Thom space

A topological space associated with a vector (or sphere) bundle or spherical fibration.

Let $ \xi $ be a vector bundle over a CW-complex $ X $. Suppose that one is given a Riemannian metric on it, and consider the unit-disc bundle $ D ( \xi ) $ associated with $ \xi $. In $ D ( \xi ) $ lies the unit-sphere subbundle $ S ( \xi ) $; the quotient space $ D ( \xi )/S ( \xi ) $ is the Thom space of the bundle $ \xi $, denoted by $ T ( \xi ) $. For a compact base space $ X $, the Thom space can also be described as the one-point compactification of the total space of the bundle $ \xi $. Moreover, the Thom space is the cone of the projection $ S ( \xi ) \rightarrow X $ and in this way one can define the Thom space of any spherical fibration. Of course, the Thom space is also defined for any bundle with fibre $ \mathbf R ^ {n} $.

Let $ O _ {k} $ be the group of orthogonal transformations of the space $ \mathbf R ^ {k} $. Over its classifying space $ \mathop{\rm BO} _ {k} $ there is the $ k $- dimensional vector bundle $ \gamma _ {k} $, associated with the universal $ O _ {k} $- bundle. The Thom space $ T \gamma _ {k} $ is often denoted by $ \mathop{\rm MO} _ {k} $ or $ \mathop{\rm TBO} _ {k} $, and is called the Thom space of the group $ O _ {k} $. Analogously one introduces the Thom spaces $ \mathop{\rm MU} _ {k} $, $ \mathop{\rm MSp} _ {k} $, etc., where $ U _ {k} $ and $ \mathop{\rm Sp} _ {k} $ are the unitary and symplectic groups, respectively.

The role of the Thom space is to allow one to reduce a series of geometric problems to problems in homotopic topology, and hence to algebraic problems. Thus, the problem of computing a bordism group reduces to the problem of computing a homotopy group of a Thom space $ \mathop{\rm MO} _ {k} $, $ \mathop{\rm MSO} _ {k} $, etc. (cf. [1], [2], and also Cobordism). The problem of classifying smooth manifolds reduces to the study of the homotopy properties of the Thom space of the normal bundle (cf. [3]). The problem of realizing cycles by submanifolds (cf. Steenrod problem) reduces to the study of the cohomology of the Thom spaces $ \mathop{\rm MSO} _ {k} $ and $ \mathop{\rm MO} _ {k} $, etc. (see also Transversal mapping; Tubular neighbourhood).

The construction of Thom spaces is natural on the category of bundles: Any morphism of (vector) bundles $ f: \xi \rightarrow \eta $ induces a continuous mapping $ T ( f ): T ( \xi ) \rightarrow T ( \eta ) $. In particular, the Thom space of an $ n $- dimensional bundle over a point is $ S ^ {n} $, and hence for any $ n $- dimensional bundle $ \xi $ over $ X $ and any point $ x \in X $ there is an inclusion $ j _ {x} : S ^ {n} \rightarrow T ( \xi ) $( induced by the inclusion of the fibre over $ x $). If $ X $ is path connected, then all such inclusions are homotopic, and one can talk about the mapping $ j: S ^ {n} \rightarrow T ( \xi ) $, which is unique up to homotopy.

For vector bundles $ \xi $ and $ \eta $ over $ X $ and $ Y $, respectively, one can define the bundle $ \xi \times \eta $ over $ X \times Y $. Then $ T ( \xi \times \eta ) = T ( \xi ) \wedge T ( \eta ) $( cf. [4]). In particular, for the trivial bundle $ \theta ^ {n} $ one has $ T ( \xi \oplus \theta ^ {n} ) = S ^ {n} T ( \xi ) $, where $ S $ is the suspension operator, so that $ T ( \theta ^ {n} ) = S ^ {n} ( X \cup \mathop{\rm pt} ) $. This circumstance allows one to construct spectra of Thom spaces, cf. Thom spectrum.

For a multiplicative generalized cohomology theory $ E $( cf. Generalized cohomology theories) there is a pairing

$$ E ^ {*} ( D ( \xi )) \otimes E ^ {*} ( D ( \xi ), S ( \xi )) \rightarrow \ E ^ {*} ( D ( \xi ), S ( \xi )). $$

There arises a pairing

$$ E ^ {*} ( X) \otimes \widetilde{E} {} ^ {*} ( T \xi ) \rightarrow \ \widetilde{E} {} ^ {*} ( T \xi ), $$

so that $ \widetilde{E} {} ^ {*} ( T \xi ) $ is an $ E ^ {*} ( X) $- module, and this is used in constructing the Thom isomorphism.

The following Atiyah duality theorem is important and often used (cf. [4], [5]): If $ M $ is a smooth manifold with boundary $ \partial M $( possibly empty) and $ \nu $ is its normal bundle, then the Thom space $ T( \nu ) $ is in $ S $- duality with $ M/ \partial M $.

References

[1]	R. Thom, "Quelques propriétés globales des variétés différentiables" Comm. Math. Helv. , 28 (1954) pp. 17–86
[2]	R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)
[3]	W.B. Browder, "Surgery on simply-connected manifolds" , Springer (1972)
[4]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[5]	M. Atiyah, "Thom complexes" Proc. London Math. Soc. , 11 (1961) pp. 291–310

References

[a1]	J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)