The rate of expansion is expressed by the Hubble Constant, H o , while the strength of gravity depends on the density and pressure of the matter in the universe. If the pressure of the matter is low, as is the case with most forms of matter of which we know, then the fate of the universe is governed by the density.
If the density of the universe is less than the "critical density", which is proportional to the square of the Hubble constant, then the universe will expand forever. If the density of the universe is greater than the "critical density", then gravity will eventually win and the universe will collapse back on itself, the so called "Big Crunch".
However, the results of the WMAP mission and observations of distant supernova have suggested that the expansion of the universe is actually accelerating, which implies the existence of a form of matter with a strong negative pressure, such as the cosmological constant.
This strange form of matter is also sometimes referred to as "dark energy". If dark energy in fact plays a significant role in the evolution of the universe, then in all likelihood the universe will continue to expand forever. The density of the universe also determines its geometry. Go ahead, dig it out of the trash. Wrap one end around to meet the other, making a cylinder.
Carefully observe the parallel lines — they remain parallel, don't they? That's because cylinders are flat. There's an important distinction between geometry, the behavior of parallel lines, and topology, the way a space can get all twisted up. While the geometry of the universe is very well measured again, it's flat , the topology is not. And here's a bonus fact: not only can we not determine the topology of the universe from observations, but there are also no laws of physics that predict or restrict the topology.
With your 2D piece of paper, you can connect the ends a few different ways. Connect one of the dimensions normally and you have a cylinder. Flip one edge over before connecting and you've made a Mobius strip. Connect two dimensions, the top to the bottom and one side to the other, and you have a torus aka a donut.
In our 3D universe, there are lots of options — 18 known ones, to be precise. Mobius strips, Klein bottles and Hantzsche-Wendt space manifolds are all non-trivial topologies that share something in common: if you travel far enough in one direction, you come back to where you started. In the case of flipped dimensions, when you come back to your starting point, you'll find yourself upside down without having tried to do so at all.
Of course we've looked to see if our universe is connected like this; we don't see any copies of galaxies, and we don't see the cosmic microwave background intersecting itself. If the universe is pretzeled-up, it's on scales far, far larger than what we can observe.
So don't get too excited by possibility of living in a real-life version of the "Asteroids" game — which, as you now know, is played on the surface of a donut.
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