a real vector bundle of rank 1) that is not isomorphic to a trivial bundle. This line bundle is the Möbius strip (which is a fiber bundle whose fibers can be equipped with vector space structures in such a way that it becomes a vector bundle). The cohomology group has just one element other than 0. This element is the first Stiefel–Whitney class of . Since the trivial line bundle over has first Stiefel–Whitney class 0, it is not isomorphic to . Two real vector bundles and which have the same Stiefel–Whitney class are not necessarily isomorphic. This happens for instance when and are trivial real vector bundles of different ranks over the same base space . It can also happen when and have the same rank: the tangent bundle of the 2-sphere and the trivial real vector bundle of rank 2 over have the same Stiefel–Whitney class, but they are not isomorphic. But if two real line bundles over have the same Stiefel–Whitney class, then they are isomorphic. Origins The Stiefel–Whitney classes get their name because Eduard Stiefel and Hassler Whitney discovered them as mod-2 reductions of the obstruction classes to constructing everywhere linearly independent sections of the vector bundle restricted to the i-skeleton of X. Here n denotes the dimension of the fibre of the vector bundle . To be precise, provided X is a CW-complex, Whitney defined classes in the i-th cellular cohomology group of X with twisted coefficients. The coefficient system being the -st homotopy group of the Stiefel manifold of linearly independent vectors in the fibres of E. Whitney proved that if and only if E, when restricted to the i-skeleton of X, has linearly-independent sections. Since is either infinite-cyclic or isomorphic to , there is a canonical reduction of the classes to classes which are the Stiefel–Whitney classes. Moreover, whenever , the two classes are identical. Thus, if and only if the bundle is orientable. The class contains no information, because it is equal to 1 by definition. Its creation by Whitney was an act of creative notation, allowing the Whitney sum Formula to be true. Definitions Throughout, denotes singular cohomology of a space with coefficients in the group . The word map means always a continuous function between topological spaces. Axiomatic definition The Stiefel-Whitney characteristic class of a finite rank real vector bundle E on a paracompact base space X is defined as