One way to construct this space is as follows. Let
be a rank nrealvector bundle over the
paracompact spaceB. Then for each point b in B, the
fiber is an -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 be the unit ball bundle with respect to our orthogonal structure, and let be the unit sphere bundle, then the Thom space is the quotient of topological spaces. is a
pointed space with the image of in the quotient as basepoint. If B is compact, then is the one-point compactification of E.
For example, if E is the trivial bundle , then and . Writing for B with a disjoint basepoint, is the
smash product of and ; that is, the n-th reduced
suspension of .
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 (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:
Thom isomorphism —
Let be a ring and 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 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 (we drop the ring )
The standard reference for the Thom isomorphism is the book by Bott and Tu.
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
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 (mod 2) are
natural transformations
defined for all nonnegative integers m. If , then coincides with the cup square. We can define the ith Stiefel–Whitney class of the vector bundle 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 -manifolds are cobordant if there is an -manifold with boundary 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 . This can be shown[2] by using a cobordism and finding an embedding to some which gives a homotopy class of maps to the Thom space defined below. Showing the isomorphism of
^Proof of the isomorphism. We can embed B into either as the zero section; i.e., a section at zero vector or as the infinity section; i.e., a section at infinity vector (topologically the difference is immaterial.) Using two ways of embedding we have the triple:
.
Clearly, deformation-retracts to B. Taking the long exact sequence of this triple, we then see:
Milnor, John. Characteristic classes. is another standard reference for the Thom class and Thom isomorphism. See especially the paragraph 18.
May, J. Peter (1999). A Concise Course in Algebraic Topology.
University of Chicago Press. pp. 183–198.
ISBN0-226-51182-0. This textbook gives a detailed construction of the Thom class for trivial vector bundles, and also formulates the theorem in case of arbitrary vector bundles.