My understanding is that it is "flat," in that if there were three devices, each shining a laser beam at each of the other two devices, the interior angles of the triangle sum to 180°. (of course there are some issues with local curvature due to gravitational wells, but on the cosmic scale (supposedly) it is flat.

One thing I have often wondered, though, how to treat "holes". Black holes, which as far as I can tell can be treated as single holes in the fabric of spacetime, and wormholes, which have two ends that connect to different locations in spacetime (like a bridge). I know that surfaces with different numbers of holes and bridges are topologically distinct, and fall into classes based on whether the number is even or odd etc. So I have wondered if it is possible to determine something like: "are there an even number of black holes in the universe?" without actually needing to count them.

In theory, many fields and waves extend "to infinity," and one could imagine that the interference pattern could be characteristic of the topology of the universe (especially if there are wormholes), but I have neither the time nor expertise to think this one through much further...