questions: ---------- 1) does the cosmological principle introduce any restrictions on the topology of the universe. 2) does the topology science relies on some multidimentional theory - like string theory
now look here,
the problem: ------------
I finally figured it out. It is about that transparency problem. The problem was how it is possible to roll up the transparency into a torus not stretching it so it's curvature remained the same. I was stubborn to say that this was not possible, so we came to the conclusion that the 2-torus in three dimenitonal eucliean space is not an good example of two dimentional closed space of zero curvature. But if we introduced the fourth spital dimention than it becomes possible. Like for example we imagine a string which has one pair of zero dimentional ends, we need at least one more dimentional space (it's two) to make it closed (to joing the ends), so in case of a flat plane (or paper) which has two pairs of one dimentional ends which we have to join, then we need two more dimentions to do that. That would be 4 dimentions. Thus in four dimenitons matematically we can do with the topology of the space whatever we want and it's curvature will still be the same.
But the problem now is how to put this idea into the real world. All these considerations about the topology of space were made from "the higer groud" - I mean from the point of view of a spce with at lest two dimentions more that the space (earlier a transparency or just a piece of paper) we were thinking of. The problem appeares if we talk about the topology of the Universe where there are only three spital dimentions. To make it close like 3-torus it is essential to introduce 3 more dimentions (it's six) to make it possible. Then I say that, that all topoloy thing implies that out phisical space is in fact more than three dimentional, otherwise we would stand before question: what is that thing that out space (universe) lies in. Since this question is considered to be sensless, or at least redundant, the only solution is to assume the first option. It leads to the link with the science of the nature of the space itself - like in the string theory, which assumes that our spacetime is 10-dimentional - time plus three known spital dimentions and 6 other which were reduced to the lenght of Planck during the very early stages of evolution of the universe. In that case that bending our unverse into a 3-torus around those additional axies could be easily (al least theoreticaly) done but the "radius" of that bend would be very small. But whatever. Is this way of thinking of topology ok ?
further questions: ------------------
For the curvature matter, there is general theory of relativity which connects the energy density with curvature. What about the topology ? What is that force that shapes the topology of space, and can the topology change in time ?
pozdrawiam Bartek