Stiefel-Whitney Class

Characteristic class w _i (ξ) which takes value in the co homology group H ⁱ(M(ξ );Z ₂) and is determined for real vector bundles ξ over the base . The Stiefel-Whitney class have the following characteristic properties: 1. For any two real vector bundles ξ , η over base

2. For the one-dimensional universal bundle ζ₁ over ℝP ^∞ the following equality is valid:

where y is a nonzero element of group H ¹(ℝP ^∞ ;ℤ₂)= ℤ₂, and w = 1 + w ₁ + w ₂ is a complete S-WC.

The S-WC allows to formulate the definition of orientable bundle. There exists an exact sequence of groups

The mapping

where is a cohomology of -bundle with the base , correlates any real vector bundle ξ and the first w ₁(ξ). As follows from a corresponding cohomological sequence, the bundle ξ with the base is orientable exclusively when w ₁(ξ)=0.

The S-WC allows to formulate the definition of spin bundle. There exists an exact sequence of groups

The mapping

of the corresponding cohomology sequence correlates every real vector bundle...