Editing Thom space (section)

==Significance of Thom's work==

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 Thom spaces ''MG''(''n''). The proof depends on and is intimately related to the [[transversality (mathematics)|transversality]] properties of [[smooth manifolds]]—see [[Thom transversality theorem]]. By reversing this construction, [[John Milnor]] and [[Sergei Novikov (mathematician)|Sergei Novikov]] (among many others) were able to answer questions about the existence and uniqueness of high-dimensional manifolds: this is now known as [[surgery theory]]. In addition, the spaces ''MG(n)'' fit together to form [[spectrum (homotopy theory)|spectra]] ''MG'' now known as '''Thom spectra''', and the cobordism groups are in fact [[stable homotopy theory|stable]]. Thom's construction thus also unifies [[differential topology]] and stable homotopy theory, and is in particular integral to our knowledge of the [[stable homotopy groups of spheres]].

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

:<math>Sq^i : H^m(-; \Z_2) \to H^{m+i}(-; \Z_2),</math>

defined for all nonnegative integers ''m''. If <math>i=m</math>, then <math>Sq^i</math> coincides with the cup square. We can define the ''i''th Stiefel–Whitney class <math>w_i(p)</math> of the vector bundle <math>p: E\to B</math> by:

:<math>w_i(p) = \Phi^{-1}(Sq^i(\Phi(1))) = \Phi^{-1}(Sq^i(u)).</math>