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Let $h^*$ be a multiplicative generalized cohomology theory (for example, let $h^*=H^*$ , singular cohomology with integer coefficients). Let $\xi\to X$ be a vector bundle of dimension $d$ over a topological space $X$ . Assume for convenience that $\xi$ has a Riemannian metric, so that we may speak of its associated sphere and disk bundles, $S(\xi)$ and $D(\xi)$ respectively.
Let $x\in X$ , and consider the fibers $S(\xi_x)$ and $D(\xi_x)$ . Since $D(\xi_x)/S(\xi_x)$ is homeomorphic to the $d$ -sphere, the Eilenberg-Steenrod axioms for $h^*$ imply that $h^{*+d}(D(\xi_x),S(\xi_x))$ is isomorphic to the coefficient group $h^*(\pt)$ of $h^*$ . In fact, $h^*(D(\xi_x),S(\xi_x))$ is a free module of rank one over the ring $h^*(\pt)$ .
Definition 1 An element $\tau\in h^*(D(\xi),S(\xi))$ is said to be a Thom class for $\xi$ if, for every $x\in X$ , the restriction of $\tau$ to $h^*(D(\xi_x),S(\xi_x))$ is an $h^*(\pt)$ -module generator.
Note that $\tau$ lies necessarily in $h^d(D(\xi),S(\xi))$ .
Definition 2 If a Thom class for $\xi$ exists, $\xi$ is said to be orientable with respect to the cohomology theory $h^*$ .
Remark 1 Notice that we may consider $\tau$ as an element of the reduced $h^*$ -cohomology group $\rh^*(X^\xi)$ , where $X^\xi$ is the Thom space $D(\xi)/S(\xi)$ of $\xi$ . As is the case in the definition of the Thom space, the Thom class may be defined without reference to associated disk and sphere bundles, and hence to a Riemannian metric on $\xi$ . For example, the pair $(\xi,\xi-X)$ (where $X$ is included in $\xi$ as the zero section) is homotopy equivalent to $(D(\xi),S(\xi))$ .
Remark 2 If $h^*$ is singular cohomology with integer coefficients, then $\xi$ has a Thom class if and only if it is an orientable vector bundle in the ordinary sense, and the choices of Thom class are in one-to-one correspondence with the orientations.
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"Thom class" is owned by antonio.
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See Also: orientation
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orientability with respect to a generalized homology theory |
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Cross-references: orientations, one-to-one correspondence, homotopy equivalent, zero section, sphere bundles, reference, Thom space, reduced, orientable, generator, restriction, ring, rank, free module, group, isomorphic, imply, axioms, homeomorphic, fibers, disk bundles, sphere, Riemannian metric, topological space, dimension, vector bundle, coefficients, integer, singular, theory, cohomology
There is 1 reference to this entry.
This is version 2 of Thom class, born on 2006-02-15, modified 2006-02-15.
Object id is 7621, canonical name is ThomClass.
Accessed 2319 times total.
Classification:
| AMS MSC: | 55-00 (Algebraic topology :: General reference works ) |
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Pending Errata and Addenda
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