It's enlightening (maybe) to consider the simplest nontrivial topology: A cylinder.
So our cylindrical spacetime is 2D, and can be parametrized by two coordinates: (x,t) but with the constraint that the point (x+L,t) is the same point as (x,L). So flying a distance L in the x-direction brings you back to where you started.
For all experiments taking place in a small region (a region of size \ll L), the topology doesn't matter---this looks just like ordinary Minkowsky spacetime. So locally, you can perform Lorentz transformations and all frames look equivalent. But for large-scale phenomena, things look very weird in a boosted frame.
Let me illustrate. Suppose you're an astronomer in this cylindrical universe. You're on Earth (assumed to be at rest in the coordinates x,t), and looking in the +x direction through a powerful telescope. What you would see, at a distance L away, is another copy of Earth (it's actually your Earth, but it looks like a copy). Not only that, you will see another copy of Earth at a distance 2L away, and another copy at a distance of 3L away, etc. After correcting for the travel time of light, you discover that these Earths are all the same age. You can interpret this as good evidence that the universe is cylindrical, and these are not "copies" at all, but the real thing. But observationally, the situation is indistinguishable from an infinite universe that happens to be periodic: all conditions repeat in space with period L.
Now, hop into a rocket that is moving at speed v relative to the Earth in the +x direction. Now, when you look through a telescope, you will again see a line of copies of Earth. But in this frame:
- The distance between copies is L/\gamma, rather than L (since Earth is moving relative to this frame, distances are Lorentz-contracted).
- The copies of Earth are not the same age! After taking into account the travel time for light, the astronomer on the rocket would conclude that there is a second Earth that is a distance L/\gamma away, and that this Earth is older than ours by an amount \frac{vL}{c^2} (assuming I've applied the Lorentz transformations correctly). At a distance of 2L/\gamma, there is yet another Earth, and this one is \frac{2vL}{c^2} older than ours. So all the infinitely many copies of Earth are different ages.
So that's the sense in which there is no global notion of time for the rocket frame. He can set up a coordinate system locally that works out fine. But if he tries to have a global notion of time, then he'll find that if some event happens at time t', it also happens at infinitely many other times: t' + \frac{n vL}{c^2} for every integer value of n. So there is no unique time for events.
