for line bundles implies that any functor θ satisfying the four axioms above is equal to w, by the following argument. The second axiom yields θ(γ1) = 1 + θ1(γ1). For the inclusion map i : P1(R) → P∞(R), the pullback bundle is equal to . Thus the first and third axiom imply Since the map is an isomorphism, and θ(γ1) = w(γ1) follow. Let E be a real vector bundle of rank n over a space X. Then E admits a splitting map, i.e. a map f : X′ → X for some space X′ such that is injective and for some line bundles . Any line bundle over X is of the form for some map g, and by naturality. Thus θ = w on . It follows from the fourth axiom above that Since is injective, θ = w. Thus the Stiefel–Whitney class is the unique functor satisfying the four axioms above. Non-isomorphic bundles with the same Stiefel–Whitney classes Although the map is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle for n even. With the canonical embedding of in , the normal bundle to is a line bundle. Since is orientable, is trivial. The sum is just the restriction of to , which is trivial since is contractible. Hence w(TSn) = w(TSn)w(ν) = w(TSn ⊕ ν) = 1. But, provided n is even, TSn → Sn is not trivial; its Euler class , where [Sn] denotes a fundamental class of Sn and χ the Euler characteristic. Related invariants Stiefel–Whitney numbers 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 . 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