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

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

$X$ 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

$$x\stackrel{{\mathrm{\Delta }}_n}{⟼}(x,x,\mathrm{\dots },x).$$

Why call it the diagonal map?

Picture $ℝ$. Its diagonal embedding into the Cartesian plane $ℝ\times ℝ$ 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.