Everyday things often follow this pattern: their small-scale structure is chaotic but their large-scale structure is smooth.
Think of the surface of a glass of water. At human scale, it looks smooth, flat, and basically featureless. We can describe it as a flat circle. Maybe if you’re really observant, you’ll notice that at the edges where the water meets the glass, the water adheres and curves upwards forming a meniscus. But the curvature is gradual, and the overall shape is still smooth.
You can play with the shape of this surface. Lightly rest a finger on top of the water in the middle of the glass—it will adhere, and wherever you move your finger around the water’s surface, the water seems to rise up to meet it. You can even produce discontinuities—just lift your finger rapidly and a droplet will form, fall into the water and disrupt it for a little while before it settles back down.
As much fun as it is to play with the water’s surface, its small-scale structure is much crazier. Molecules are constantly bouncing around, their polarizations causing them to briefly attract nearby molecules before being violently knocked away by a high-energy neighbor. Some molecules leave the party and some come back in through the processes of evaporation and condensation. There are invisible microcurrents too subtle for the human eye, guiding huge numbers of molecules to new locations. I think it’s safe to say that the small-scale structure here is more crazy than the large.
Manifolds are a kind of mathematical object that provides a sort of counterpoint to this common pattern. The structure of a manifold at small scale is good old Euclidean space—the kind with three flat dimensions and not much going on descriptively besides maybe a Cartesian coordinate system and a basis for that system.
Manifolds have simple small structure but can have crazy large structure. One simple example is the torus, or donut shape. Near every point on a toroidal surface, we can describe what’s going on in terms of a local 2-dimensional space—just a plane with x and y axes. However, the larger-scale structure of a torus curves around back to itself. If we just use a simple 2d coordinate system to navigate, we could find ourselves doing things that are impossible in what we normally think of as 2d space—for example ending up at the same point you started from after only traveling in one constant direction.
One convenient way to get an intuition for 2-manifolds is to construct them from an ordinary 2d piece of paper. The surface of the paper itself provides our physical constraint of having 2 dimensions, but we’re allowed to move that sheet around in 3d space to get global curvatures that don’t affect the local boringness of the paper’s structure. A torus, for example, can be constructed by first rolling the paper into a cylinder, and then curving the ends of the cylinder around and joining them together.
A Möbius strip is another example of a cool manifold to think about. Its paper construction is also relatively simple. Take a 2d strip of paper, give it a single twist, and join the ends of the paper together. From a 2d perspective, there’s nothing weird going on (except that traversing the length of the paper eventually gets you back to where you started from). However, from a 3d perspective, the paper now only has one side, in a sense, but certainly has two sides in another!
Another classic example of a manifold is a Klein bottle. Unfortunately, we can’t construct a true Klein bottle in our 3d universe—we need four dimensions to be able to do so. The bottle’s gimmick is that while, like our other 2-manifolds, it only has two dimensions on each point of its surface, the bottle passes through itself (without a hole to pass through). Klein bottles have zero volume, are one-sided, are locally 2-dimensional, have an interesting 3d immersion, and can only exist in four (or more) dimensions!
Our universe can be modeled as a 3-manifold. Locally, things exist in 3d space but there could be a more-interesting large-scale structure. One example of a more-interesting possible shape for the universal manifold is a Poincaré homology sphere. While it’s hard to visualize, the construction of this space could be done by gluing faces of a dodecahedron together, or just by doing some Dehn-tistry. Because of this gluing-together process (and its being a quotient space), if this manifold is the shape of the universe then maybe it’s possible to travel so far in one direction that you end up where you started.
I currently believe that the space in our universe has the trivial topology—that is, that the universe is basically just a big blob with no way to cut up the space inside it besides “all or nothing”. I’m not sure what to think about its shape, but it’s probably some kind of manifold. It’s fun to imagine travelling so far in one direction that you end up where you started, but just because something is fun doesn’t make it true (imagine living in that kind of universe!).
I also don’t know what to think about its curvature—this has implications for whether it’s finite, for example. The cosmic topology, curvature, and shape are all still up for grabs in my mind. It’d be fascinating to learn that the universe has spatial holes in it, or maybe has separate discontinuous pockets with no way to communicate between them (there’d be no space between them through which to communicate), but for now the big-curvy-locally-Euclidean-possibly-infinite-blob model satisfies my intuitions pretty well.