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/