haggai (relation) claims in which 3-manifold do we live in? that looking at spacetime merely adds some differential structure. But actually the situation is a lot more complex than that.
The Universe is (as far as we know, and in accordance with General Relativity (GR)) a 4-dimensional space which is a quasi-metric space ("distances" in it are properly measured as sqrt(x^2+y^2+z^2-t^2) (or, equivalently, sqrt(x^2+y^2+z^2+(it)^2)), which can be negative). And it's also required to have some time-like property (not space-like, which is the antonym of time-like), which equates with not being able to exceed the speed of light and/or (equivalently, given the assumptions of relativity) not being able to go backwards in time. But the equations of GR are differential equations, hence they have a purely local quality. And they are what prohibit space-like paths.
Every point in spacetime has a light cone surrounding it; this is the set of geodesic ("straight") paths at lightspeed going though it. Divide the light cone into its "forward" half (the light-cone of light rays moving from the point, "forwards in time") and its "backward" half (the light-cone of light rays converging on the point, or "moving backwards in time" from it). The time-like property of spacetime means that there do not exist time-like paths from the "forwards" half to the "backwards" half. But since the GR equations are local, this prohibition is also only local: in a small enough neighborhood of every point in spacetime (say 106 lightyears and 106 years around us) no path goes from a point in our forwards light cone to our backwards light cone.
But all this says nothing about the global case!
It is entirely possible to imagine a spacetime which has time-like paths which are globally space-like, in the sense that they start in our forward light cone and end in our backward light cone, while never exceeding the speed of light! This is a path were you leave Earth 10 years from now, and travel (for a long while) until you reach the Earth 100 years ago. It's also known as time travel. (Time travel into our future is trivial: you do it all the time, and by travelling rapidly you can "shift" your rate of the advance of Earth-time from the Earth's rate of advance of Earth-time, thereby travelling into the future; time travel into the past is something so weird that it's hard to see it as possible). For instance, if spacetime were like a 4-torus we could observe such paths. Of course, we know it isn't (since we have evidence for a Big Bang), but more complicated structures could have similar pathologies.
This would have an important bearing on the structure of our Universe. For one thing, the concept of causality would collapse, or at least be a local concept. All of our experience is limited to a certain region of spacetime; even if we include the evidence of our radio telescopes, we've still seen only a small region. And of course, we've never seen events in our forward light cone (this is what we think of as our future). One way to get evidence of such a globally space-like structure would be to see if our telescopes (which point into our backwards light cone) can "see" images in our future (our forwards light cone). But our telescopes have a limited range, and any light rays moving in a globally space-like path might well be distorted beyond recognition, so not receiving such an indication means nothing.
Or we can go the quantum boundary way suggested in which 3-manifold do we live in?. Unfortunately, GR and Quantum Mechanics are incompatible; nobody knows how to reconcile the 2 theories (or, indeed, whether this is even possible). And understanding, maybe even formulating, the boundary problems of the wave equation across such a globally space-like path would require (at the very least) such a unified theory.
Project Wells will have to wait at least another century before we can even start thinking about how to do it.
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