Questions from Kilian Walsh (NYU) today reminded me of an old, abandoned idea: Look for evidence of a periodic universe (topological non-triviality) in the large-scale structure of galaxies. Papers by Starkman (CWRU) and collaborators (one of several examples is here) claim to rule out most interesting topologies using the CMB alone. I don't doubt these papers but (a) they effectively make very strong predictions for the large-scale structure and (b) if CMB (or topology) theory is messed up, maybe the constraints are over-interpreted.
The idea would be to take pairs of finite patches of the observed large-scale structure and look to see if there are shifts, rotations, and linear amplifications (to account for growth and bias evolution) that make their long-wavelength (low-pass filtered) density fields match. Density field tracers include the LRGs, the Lyman-alpha forest, and quasars. You need to use (relatively) high-redshift tracers if you want to test conceivably relevant topologies.
Presumably all results would be negative; that's fine. But one nice side effect would be to find structures (for example clusters of galaxies) residing in very similar environments, and by
similar I mean in terms of full three dimensional structure, not just mean density on some scale. That could be useful for testing non-linear growth of structure.