is the Bockstein homomorphism, corresponding to reduction modulo 2, Z → Z/2Z: For instance, the third integral Stiefel–Whitney class is the obstruction to a Spinc structure. Relations over the Steenrod algebra Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth manifold (defined as the Stiefel–Whitney classes of the tangent bundle) are generated by those of the form . In particular, the Stiefel–Whitney classes satisfy the , named for Wu Wenjun: See also Characteristic class for a general survey, in particular Chern class, the direct analogue for complex vector bundles Real projective space References Dale Husemoller, Fibre Bundles, Springer-Verlag, 1994. External links Wu class at the Manifold Atlas Characteristic classes