# diagonal embedding

Given a topological space^{} $X$, the *diagonal embedding*, or *diagonal map* of $X$ into $X\times X$ (with the product topology) is the map

$$x\stackrel{\mathrm{\Delta}}{\u27fc}(x,x).$$ |

$X$ is homeomorphic^{} to the image of $\mathrm{\Delta}$ (which is why we use the word “embedding”).

We can perform the same construction with objects other than topological spaces: for instance, there’s a diagonal map $\mathrm{\Delta}:G\to G\times G$, from a group into its direct sum^{} with itself, given by the same . It’s sensible to call this an embedding, too, since $\mathrm{\Delta}$ is a monomorphism^{}.

We could also imagine a diagonal map into an n-fold product given by

$$x\stackrel{{\mathrm{\Delta}}_{n}}{\u27fc}(x,x,\mathrm{\dots},x).$$ |

## Why call it the diagonal map?

Picture $\mathbb{R}$. Its diagonal embedding into the Cartesian plane $\mathbb{R}\times \mathbb{R}$ is the diagonal line $y=x$.

## What’s it good for?

Sometimes we can use information about the product space $X\times X$ together with the diagonal embedding to get back information about $X$. For instance, $X$ is Hausdorff^{} if and only if the image of $\mathrm{\Delta}$ is closed in $X\times X$ [proof (http://planetmath.org/ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed)]. If we know more about the product space than we do about $X$, it might be easier to check if $\mathrm{Im}\mathrm{\Delta}$ is closed than to verify the Hausdorff condition directly.

When studying algebraic topology, the fact that we have a diagonal embedding for any space $X$ lets us define a bit of extra structure in cohomology^{}, called the cup product^{}. This makes cohomology into a ring, so that we can bring additional algebraic muscle to bear on topological questions.

Another application from algebraic topology: there is something called an $H$-space, which is essentially a topological space in which you can multiply two points together. The diagonal embedding, together with the multiplication, lets us say that the cohomology of an $H$-space is a Hopf algebra^{}; this structure lets us find out lots of things about $H$-spaces by analogy to what we know about compact Lie groups.

Title | diagonal embedding |
---|---|

Canonical name | DiagonalEmbedding |

Date of creation | 2013-03-22 14:20:41 |

Last modified on | 2013-03-22 14:20:41 |

Owner | waj (4416) |

Last modified by | waj (4416) |

Numerical id | 8 |

Author | waj (4416) |

Entry type | Definition |

Classification | msc 54B10 |

Classification | msc 18A05 |

Synonym | diagonal map |

Related topic | ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed |