# The Unapologetic Mathematician

## The Dimension of a Manifold

One thing that may seem clear at first blush about manifolds actually takes a little thought to be sure about. Specifically, our definition says that each point $p\in M$ is contained in some coordinate patch $U$, which comes with a coordinate map $\phi:U\to\mathbb{R}^n$. But does the $n$ here have to be the same for all points $p$?

Well, let’s start by considering what happens if two coordinate patches intersect each other. Let $U_1\subseteq M$ and $U_2\subseteq M$ be open subsets with $U_1\cap U_2$ not empty, and with coordinate maps $\phi_1:U_1\to\mathbb{R}^{n_1}$ and $\phi_2:U_2\to\mathbb{R}^{n_2}$. Now we can restrict our coordinate patches to the intersection, getting $\left(U_1\cap U_2,\phi_1\vert_{U_1\cap U_2}\right)$ and $\left(U_1\cap U_2,\phi_2\vert_{U_1\cap U_2}\right)$. These are two different coordinate patches with two different coordinate maps on the same open subset $U_1\cap U_2\subseteq M$.

So what? So now we can use $\phi_2$ in reverse to lift up from $\phi_2(U_1\cap U_2)\subseteq\mathbb{R}^{n_2}$ into $U_1\cap U_2\subseteq M$, and then use $\phi_1$ forward to drop down into $\phi_1(U_1\cap U_2)\subseteq\mathbb{R}^{n_1}$. This composition $\phi_1\circ\phi_2^{-1}$ is thus a homeomorphism from an open region in $\mathbb{R}^{n_2}$ to an open region in $\mathbb{R}^{n_1}$. And this is absolutely impossible unless $n_1=n_2$.

So now we know that any two coordinate patches that intersect must use the same value of $n$. Does this mean that we always have to use the same value of $n$? Well, not quite.

Take two distinct natural numbers $m$ and $n$. For each one, we can come up with all the coordinate patches with that dimension, and take their unions $U_m$ and $U_n$. Since the union of any collection of open sets is open, each of these sets must be open. But they can’t intersect, or else some coordinate patch with dimension $m$ and some other with dimension $n$ would have to intersect, and we just saw that they can’t.

The only way this is possible is for $U_m$ and $U_n$ to live in different connected components of $M$. And, indeed, our definition so far doesn’t rule out this possibility. We could have a two-dimensional sphere and a one-dimensional circle floating next to each other, never touching, and they would count as a manifold according to what we’ve said so far.

There are two ways around this. One is to only ever talk about connected manifolds, which automatically have a unique dimension since they only have one connected component. However, this imposes restrictions, like making it difficult to take intersections of manifolds and have the result still be a manifold, since it may suddenly be disconnected. The other alternative, which we will use, is simply to assert that all connected components of a manifold must have the same dimension.

Either way, we can specify this dimension by saying that $M$ is an $n$-dimensional manifold, or an $n$-manifold for short.

February 24, 2011