Is topology of universe observable? There is an idea that the geometry of physical space is not observable(i.e. it can't be fixed by mere observation). It was introduced by H. Poincare. In brief it says that we can formulate our physical theories with the assumption of a flat or curved space by changing some assumptions and these two formulations are empirically indistinguishable. Here is a topic related to it:
Can general relativity be completely described as a field in a flat space? . This idea is also accepted by many contemporary physicists including Kip Thorne, see:
Thorne, K. S. 1996. Black Holes and Time Warps, New York: W.W. Norton, pp.400-2,
But here is my question:
Is the topology of the universe observable? By observable, I mean can we find it by observations, or is  the topology also subject to indeterminacy by experiment (similar to geometry)? What do physicists claim about the topology of the universe?
This question is of importance because when we move from a formulation in a curved and closed geometry to a flat and infinite geometry and we claim both formulations are equivalent, then it seems that, at least ,global topological properties like boundedness are not observable. In the post mentioned it has been stated that only differential topology is observable, in agreement to what I said.
I want to know what physicists say about observability of topology, and why?
(I am not asking what is the topology of space, I know that question was asked before. I am asking about a deeper issue: whether it can be observed in principle or not?)
 A: I'm not going to provide a full answer here, because I don't know the answer, but I want to give some statements that illustrate quite nicely the kind of problems one would face when determining topology of anything:
We know spacetime is a manifold. That means, locally, it looks just like $\mathbb{R}^4$. That's already a bummer. We can't do jack at one place to find out anything about topology. But, as soon as we move, we get into all the complications of reference frames and whatnot. So, experimentally, whether or not we can principally detect topology, it's going to be one hell of a challenge.
But it gets worse. You know how we always suppose that fields fall off at infinity? That's one of the natural reasons principal bundles arise in gauge theories. If we want to make precise the notion of a field $A$ falling off at infinity, we say it has to be a smooth function and have a well-defined value $A(\infty)$. And what's $\mathbb{R}^n$ together with $\infty$? The one-point compactification, also known as the sphere $S^n$. But it is not quite feasible to find global solutions to the equations of motion of a gauge theory on $S^n$, thanks to the hairy ball theorem and others.  So we say: Alright, let's solve the e.o.m. locally on some open sets $U_\alpha,U_\beta$ homeomorphic to the disk (think of the hemispheres overlapping a bit at the equator), and patch the solutions $A_\alpha,A_\beta$ together on the overlap by a gauge transformation on $U_\alpha \cap U_\beta$. Now, we've got our field living naturally on the sphere $S^n$ if we want a global solution. Does this mean that we actually live on an $S^n$, or just that we are inept to find a coherent description of the physics on $\mathbb{R}^n$? What would that even mean?
I can hear the people saying "We can always examine what the curvature is - $S^n$ has non-vanishing one, $\mathbb{R}^n$ has vanishing one.". That's alright, but the above gauge argument forces us the either accept that there is no globally well-defined gauge potential $A$ on $\mathbb{R}^n$ or to think of some $S^n$ on which a patched-together solution lives. What's more real? What would it even mean to say one of these views is more meaningful than the other?
So, you might be inclined to say: "Screw these weird gauge potentials, we're living on a spacetime, and not some bundle!" But there are topological effects of these bundles such as instantons or the Aharonov-Bohm effect. Spacetime alone is not enough. And what would be a meaningful distinction between "These bundles are not where we live, they're 'above' spacetime" and "We live on the bundles, and most often only experience the projection on spacetime"?
What I am trying to say is that it is not even clear what we should regard as the universe we live in. The ordinary, 4D spacetime is not enough to account for all the strange things that might happen.
And as I said in the beginning, don't take this as an answer. I am biased from being immersed in gauge theories, and only having superficial knowledge of the intricacies of GR. But from what I see, all "non-trivial topology" can also be seen as arising from patching together local solutions to the physical laws that otherwise don't match well.
A: As has been discussed in many questions around here (e.g. here), relativity tells us only about local properties and behavior of a space-time. There are some exceptions when we make global assumptions - if we have a space of globally and strictly constant positive curvature, non-trivial topology is imminent because the space has to be the 3-sphere $\mathbb{S^3}$. 
