Quote by Mike2
Was this paper an attempt to justify or perhaps even derive the very overall topology of spacetime?

Not really. The main goal was to get a theory of quantum gravity in 3+1 dimensions that
works  meaning that it reduces to general relativity at length scales much larger than the Planck scale. They didn't prove their model works, but they produced some impressive evidence that it might.
But, there is something to say about topology here.
In their model, you can take space at a given time to have any topology you want  any compact 3dimensional manifold, that is. The model then ensures that the topology of space will remain the same at all other times.
In other words, the model forbids "topology change".
They wanted this, because in very similar models (dynamical triangulation models) that don't forbid topology change, there's a strong tendency for all hell to break loose: typical spacetimes are either "crumpled" or "branched polymers". This problem had afflicted the subject for decades! This is what Ambjorn, Jurkiewicz and Loll seem to have gotten around!
I wonder if causality is the key to the topology of spacetime.

I'd prefer to say it's the key to preventing topology change. This is wellknown in classical general relativity, where one can prove there's no topology change if spacetime is "globally hyperbolic"  that is, very roughly, if it has a wellbehaved concept of causality.