Thom space

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.
Construction of the Thom space.
One way to construct this space is as follows. Let
be a rank "n" real vector bundle over the paracompact space "B". Then for each point "b" in "B", the fiber formula_2 is an formula_3-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let formula_4 be the unit disk bundle with respect to our orthogonal structure, and let formula_5 be the unit sphere bundle, then the Thom space formula_6 is the quotient formula_7 of topological spaces. formula_6 is a pointed space with the image of formula_5 in the quotient as basepoint. If "B" is compact, then formula_6 is the one-point compactification of "E".
For example, if "E" is the trivial bundle formula_11, then formula_12 and formula_13. Writing formula_14 for "B" with a disjoint basepoint, formula_6 is the smash product of formula_14 and formula_17; that is, the "n"-th reduced suspension of formula_14.
The Thom isomorphism.
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 formula_19 coefficients to avoid complications arising from orientability; see also Orientation of a vector bundle#Thom space.)
Let formula_20 be a real vector bundle of rank "n". Then there is an isomorphism, now 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 "k"th suspension of formula_14, "B" with a disjoint point added (cf. #Construction of the Thom space.) 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 formula_23 as a right formula_24-module. The class "u" is usually called the Thom class of "E". Since the pullback formula_25 is a ring isomorphism, formula_26 is given by the equation:
In particular, the Thom isomorphism sends the identity element of formula_28 to "u". Note: for this formula to make sense, "u" is treated as an element of (we drop the ring formula_29)
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"("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 formula_32, then formula_33 coincides with the cup square. We can define the "i"th Stiefel–Whitney class formula_34 of the vector bundle formula_20 by:
Consequences for differentiable manifolds.
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.
Thom spectrum.
Real cobordism.
There are two ways to think about bordism: one as considering two formula_3-manifolds formula_38 are cobordant if there is an formula_39-manifold with boundary formula_40 such that
Another technique to encode this kind of information is to take an embedding formula_42 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 formula_44. This can be shown by using a cobordism formula_40 and finding an embedding to some formula_46 which gives a homotopy class of maps to the Thom space formula_47 defined below. Showing the isomorphism of
requires a little more work.
Definition of Thom spectrum.
By definition, the Thom spectrum is a sequence of Thom spaces
where we wrote formula_50 for the universal vector bundle of rank "n". The sequence forms a spectrum. A theorem of Thom says that formula_51 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.