a question on topological manifolds and what topology provides

Question

When one talks of a topological manifold being locally homeomorphic to $\mathbb{R}^{n}$ is it meant that the topology of the manifold is locally identical to a Euclidean topology such that we can represent points on the manifold (locally) in the same manner that we can for $\mathbb{R}^{n}$, i.e. as $n$-tuples of real numbers?

I must admit I find the whole concept of topology quite confusing, I've read things such as "a topology on a manifold endows it with a primitive notion of geometrical structure without the need to introduce the notion of a metric etc." What exactly is meant by this? In physics, particularly general relativity, space time is taken as a topological manifold, and I know that the manifold has a topology defined on it is key, but I don't really understand the conceptual significance of it?

Answer 1

When we say that a topological space $X$ is a topological manifold being locally homeomorphic to $\mathbb{R}^n$ if for each $x\in X$ there exists an open set $U$ containing $x$ with a continuous function $\phi_U:U\rightarrow \mathbb{R}^n$ such that $\phi_U$ induces an homeomorphism between $U$ and $\phi_U(U)$. This is the ground definition. The $(U,\phi_U)$ are called charts.

Now one usually requires additional "properties" (they are quite technical, in most cases they are obvious) namely the topology on $X$ should be Hausdorff and countable at infinity.

The idea behind your quote is (IMO) that generally you just don't want to work with "topological" manifold but you want to work on "differentiable" manifold or "holomorphic" manifold (to name but a few). Now those additional structures are quite hard to define out of nowhere.

If $X$ is a topological manifold with a set of charts $\{(U,\phi_U)\}$ we say that they define a "differentiable" manifold if for any chart $U,V$ such that $U\cap V\neq \emptyset$ we have that :

$$\phi_U\circ\phi_V^{-1}:\phi_V(U\cap V)\rightarrow \phi_U(U\cap V) $$

is a diffeomorphism. Remark that we already know that this is a homeomorphism but because $\phi_V(U\cap V)$ and $ \phi_U(U\cap V)$ are both included in $\mathbb{R}^n$ we can talk about diffeomorphism.

Hence we see that to define a geometric structure on some space $X$ we need to get a structure of topological manifold for $X$ and then require all the transition to respect the geometric structure in question.

That is why the topological manifold is the "primitive notion of geometric structure" it is a tool that allows us to put some significant geometric structure (differentiable, holomorphic...) on a space.

Answer 2

In physics one works with coordinate systems all the time, and one changes coordinate systems all the time.

Before relativity theory, in the Newtonian universe, it seemed that space might have one more-or-less unique "natural" coordinate system, making space a real-life copy of the mathematical abstraction $\mathbb{R}^3$ --- unique, that is, up to translation and rotation of coordinates, and up to uniform expansion of coordinates which amounts to a choice of unit distance.

After relativity no-one expects this kind of uniqueness of coordinate systems anymore. Nonetheless, space-time coordinate systems still remain a key tool of (non-quantum-mechanical) physics.

So how do you speak about space-time if you don't have just one gigantic global natural coordinate system? Well you speak about it as patching together zillions of small scale coordinate systems, which might or might not be natural in some "local" sense. But to patch together coordinate systems like that, you need a mathematical theory, and that's precisely what topology gives you.

Space-time is a topological space, which gives it a structure of "nearness" in a rather coarse sense, or you can think of it as a structure of "convergent sequences", and there are various other ways that we use our intuition to understand topological spaces in general. Mathematically, you are simply specifying what subsets are "open", and you are specifying a few rock bottom axioms that open sets should satisfy (see the definition of a topology).

Next, space-time has coordinate systems; the rock-bottom minimum feature is that the coordinate systems are given by homeomorphisms $\phi : U \to V \subset \mathbb{R}^4$ where $U$ is an open subset of space-time and $V$ is an open subset of the coordinate space $\mathbb{R}^4$. The set of domains $U$ for these coordinate systems should cover all of space-time. Open-ness is important for many reasons, the most basic theoretical reason being that multivariable calculus only really makes sense when working with functions whose domains are open subsets of Euclidean space.

Next, although space-time coordinate systems are not assumed to be "unique" or "natural", nonetheless if you have two space-time coordinate systems on overlapping open sets $U_1,U_2$ it is required that the overlap map between those two systems be nice in some manner. The domain of the overlap map is $\phi_1(U_1 \cap U_2)$ which is an open subset of $V_1$ and therefore an open subset of $\mathbb{R}^4$, its range is $\phi_2(U_1 \cap U_2)$ which is an open subset $V_2$ and therefore an open subset of $\mathbb{R}^4$, and the formula for this map is $\phi_2 \circ \phi_1^{-1}$. Again the rock bottom minimum requirement on this overlap map is that it be infinitely differentiable, meaning all partial derivatives of all orders exist at all points (here is where open-ness is most important, in order to apply the tools of multivariable calculus).

Next, depending on what portion of physics you are working in, you will want to have still more structure on your coordinate systems, and your overlap maps should respect this structure. For instance in general relativity you want the extra structure of various tensors such as a Lorentz metric and a stress-energy tensor, and so on.