This is one of the common misconceptions about the "shape of the universe", or manifolds in general, i.e. that a manifold is just a sort of cut-out from flat eucledian 3D space. Like if we were living in a doughnut, we could just eat ourselves to the inner edge and find the 'hole'. But manifolds can exist in a meaningful way without reference to an 'outside space' in which they live. You can even define an 'intrinsic' curvature on them, without having to give an outside space in which they curve into. The idea that the 'universe is toroidal' has much more to do with topology than with geometry (though obviously geometry plays an important role). Think of a video game, where when you cross the upper side of the screen you appear at the lower side, and when you cross the right side of the screen you appear on the left side. It might not look like it, but this map has the essential structure of a torus (going along the wide ring of a torus, you'd eventually end up where you started, and the same is true for going along the small ring). Yet, there is no 'hole' to speak of. This is an example of a universe with toroidal topology (but no curvature). When physicists or mathematicians try to sell you on the idea of odd shaped universes, think of it as being a bug on the surface of such an object. You don't know the shape of it, you only can walk along it its principal directions, and for certain shapes and directions you find you end up where you started if you walk long enough, or that you cross a path you've already walked on. And this knowledge sort of defines the 'true shape' of that object, even though it doesn't necessarily exist in that way in 3D space. More on reddit.com

It is possible for the topology to be non-trivial, but afaik we don’t know for sure; if the universe does loop back on itself, then it would be on large length scales that we can’t measure. We are fairly confident the universe is flat (has 0 curvature) at least. Note that this means that the universe probably doesn’t literally look like a donut: physical donuts have curvature, whereas topological tori need not have curvature. This is because a donut is a 2-torus embedded in 3D space, which isn’t equivalent to 2D space having the same topology as a 2-torus. A simple analogy for those who aren’t aware: an example of a 2D universe with non-trivial topology but flat geometry is a Pac-Man level, which is essentially a flat 2-torus. The curvature is flat everywhere, but it loops back on itself at the edges. More on reddit.com

It seems to me that the thing you get in this way is not a torus, at least not the way you're describing it. By starting with a square and collapsing the boundary to a single point, we get a 2-sphere (meaning the surface is 2D). Then you want to also identify the boundary with a point in the center of the square, so it sounds like you must get the same outcome, at least topologically, as if you identified two points on a sphere. This is the wedge sum of a sphere and a circle, which is explained in Examples 0.8 and 0.11 of Hatcher's Algebraic Topology. If you start with a cube instead of a square, I believe that something similar applies one dimension up. On the other hand, if you don't mean to collapse the boundary to a single point, but to puncture out a disk around the singularity in the center of the square and identify the boundary of that disk with the boundary of the square, then I do believe you will get a torus, and the same likely does apply one dimension up, which we can probably verify by the Siefert-van Kampen theorem (as open neighborhoods, we can use the top and bottom halves of the input square/cube, overlapping at the equator, so long as the boundary of the inner puncture is completely contained in both in order to make the intersection path connected (that is probably only necessary in dimension 2). This is probably what you are seeing. A torus has zero curvature, so as far as I know, I don't think our physical observations can distinguish a homotopy trivial universe from one which is a torus, but where the length of the shortest nontrivial element of the fundamental group is larger than the diameter of the observable universe. Mods, I hesitate to say this considering I am not trained as a physicist (but rather a mathematician who works and publishes with physicists), but I'm not convinced this is completely "crackpot physics" as defined in the sidebar. The topology is intelligible, even if it is not stated the way an expert would put it, and I expect that numerically simulating the solutions of the right differential equations on a torus might have some physical relevance, at least as a toy model, though it really is more mathematics than physics. OP, I would suggest reading Hatcher. I feel like you're the kind of person who would really enjoy it. More on reddit.com

That being said and my mistake acknowledged, I still think this is very interesting and worth the post for the discussion I hope it encourages. Holographic universe theory has suggested that the universe is toroidal in shape and nature, and the video I linked is a great intro into exactly that ... More on reddit.com