diagonal embedding

Given a topological spaceMathworldPlanetmath X, the diagonal embedding, or diagonal map of X into X×X (with the product topology) is the map


X is homeomorphicMathworldPlanetmath to the image of Δ (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 Δ:GG×G, from a group into its direct sumPlanetmathPlanetmath with itself, given by the same . It’s sensible to call this an embedding, too, since Δ is a monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

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


Why call it the diagonal map?

Picture . Its diagonal embedding into the Cartesian plane × is the diagonal line y=x.

What’s it good for?

Sometimes we can use information about the product space X×X together with the diagonal embedding to get back information about X. For instance, X is HausdorffPlanetmathPlanetmath if and only if the image of Δ is closed in X×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 ImΔ 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 cohomologyPlanetmathPlanetmath, called the cup productMathworldPlanetmath. 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 algebraPlanetmathPlanetmath; 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