Topologists study geometric shapes, disregarding lengths, angles, areas, and volumes. This is because they are interested in properties that remain unchanged when the shapes are deformed, such as bending, twisting, and stretching. A square, for example, can be turned into a circle or a triangle, but a circle cannot be turned into two circles without breaking it. Similarly, a ball can be turned into a cube, but a donut cannot be turned into a ball without taking a bite out of it. The objects that topologists mostly focus on are called manifolds. These objects look from any given point like flat space or n-dimensional hyperspace. An example of a 2-dimensional manifold is the surface of the Earth, which looks like a plane around any point. Other examples of 2-dimensional manifolds are the surface of a donut or the Klein bottle. 1-dimensional manifolds are circles, and 3-dimensional manifolds are our ordinary space or the universe. It’s a 3-dimensional manifold, which could be hyperspace or a 3-dimensional hypersphere. Though no one has done it, it is possible that if one goes very high up, they would come back to where they are. We don’t know if it is possible or not, similar to the surface of the Earth when one goes around and comes back.
No, there are other ways to test it. People are looking at cosmic microwave background radiation and have put forward various theories about the shape of the universe, but there is no definite theory that everyone accepts. It is not yet known what shape it is, but there are some hints from the universe.
We have 3-dimensional objects and then we can think about 4-dimensional and so on. A good example is the hypersphere, which is a 1-dimensional manifold given by the equation x² + y² = 1 and a sphere, which is a 3-dimensional manifold given by the equation x² + y² + z² = 1. We can generalize this, talking about the n-dimensional hypersphere, given by the equation x1² + ... + xn² + xn+1² = 1, living in n+1-dimensional space and it’s an n-dimensional manifold.
Exotic objects are topologically the same but smoothly different. A circle can be turned into a square, and in the process a corner must be made. The circle is smooth and can be turned into other smooth curves, like an ellipse, but it could be that two smooth objects can be deformed into each other only by making corners somewhere in the middle. An example is a figure eight, which can be turned into a circle if one of the loops is made smaller and smaller, but at some point a corner must be made and then it becomes a circle. Exotic objects don’t exist in dimensions 1, 2 and 3, but they start existing in dimension 4 and higher. The first examples were found in 1956 in dimension 7, they are exotic hyperspheres of 7 dimensions. Now in dimensions 5 and higher, you can use this trick to separate disks.You can take a donut and separate it into two pieces.And that’s not possible in dimensions 4 and lower.So this is why dimension 4 is different. Mathematicians are studying objects by looking at loops and how to contract them, spanning a 2-dimensional disc. To make it easier to study the objects, they are splitting the disks apart. This is done by using lines, which are 1-dimensional objects. To separate disks, they need one extra dimension, which is why they use 5 and higher dimensions. This is called the Whitney Trick and it led to the classification of smooth manifolds in higher dimensions back in the 1960s. Unfortunately, this trick cannot be used in dimension 4, making it the hardest to understand. In dimension 4, there are infinitely many exotic hyperspaces, unlike in any other dimensions where there are only finitely many. Whether dimension 4 should be studied more or not depends on the perspective of the mathematician. In many theorems, it is said that this is happening in all dimensions except 4 and there is still much to be discovered in this dimension.