over X. The Grassmannian of lines is just the infinite projective space which is doubly covered by the infinite sphere by antipodal points. This sphere is contractible, so we have Hence P∞(R) is the Eilenberg-Maclane space . It is a property of Eilenberg-Maclane spaces, that for any X, with the isomorphism given by f → f*η, where η is the generator . Applying the former remark that α : [X, Gr1] → Vect1(X) is also a bijection, we obtain a bijection this defines the Stiefel–Whitney class w1 for line bundles. The group of line bundles If Vect1(X) is considered as a group under the operation of tensor product, then the Stiefel–Whitney class, w1 : Vect1(X) → H1(X; Z/2Z), is an isomorphism. That is, w1(λ ⊗ μ) = w1(λ) + w1(μ) for all line bundles λ, μ → X. For example, since H1(S1; Z/2Z) = Z/2Z, there are only two line bundles over the circle up to bundle isomorphism: the trivial one, and the open Möbius strip (i.e., the Möbius strip with its boundary deleted). The same construction for complex vector bundles shows that the Chern class defines a bijection between complex line bundles over X and H2(X; Z), because the corresponding classifying space is P∞(C), a K(Z, 2). This isomorphism is true for topological line bundles, the obstruction to injectivity of the Chern class for algebraic vector bundles is the Jacobian variety. Properties Topological interpretation of vanishing wi(E) = 0 whenever i > rank(E). If Ek has sections which are everywhere linearly independent then the top degree Whitney classes vanish: . The first Stiefel–Whitney class is zero if and only if the bundle is orientable. In particular, a manifold M is orientable if and only if w1(TM) = 0. The bundle admits a spin structure if and only if both the first and second Stiefel–Whitney classes are zero. For an orientable bundle, the second Stiefel–Whitney class is in the image of the natural map H2(M, Z) → H2(M, Z/2Z) (equivalently, the so-called third integral Stiefel–Whitney class is zero) if and only if the bundle admits a spinc structure. All the Stiefel–Whitney numbers (see below) of a smooth compact manifold X vanish if and only if the manifold is the boundary of some smooth compact (unoriented) manifold (Warning: Some Stiefel-Whitney class could still be non-zero, even if all the Stiefel Whitney numbers vanish!) Uniqueness of the Stiefel–Whitney classes The bijection above