Thom space

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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[edit]

One way to construct this space is as follows. Let

p : EB

be a rank k real vector bundle over the paracompact space B. Then for each point b in B, the fiber Fb 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.[further explanation needed] 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 Thom isomorphism[edit]

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, now called a Thom isomorphism

\Phi \colon H^i(B; \mathbf{Z}_2) \to \tilde{H}^{i+k}(T(E); \mathbf{Z}_2),

for all i greater than or equal to 0, where the right hand side is reduced cohomology.

We can loosely interpret the theorem as being a generalization of the suspension isomorphism on (co)homology, because the Thom space of a trivial bundle on B of rank k is isomorphic to the kth suspension of B+, B with a disjoint point added.

This theorem was formulated and proved by René Thom in his 1952 thesis.

The Thom class[edit]

The isomorphism of the theorem is explicitly known: there is a certain cohomology class, the Thom class, in the kth cohomology group of the Thom space. Denote this Thom class by U. Then for a class b in the cohomology of the base, we can compute the Thom isomorphism via the pullback of the bundle projection and the cohomology cup product:

\Phi(b) = p^*(b) \smile U.

In particular, the Thom isomorphism sends the identity element of H*(B) to U.

Note: for this formula to make sense, U is treated as an element of H^k(D(E),\operatorname{Sph}(E);\mathbf{Z}_2)\cong \tilde H^k(T(E);\mathbf{Z}_2), where D(E) is the associated disk bundle, so we have a cup product

H^i(D(E);\mathbf{Z}_2)\otimes H^k(D(E),\operatorname{Sph}(E);\mathbf{Z}_2)\to H^{i+k}(D(E),\operatorname{Sph}(E);\mathbf{Z}_2)\cong \tilde H^k(T(E);\mathbf{Z}_2).

Significance of Thom's work[edit]

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

Sq^i \colon H^m(-; \mathbf{Z}_2) \to 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 ith Stiefel-Whitney class wi (p) of the vector bundle p : EB by:

w_i(p) = \Phi^{-1}(Sq^i(\Phi(1))) = \Phi^{-1}(Sq^i(U)).\,

Consequences for differentiable manifolds[edit]

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.

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