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p be a rank ''k'' Real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the fiber ''F''''b'' is a ''k''-dimensional real Vector Space . We can form an associated Sphere Bundle Sph(''E'') → ''B'' by taking the One-point Compactification of each fiber separately. Finally, from the total space Sph(''E'') we obtain the Thom complex ''T''(''E'') by identifying all the new points to a single point , which we take as the Basepoint of ''T''(''E''). The significance of this construction begins with the following result, which belongs to the subject of Cohomology of fiber bundles. (We have stated the result in terms of Z2 Coefficients to avoid complications arising from Orientability .) Let ''B'', ''E'', and ''p'' be as above. Then there is an isomorphism, now called a Thom isomorphism :, for all ''i'' greater than or equal to 0, where the Right Hand Side is Reduced Cohomology . We can loosely interpret the theorem in the following geometric sense. Since ''E'' is a vector bundle it Retracts onto the base ''B''. So we might suppose that ''E'' would be cohomologically equivalent to ''B''. In a way, the theorem bears out this expectation. This theorem was formulated and proved by :
In his 1952 paper, Thom showed that the Thom class, the Stiefel-Whitney Class es, and the Steenrod Operation s were all related. He used these ideas to prove in the 1954 paper ''Quelques propriétés globales des variétés differentiables'' that the Cobordism groups could be computed as the Homotopy Groups of certain spaces ''MSO''(''n''). The spaces ''MSO(n)'' themselves arise as Thom spaces and comprise a Spectrum ''MSO'' that is now called a ''Thom spectrum'' (along with other related spectra). This was a major step toward modern Stable Homotopy Theory . If the Steenrod operations are available, we can use them and the isomorphism of the theorem to construct the Stiefel-Whitney classes. Recall that the Steenrod operations (mod 2) are Natural Transformation s :, defined for all nonnegative integers ''m''. If ''i'' = ''m'', then ''Sqi'' coincides with the cup square. We can define the ''i''th Stiefel-Whitney class ''w''''i'' (''p'') of the vector bundle ''p'' : ''E'' → ''B'' by: : SEE ALSO REFERENCES
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