# Thom space

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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