a Stiefel-Whitney class
/ SHTEE-ful HWIT-nee /
Take a real vector bundle E and ask the most basic global questions: can you comb it (find a nowhere-zero section)? Is it orientable? Does it carry a spin structure? The Stiefel-Whitney classes are a sequence of mod-2 cohomology classes w_i(E) in H^i(X; Z/2), i = 0, 1, 2, ..., that answer exactly such yes/no obstruction questions, where the only arithmetic available is even-versus-odd because we work mod 2.
They are defined cleanly through the classifying space: H^*(BO(n); Z/2) is a polynomial ring Z/2[w_1, ..., w_n], and the Stiefel-Whitney classes of a bundle E are the pullbacks of these generators along the classifying map X -> BO(n). The total class w(E) = 1 + w_1 + w_2 + ... is multiplicative for direct sums, w(E + E') = w(E) w(E') (the Whitney sum formula), and can be computed by the splitting principle. Two of the low classes have crisp meaning: w_1(E) in H^1(X; Z/2) vanishes if and only if E is orientable, and for an orientable E, w_2(E) in H^2(X; Z/2) is the obstruction to a spin structure.
Their power is that they are computable, defined over any base (only mod-2 cohomology is needed, no orientation or complex structure required), and detect real phenomena — the nonexistence of certain immersions and embeddings, the parallelizability of spheres, the failure of a manifold to bound. The honest limits: they are mod-2 invariants, so they are blind to any finer integral information, and they can vanish for nontrivial bundles. The tangent bundle of a manifold has Stiefel-Whitney classes that are diffeomorphism invariants of the manifold (Stiefel-Whitney numbers), but two bundles with identical w_i can still be non-isomorphic.
For the tangent bundle of the real projective plane RP^2, the total Stiefel-Whitney class is w(T RP^2) = (1 + a)^3 = 1 + a + a^2, where a is the generator of H^1(RP^2; Z/2). Since w_1 = a is nonzero, RP^2 is non-orientable; since w_2 = a^2 is nonzero, RP^2 does not bound a 3-manifold. Both facts fall straight out of the mod-2 arithmetic.
w_1 detects non-orientability, w_2 detects that RP^2 is not a boundary — both read off mod 2.
Stiefel-Whitney classes live in mod-2 cohomology and carry only 2-torsion-level information; they cannot see anything an integral or rational invariant would. Equal Stiefel-Whitney classes do NOT imply isomorphic bundles. They are obstructions, not a complete invariant.