But we can also add nontrivial topology without much constraints. The full richness can be explored e.g. through quotienting the "canonical" space-slices $\mathbb{E^3},\mathbb{S^3},\mathbb{H}^3$ (flat euclidean, 3-sphere, 3-hyperbolic) by a discrete symmetry group $\Gamma$. I.e. for $\Gamma$ a group of discrete translations in all directions "cutting up" $\mathbb{E}^3$, we get a topological 3-torus $\mathbb{E^3}/\Gamma = \mathbb{T^3}$.
The intuitive picture is that the space looks locally exactly as our good olde' flat euclidean space $\mathbb{E^3}$, but after a certain distance (the translation), we get to the same place. Naturally, as this is the same place, we should find the same things at these places up to their movement and evolution during the time we weren't there. 
As for cosmological observation, if we are to detect nontrivial topology with current methods, the space or the non-trivialities must be "sufficiently small". Imagine we are on a sphere and we are restricted by observation to see a very very small patch of it - there will be no way we can conclude it is a sphere. 
If we however see beyond say one of the discrete translations of $\Gamma$, we should be in principle able to detect multiple images of the same object. The problem is, since light took longer to travel from the image farther away, we will see the more distant object to be "younger" than the closer one and most probably under a different angle. For a decently large universe with other effects like redshift and obscuring, this is probably a deal-breaker.
Nevertheless, the endurance of scientists is endless. We may detect a repetition in the images when collecting large amounts of data for all visible objects and using certain correlation methods to evaluate them. Certain types of topological non-triviality would then be visible as peaks or "spikes" in the correlation indicators. 
The deepest image of the universe is the CMB, of which we have a very detailed dataset. CMB can be viewed as a snapshot of a large sphere at luminous distance $\chi_{CMB}$. If this sphere intersects a topological non-triviality, we should see "circles" or certain pattern repetitions in the CMB. So long, the few tests of the data however did not reveal any of this. If anything, an $\mathbb{R^2}\times \mathbb{S^1}$ (a "3-tube") topology is conjectured in association with the slightly preferred direction of the CMB.

There are possible indirect tests suggested by the discussion of ACuriousMind - a non-trivial topology imposes different boundary conditions on fundamental fields and other possible objects. Note however that the theory we develop means "very far" by $\infty$. That is e.g. in particle experiments, "very far" may be a distance of few meters, not cosmological scales. Effects due to topologically different boundary conditions would most probably play an important role in the very early universe and might provide indirect tests of the cosmic topology.

My main source for this answer are this and this review article.  
A: Every single comment and answer was very useful. I think i have found an answer in an old paper by Clark Glymour (Minnesota studies in philosophy of science, volume III, pp.50-60):
"It has recently been noted (Ellis, 1971; Dautcourt, 1971; Ellis and Sciama, 1972; Glymour, 1972; Trautman, 1965) that in some general relativistic cosmologies various global features of space-time may necessarily escape determination. In contrast to classical space-time theories, the fundamental group of space-time may itself be such a feature in a relativistic space-time. A precise account of what it means for two space-times to be "indistinguishable" will permit us to prove some elementary propositions concerning the classification of indistinguishable space-times which have distinct global topologies".
It says that it is possible for two space times to be empirically equivalent and yet have different topological properties, so it seems that the topology is also subject to indeterminacy by experiment at least in some of its aspects. The question of interest could be which aspects of topology are observable and which one unobservable?, 
I wasn't able to get access to Ellis's paper but here is its address:
Ellis, G. F. R. (1971). "Topology and Cosmology," General Relativity and Gravitation,
vol. 2, p. 7., if someone has it ,then i would be grateful if he explains it here for us,
A: Here is a simple thought experiment to help visualize the shape of a Big Bang universe. All directions point back in time. Theoretically if one could see far enough back in time, one is looking towards the Origin, a single point, the only point we all have in common. Therefore all directions ultimately point toward the Origin, which in some sense can be considered the center of the universe, and the only boundary. To understand the shape of the entire universe, one must consider ALL the domains of time and space. It makes no sense to ask what is the shape "now" (which is generally said to be almost flat.) So, if all directions from any location in the universe lead to a single common point, it gives us a few possibilities. The simplest is the torus, with a point sized (or very small) center. The torus has been shown in many ways to be the most efficient energy system in physics.
