spaces with a specified fundamental group - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T20:35:24Z http://mathoverflow.net/feeds/question/102845 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102845/spaces-with-a-specified-fundamental-group spaces with a specified fundamental group Steven 2012-07-22T00:31:35Z 2012-07-22T00:31:35Z <p>Given an arbitrary group, how hard is it to come up with a space which has this group as its fundamental group? In particular, is there a known space which has $\hat{\mathbb{Z}}$ as its fundamental group? Is this space too complicated to be worth studying? And what if I also specify that I want the higher homotopy groups to be 0? (I feel that perhaps I've heard somewhere that such a space exists for all groups- is this true?)</p>