Stiefel-Whitney class

Stiefel-Whitney classes arise in , usually thought of as having fibres [0,1]. The cohomology group :H^1(S^1,mathbb Z/2mathbb Z) has just one element other than 0, this element being the first Steifel-Whitney class, w_1, of that line bundle.

=Axioms=

Throughout, H^i(;cdot;;G) denotes singular cohomology with coefficient group (mathematics) G.

# For every real vector bundle E ightarrow X, there exist w_i(E) in H^i(X;mathbb Z/2mathbb Z) which are natural transformation, i.e., characteristic classes. # w_0(E)=1 in H^0(X;mathbb Z/2mathbb Z). # w_ i(E)=0 whenever i>mathrm{rank}(E). # w_1(gamma^1)=x in H^1(mathbb RP^1;mathbb Z/2mathbb Z)=mathbb Z/2mathbb Z (normalization condition). Here, gamma^n is the canonical line bundle. # w_k(Eoplus F)=sum_{i+j=k}w_i(E)cup w_j(F). # If E^k has s_1,ldots,s_{ell} fiber bundle#sections which are everywhere linearly independent then w_{k-ell+1}=cdots=w_k=0.

Some work is required to show that such classes do indeed exist and are unique.

=Properties=

The first Stiefel-Whitney class is zero if and only if the bundle is orientability.

The second Stiefel-Whitney class is zero if and only if the bundle admits a spin structure.

=See also=

*Singular homology for the definition of (co)homology with coefficients .

=References=

J. Milnor & J. Stasheff, Characteristic Classes , Princeton, 1974.