If we work on a manifold of dimension "n", then any product of Stiefel–Whitney classes of total degree "n" can be paired with the Z/2Z-fundamental class of the manifold to give an element of Z/2Z, a Stiefel–Whitney number of the vector bundle. For example, if the manifold has dimension 3, there are three linearly independent Stiefel–Whitney numbers, given by formula_44. In general, if the manifold has dimension "n", the number of possible independent Stiefel–Whitney numbers is the number of partitions of "n". The Stiefel–Whitney numbers of the tangent bundle of a smooth manifold are called the Stiefel–Whitney numbers of the manifold. They are known to be cobordism invariants. It was proven by Lev Pontryagin that if "B" is a smooth compact ("n"+1)–dimensional manifold with boundary equal to "M", then the Stiefel-Whitney numbers of "M" are all 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 (4"k"+1)-dimensional manifold, formula_45  The Stiefel–Whitney classes "wk" are the Steenrod squares of the Wu classes "vk", defined by Wu Wenjun in . Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: "Sq"("v") = "w". 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 "n-k", formula_46. Or more narrowly, we can demand formula_47, again for cohomology classes "x" of degree "n-k". The element formula_48 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.