One way to construct this space is as follows. Let be a rank nreal vector bundle over the paracompact space B. Then for each point b in B, the fiber is an n-dimensional real vector space. We can form an n-sphere bundle by taking the one-point compactification of each fiber and gluing them together to get the total space. Finally, from the total space we obtain the Thom space as the quotient of by B; that is, by identifying all the new points to a single point, which we take as the basepoint of. If B is compact, then is the one-point compactification of E. For example, if E is the trivial bundle, then is and, writing for B with a disjoint basepoint, is the smash product of and ; that is, the n-th reduced suspension of. Alternatively, since B is paracompact, E can be given a Euclidean metric and then can be defined as the quotient of the unit disk bundle of E by the unit -sphere bundle of E.
The Thom isomorphism
The significance of this construction begins with the following result, which belongs to the subject of cohomology of fiber bundles. Let be a real vector bundle of rank n. Then there is an isomorphism, now called a Thom isomorphism for all kgreater than or equal to 0, where the right hand side is reduced cohomology. This theorem was formulated and proved by René Thom in his famous 1952 thesis. We can interpret the theorem as a global generalization of the suspension isomorphism on local trivializations, because the Thom space of a trivial bundle on B of rank k is isomorphic to the kth suspension of, B with a disjoint point added This can be more easily seen in the formulation of the theorem that does not make reference to Thom space: In concise terms, the last part of the theorem says that u freely generates as a right -module. The class u is usually called the Thom class of E. Since the pullback is a ring isomorphism, is given by the equation: In particular, the Thom isomorphism sends the identity element of to u. Note: for this formula to make sense, u is treated as an element of
Significance of Thom's work
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 Thom spaces MG. The proof depends on and is intimately related to the transversality properties of smooth manifolds—see Thom transversality theorem. By reversing this construction, John Milnor and Sergei Novikov 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 fit together to form spectraMG now known as Thom spectra, and the cobordism groups are in fact 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 are natural transformations defined for all nonnegative integersm. If, then coincides with the cup square. We can define the ith Stiefel–Whitney class of the vector bundle by:
If we take the bundle in the above to be the tangent bundle of a smooth manifold, the conclusion of the above is called the Wu formula, and has the following strong consequence: since the Steenrod operations are invariant under homotopy equivalence, we conclude that the Stiefel–Whitney classes of a manifold are as well. This is an extraordinary result that does not generalize to other characteristic classes. There exists a similar famous and difficult result establishing topological invariance for rational Pontryagin classes, due to Sergei Novikov.
By definition, the Thom spectrum is a sequence of Thom spaces where we wrote for the universal vector bundle of rank n. The sequence forms a spectrum. A theorem of Thom says that is the unoriented cobordism ring; the proof of this theorem relies crucially on Thom’s transversality theorem. The lack of transversality prevents from computing cobordism rings of, say, topological manifolds from Thom spectra.
