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Dear Professor

You have explained compact surfaces. But how does this get tie up with general theory of relativity or it's derivative Friedman equation. Or any other physical theory.
There has to be another paper.

1. Let us take a look at Friedman equation. It has the Hubble constant and cosmological constant. Both are non zero entities. How would the compact topology work with this situation. Also einstein's field equations describes a dynamic universe, even without the cosmological constant. The universe was born this way.
Did the universe born with a compact topology just after the big bang?

2. Raycahudhuri equation suggests that all geodesics will come together and form a singularity in the future if there is no cosmological constant. Since the existence of cosmological constant is proven from super nova observations the universe is expanding and also accelerating.
My question is how would the compact topology works out with accelerating universe.
3. My third question is whether there is any theory that connects particle physics with compact topology?

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