In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.
Let be a rank n real vector bundle over the paracompact space B.
by taking the one-point compactification of each fiber and gluing them together to get the total space.
[further explanation needed] Finally, from the total space
Alternatively,[citation needed] since B is paracompact, E can be given a Euclidean metric and then
-sphere bundle of E. The significance of this construction begins with the following result, which belongs to the subject of cohomology of fiber bundles.
coefficients to avoid complications arising from orientability; see also Orientation of a vector bundle#Thom space.)
be a real vector bundle of rank n. Then there is an isomorphism called a Thom isomorphism for all k greater 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
be an oriented real vector bundle of rank n. Then there exists a class where B is embedded into E as a zero section, such that for any fiber F the restriction of u is the class induced by the orientation of F. Moreover, is an isomorphism.
In concise terms, the last part of the theorem says that u freely generates
is given by the equation: In particular, the Thom isomorphism sends the identity element of
Note: for this formula to make sense, u is treated as an element of (we drop the ring
) The standard reference for the Thom isomorphism is the book by Bott and Tu.
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 properties of smooth manifolds—see Thom transversality theorem.
By reversing this construction, John Milnor and 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 spectra MG 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 (mod 2) are natural transformations defined for all nonnegative integers m. If
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.
such that Another technique to encode this kind of information is to take an embedding
and considering the normal bundle The embedded manifold together with the isomorphism class of the normal bundle actually encodes the same information as the cobordism class
which gives a homotopy class of maps to the Thom space
for the universal vector bundle of rank n. The sequence forms a spectrum.
[7] The lack of transversality requires that alternative methods be found to compute cobordism rings of, say, topological manifolds from Thom spectra.