Thom space

One way to construct this space is as follows. Let

${\displaystyle p:E\to B}$

be a rank n real vector bundle over the paracompact space B. Then for each point b in B, the fiber ${\displaystyle E_{b}}$ is an ${\displaystyle n}$-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 ${\displaystyle D(E)}$ be the unit ball bundle with respect to our orthogonal structure, and let ${\displaystyle S(E)}$ be the unit sphere bundle, then the Thom space ${\displaystyle T(E)}$ is the quotient ${\displaystyle T(E):=D(E)/S(E)}$ of topological spaces. ${\displaystyle T(E)}$ is a pointed space with the image of ${\displaystyle S(E)}$ in the quotient as basepoint. If B is compact, then ${\displaystyle T(E)}$ is the one-point compactification of E.

For example, if E is the trivial bundle ${\displaystyle B\times \mathbb {R} ^{n}}$, then ${\displaystyle D(E)=B\times D^{n}}$ and ${\displaystyle S(E)=B\times S^{n-1}}$. Writing ${\displaystyle B_{+}}$ for B with a disjoint basepoint, ${\displaystyle T(E)}$ is the smash product of ${\displaystyle B_{+}}$ and ${\displaystyle S^{n}}$; that is, the n-th reduced suspension of ${\displaystyle B_{+}}$.

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 ${\displaystyle \mathbb {Z} _{2}}$ coefficients to avoid complications arising from orientability; see also Orientation of a vector bundle#Thom space.)

Let ${\displaystyle p:E\to B}$ be a real vector bundle of rank n. Then there is an isomorphism, now called a Thom isomorphism

${\displaystyle \Phi :H^{k}(B;\mathbb {Z} _{2})\to {\widetilde {H}}^{k+n}(T(E);\mathbb {Z} _{2}),}$

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 ${\displaystyle B_{+}}$, 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 ${\displaystyle H^{*}(E,E\setminus B;\Lambda )}$ as a right ${\displaystyle H^{*}(E;\Lambda )}$-module. The class u is usually called the Thom class of E. Since the pullback ${\displaystyle p^{*}:H^{*}(B;\Lambda )\to H^{*}(E;\Lambda )}$ is a ring isomorphism, ${\displaystyle \Phi }$ is given by the equation:

${\displaystyle \Phi (b)=p^{*}(b)\smile u.}$

In particular, the Thom isomorphism sends the identity element of ${\displaystyle H^{*}(B)}$ to u. Note: for this formula to make sense, u is treated as an element of (we drop the ring ${\displaystyle \Lambda }$)

${\displaystyle {\tilde {H}}^{n}(T(E))=H^{n}(\operatorname {Sph} (E),B)\simeq H^{n}(E,E\setminus B).}$[1]

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

${\displaystyle Sq^{i}:H^{m}(-;\mathbb {Z} _{2})\to H^{m+i}(-;\mathbb {Z} _{2}),}$

defined for all nonnegative integers m. If ${\displaystyle i=m}$, then ${\displaystyle Sq^{i}}$ coincides with the cup square. We can define the ith Stiefel–Whitney class ${\displaystyle w_{i}(p)}$ of the vector bundle ${\displaystyle p:E\to B}$ by:

${\displaystyle w_{i}(p)=\Phi ^{-1}(Sq^{i}(\Phi (1)))=\Phi ^{-1}(Sq^{i}(u)).}$

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.

• Cobordism
• Cohomology operation
• Steenrod problem
• Hattori–Stong theorem

• Sullivan, Dennis (2004). "René Thom's Work on Geometric Homology and Bordism". Bulletin of the American Mathematical Society. 41 (3): 341–350. doi:10.1090/S0273-0979-04-01026-2.
• Bott, Raoul; Tu, Loring (1982). Differential Forms in Algebraic Topology. New York: Springer. ISBN 0-387-90613-4. A classic reference for differential topology, treating the link to Poincaré duality and the Euler class of Sphere bundles
• May, J. Peter (1999). A Concise Course in Algebraic Topology. University of Chicago Press. pp. 183–198. ISBN 0-226-51182-0.
• "Explanation for the Pontryagin–Thom construction". MathOverflow.
• Stong, Robert E. (1968). Notes on cobordism theory. Princeton University Press.
• Thom, René (1954). "Quelques propriétés globales des variétés différentiables". Commentarii Mathematici Helvetici. 28: 17–86. doi:10.1007/BF02566923. S2CID 120243638.
• Ando, Matthew; Blumberg, Andrew J.; Gepner, David J.; Hopkins, Michael J.; Rezk, Charles (2014). "Units of ring spectra and Thom spectra". Journal of Topology. 7 (4): 1077–1117. arXiv:0810.4535. doi:10.1112/jtopol/jtu009. MR 0286898. S2CID 119613530.

• http://ncatlab.org/nlab/show/Thom+spectrum
• "Thom space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Akhil Mathew's blog posts: https://amathew.wordpress.com/tag/thom-space/