In mathematics, the Thom space or Thom complex (named after René Thom) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. One way to construct this space is as follows. Let

: p : E → B

be a rank k real number 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 infty, 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 :Phi colon H^i(B; mathbf{Z}_2) o ilde{H}^{i+k}(T(E); mathbf{Z}_2), for all i greater than or equal to 0, where the RHS is reduced cohomology.

We can loosely interpret the theorem in the following geometric sense. Since E is a vector bundle it retraction 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 : :Phi(b) = p^*(b) smile U. In particular, the Thom isomorphism sends the identity_(mathematics) element of H *( B ) to U .

In his 1952 paper, Thom showed that the Thom class, the Stiefel-Whitney classes, and the Steenrod operations 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 (homotopy theory) 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 transformations :Sq^i colon H^m(-; mathbf{Z}_2) o H^{m+i}(-; mathbf{Z}_2), 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 wi(p) of the vector bundle p : E → B by: :w_i(p) = Phi^{-1}(Sq^i(Phi(1))) = Phi^{-1}(Sq^i(U)).

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