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. /humble physicist/
Nikolai Zaitsev
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