# Thom isomorphism theorem

Let $\xi \to X$ be a $d$-dimensional vector bundle^{} over a topological space^{} $X$, and let ${h}^{*}$ be a multiplicative generalized cohomology theory, such as ordinary cohomology. Let $\tau \in {h}^{d}(D(\xi ),S(\xi ))$ be a Thom class for $\xi $, where $D(\xi )$ and $S(\xi )$ are the associated disk and sphere bundles of $\xi $.

Since ${h}^{*}$ is a multiplicative theory, there is a generalized cup product^{} map

$${h}^{*}(D(\xi )){\otimes}_{{h}^{*}}{h}^{*}(D(\xi ),S(\xi ))\to {h}^{*}(D(\xi ),S(\xi )),$$ |

where the tensor product^{} is over the coefficient ring ${h}^{*}(\mathrm{pt})$ of the theory. Using the isomorphism^{} ${p}^{*}:{h}^{*}(X)\cong {h}^{*}(D(\xi ))$ induced by the homotopy equivalence^{} $p:D(\xi )\to X$, we obtain a homomorphism^{}

$$T:{h}^{n}(X)\to {h}^{n+d}(D(\xi ),S(\xi ))\cong {\stackrel{~}{h}}^{n+d}({X}^{\xi})$$ |

taking $\alpha $ to ${p}^{*}(\alpha )\cdot \tau $. Here ${X}^{\xi}$ stands for the Thom space $D(\xi )/S(\xi )$ of $\xi $.

Thom isomorphism theorem $T$ is an isomorphism ${h}^{*}(X)\cong {\stackrel{~}{h}}^{*+d}({X}^{\xi})$ of graded modules over ${h}^{*}(\mathrm{pt})$.

###### Remark 1

When $\xi $ is a trivial bundle^{} of dimension^{} $1$, this generalizes the suspension isomorphism. In fact, a typical proof of this theorem for compact^{} $X$ proceeds by induction over the number of open sets in a trivialization of $\xi $, using the suspension isomorphism as the base case and the Mayer-Vietoris sequence to carry out the inductive step.

###### Remark 2

There is also a homology^{} Thom isomorphism ${\stackrel{~}{h}}_{*+d}({X}^{\xi})\cong {h}_{*}(X)$, in which the map is given by cap product with the Thom class rather than cup product.

Title | Thom isomorphism theorem |
---|---|

Canonical name | ThomIsomorphismTheorem |

Date of creation | 2013-03-22 15:40:52 |

Last modified on | 2013-03-22 15:40:52 |

Owner | antonio (1116) |

Last modified by | antonio (1116) |

Numerical id | 6 |

Author | antonio (1116) |

Entry type | Theorem |

Classification | msc 55-00 |