It's a simple undeniable fact of everyday life that we live in a 3-dimensional manifold. Space around us is inherently experienced as (homeomorphic) to R3. There's clearly no possibility we'll experience anything else ("having" more or less dimensions suddenly, or experiencing a point with no surrounding open set). Of course, this just shows we live in some 3-manifold. But which one? The tacit assumption is usually that it's R3. Physicists regularly posit this (spacetime just adds a differential manifold structure), or sometimes compactify space by adding a single point at infinity; strictly speaking, this makes space a 3-sphere (homeomorphic to the surface of a four-dimensional ball), but puts any differences from R3 literally infinitely distant from us.

What if that's wrong? The surface of the Earth, locally, is obviously a 2-manifold, and for a long time people believed it therefore had to be an infinite plane (R2) or, more commonly, a disk. They did this just because those are the easiest manifolds to think of, and the scale of any non-local structure (e.g. routes which circle the globe) was larger than anything they had experienced. Eratosthenes guessed the Earth must be a sphere, but it took considerably more direct proof than his to convince mankind in general. Some misguided people still believe in a Flat Earth Theory. Could our universe be some more complicated 3-dimensional manifold, with the nearest effects too far away to have been noticed yet?

3-dimensional manifolds are very varied. Here are some things they can do, with easier-to-grasp analogies from 2-dimensional manifolds. To begin with, they can be closed, just like a 2d sphere or torus.

Like a torus, they can be closed "in different ways": on a torus, you can circle just round the tube, or go at right angles to that and go all around that way too, but there's no way to deform one path continuously into the other. (When someone nodes homotopy group properly you should read that...) Just as a torus can be built by gluing a rectangle along two opposite edges, and then along the two remaining edges (think of video-game were you leave one edge to reappear on the opposite side), a 3-torus (with 3 different directions to "go around") can be built by gluing a cube along three pairs of opposite faces (you may need a 4-dimensional friend for the last stage). Think of walking along a torus, laying out some string. Eventually you get back to the end you left behind, and tie your string into a loop. There's no way to gather up the string now (unlike doing the same thing on a sphere) -- it encloses a "hole" which has no adequate description in 2d terms; the same thing could happen if our universe is a 3-torus. In a complicated manifold you could have any number of peculiar "circular" paths and relations between them.

Next, there's no reason our universe should be orientable. If you were a 2d physicist living on a Klein bottle and you sent a top spinning away from you, it would one day come back from behind you, spinning in the opposite direction. Wham goes global conservation of angular momentum, et cetera.

The manifold we live in is patently metrisable: we experience and measure distance all the time. Is the metric bounded? In a 3-sphere it would be (go far enough anywhere and you'll get back to your starting point), and in R3 it wouldn't.

Going back to those closed paths, the "hole" (in 4 or more dimensions) can extend infinitely. Consider living on an infinitely long cylinder instead of a torus (one direction is "normal", the perpendicular direction is like one on the torus). Or, with the same topology but a different metric, what if space were a little like a 2d space composed of two planes (in R3, each with a small circular hole, the two holes connected by a narrow tube. Now think multiple planes, and multiple tubes....

What effect does all this have, and can we find out the truth? Well, obviously, like the great explorers we could, in theory, find out one day by going out in a spaceship to chart the universe. Since this will take several gigayears (there don't seem to be any non-trivial closed paths in our mmediate vicinity, or anything like that....), let's label that Plan B. As mentioned above, some physics we're used to are just plain wrong in certain 3-manifolds. But remember what we had to do with that top on the Klein bottle: send it round the universe! Anything of that sort will again take way too long. So is it hopeless?

I think not. First of all, there is in fact an experiment which has already been running for us since the birth of the universe. Light! We may not be able to travel mllions of light years to chart the universe, but the light from distant galaxies has already done that for us. If we live in a 3-sphere, for instance, we might see the same ultra-distant galaxy shining at us from two different (opposite?) directions; we'd identify it by the exact spectrum, or whatever. In a 3-torus and so on we'd see it from several directions, perhaps estimating the distances too, thus beginning to piece together a map of our manifold. I don't know that anyone has tried to compare "different" galaxies, but at least there's a chance we could handle these observations.

The other "ray of light" comes from quantum mechanics. Wouldn't it be fun if the theory of the tiniest details of the universe let us gain an insight into the grandest scale of its sturcture? Objects (in quantum mechanics) don't exist at a point, but rather have a spread-out wave function; because it is fairly localised, the position and other quantities are determined reasonably well. However the wave function satisifies a differential equation with certain boundary conditions. If the boundary conditions of a free particle are that the wave function continues to infinity (under our usual assumption of living in R3), the solutions are different than if there exist non-trivial closed paths. Just as the solutions of such an equation on a circle or sphere become discrete, instead of the continuous soluions permissible on a line or plane, so we might find certain energy-states quantised just by the characteristics of the manifold we live in. A local experiment on something with a long enough wavelength (perhaps a graviton?), observed at a small enough scale, might reveal the existence (and length) of a route to circumnavigate the universe.

They're both not experiments we're likely to be able to do tomorrow, but the theoretical numbers don't seem to be completely ludicrous (unlike taking a spaceship to "go check"). I think "Project Eratosthenes" could literally be "the biggest thing" we ever learn about the universe.