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Integral Third Stiefel–Whitney Class
Definition
Let be a paracompact Hausdorff compactly generated weak Hausdorff space of CW homotopy type, let be a numerable real vector bundle of finite rank, and use singular cohomology with the coefficient groups indicated. Assume the full Axiom of Choice required by Bockstein connecting operation, The Bockstein is independent of lift and representative, and Stiefel–Whitney classes from the projective-bundle relation. Let
be the coefficient sequence defining the integral Bockstein . Define the integral third Stiefel–Whitney class by
where is the second Stiefel–Whitney class of Stiefel–Whitney classes from the projective-bundle relation. This is an integral class; the definition does not replace it by its real image, so torsion information is retained.