Assuming FLRW is correct, can it be 3-torus? Assuming Friedmann–Lemaître–Robertson–Walker metric is correct, can space topology be 3-torus? I.e. are these two hypotheses compatible?
 A: The assumption underlying the FLRW metric is that space is homogeneous and locally isotropic.
Schur’s Theorem states that a connected Riemann manifold of dimension $n>2$ that is locally isotropic around every point has constant sectional curvature $K$.
If $(M,g)$ is a Riemann 3-manifold with sectional curvature $K=0$ everywhere (such as the flat 3-torus $T^3$), then $(M,g)$ is locally isometric to the Euclidean space $E^3$.
The FLRW models are homogeneous and (locally) isotropic solutions of Einstein's field equations, of which the spatial sections have constant sectional curvature.
Relativistic cosmological models with topology $T^3$ and relativistic cosmological models with topology $E^3$ —assuming the same cosmic components and energy densities— are described by exactly the same Robertson-Walker metric, even though $T^3$ is finite in extent while $E^3$ is infinite. In particular, the solution of the Friedmann equations —the scale factor $a(t)$— is also identical. Only the boundary conditions on the spatial coordinates are changed.
Einstein's field equations are local PDEs that relate the metric and its derivatives at a point to the density and flux of energy and momentum at that point. EFEs determine the local geometry of spacetime. Although the local constraints imposed by the EFEs are very powerful (severely restricting what the spacetime manifold can do), they stop short of having a determining effect on the global topology of space. The spatial topology of the universe cannot be determined from its spatial curvature because each of the three plausible cosmic geometries —Euclidean, spherical, hyperbolic— is consistent with many different topologies.
The problem is that in most studies in cosmology the topology of the space-like hypersurfaces of the spacetime is assumed to be simply-connected: the hypersphere $S^3$, the Euclidean 3-space $E^3$, or the hyperbolic 3-space $H^3$. However, there is no particular reason for space to have a trivial topology, that is, the global topology of space does not have to be that of $E^3$, $S^3$, or $H^3$. There are a great variety of multiply-connected topologies for $K=0$, $K>0$, and $K<0$.
To sum up, $T^3$ is not globally isotropic, but is locally isometric to $E^3$. Consequently, cosmological models with spatial topology $T^3$ are described by the same FLRW metric as cosmological models with spatial topology $E^3$. The FLRW metric is perfectly compatible with a spacetime with the spatial topology of $T^3$.
NOTE. Here the expression "topology of space" is used in the sense "GLOBAL topology of space" (i.e. properties of the space that are not "local"), not in the technical meaning of topology as a collection of open sets.
A: FLRW cosmology postulates that there exists a frame in which space is isotropic and homogenous, but a 3-torus is not globally isotropic. You can see this in that if you head along a (spatial) geodesic along a 3-torus whether you get back to your starting point and how far you have to travel before you do depends on the direction you go.
