Hi everyone, I was reading about how the universe might have a multi-connected topology, and how we might confirm that by finding opposing pairs of circles in the cosmic microwave background radiation, and a question came to my mind.

One of the postulates physics lay on is that there is no special direction in the universe, that is, physics laws are isotropic. But now, I get told that if the universe actually has a folded topology, I will be able to find matching pairs of circles in the CMB at specific places in the sky, these places depending on what exactly the large-scale topology of the universe is.

The way I understand it, these circles, if any, will be found in the same direction regardless of where you stand in space. So where has the symmetry gone?

Does the theory of a multi-connected space require the existence of an arbitrary reference set of directions that define the orientation of the base polyhedron that describes the universe’s topology? Its orientation would have at least one physically observable effect, the existence of pairs of circles in the CMB, but could there be more?

Or am I totally misguided?