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To me, one needs to look for an answer from the other end: Big Bang. If the Universe was born from a single 1-connected point and all the following expansion has no singularities, i.e. can be considered as homeomorphic transformation, then the Universe has no reason to become n-connected. Some confirmation of this is covered in the article ARXIV:1306.0533 by Maldacena and Susskind. PS. So, one has to know connectivity of the "seed" of the Universe only, because generally, physicists have good reason to "believe" that all physical processes are homeomorphic transformations. /humble physicist/