Thom space

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In mathematics, the Thom space or Thom complex (named after René Thom) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. One way to construct this space is as follows. Let

p : E → B

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. 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 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 in the following geometric sense. Since E is a vector bundle it retracts onto the base B. So we might suppose that E would be cohomologically equivalent to B. In a way, the theorem bears out this expectation.

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

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 spaces MSO(n). The spaces MSO(n) themselves arise as Thom spaces and comprise a spectrum MSO that is now called a Thom spectrum (along with other related spectra). This was a major step toward modern stable homotopy theory.

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 : E → B by:

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

[edit] See also

[edit] References

  • Dennis Sullivan, René Thom's Work on Geometric Homology and Bordism. Bull. Am. Math. Soc. 41 (2004), pp. 341-350.
  • René Thom, Quelques propriétés globales des variétés differentiables. Comm. Math. Helv. 28 (1954), pp. 17-86.
  • J.P. May, A Concise Course in Algebraic Topology. University of Chicago Press, 1999, pp. 183-198.
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