# cup product

Let $X$ be a topological space^{} and $R$ be a commutative ring. The diagonal map $\mathrm{\Delta}:X\to X\times X$ induces a chain map between singular cochain complexes^{}:

$${\mathrm{\Delta}}^{*}:{C}^{*}(X\times X;R)\to {C}^{*}(X;R)$$ |

.

Let $h:{C}^{*}(X;R)\otimes {C}^{*}(X;R)\to {C}^{*}(X\times X;R)$

denote the chain homotopy equivalence associated with the Kunneth .

Given $\alpha \in {C}^{p}(X;R)$ and $\beta \in {C}^{q}(X;R)$ we define

$\alpha \u2323\beta ={\mathrm{\Delta}}^{*}h(\alpha \otimes \beta )$.

As ${\mathrm{\Delta}}^{*}$ and $h$ are chain maps, $\u2323$ induces a well defined product^{} on cohomology groups^{}, known as the cup product^{}. Hence the direct sum^{} of the cohomology groups of $X$ has the structure^{} of a ring. This is called the cohomology ring of $X$.

Title | cup product |
---|---|

Canonical name | CupProduct |

Date of creation | 2013-03-22 15:37:42 |

Last modified on | 2013-03-22 15:37:42 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 7 |

Author | whm22 (2009) |

Entry type | Definition |

Classification | msc 55N45 |