# Thom class

Let ${h}^{*}$ be a 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 ${h}^{*}(\mathrm{pt})$ of ${h}^{*}$. In fact, ${h}^{*}(D({\xi}_{x}),S({\xi}_{x}))$ is a free module^{} of rank one over the ring ${h}^{*}(\mathrm{pt})$.

###### 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 ${h}^{*}(\mathrm{pt})$-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^{} ${\stackrel{~}{h}}^{*}({X}^{\xi})$, 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.

Title | Thom class |
---|---|

Canonical name | ThomClass |

Date of creation | 2013-03-22 15:40:48 |

Last modified on | 2013-03-22 15:40:48 |

Owner | antonio (1116) |

Last modified by | antonio (1116) |

Numerical id | 5 |

Author | antonio (1116) |

Entry type | Definition |

Classification | msc 55-00 |

Related topic | Orientation2 |

Defines | orientability with respect to a generalized homology theory |