## spaces with a specified fundamental group [closed]

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?)

-
You want to Google for "Eilenberg-Maclane space". – Steven Landsburg Jul 22 at 0:50
Steven Landsburg's comment handles your more difficult, second question. For the first question, where you don't care about the higher homotopy, there's an easier answer: Take a presentation of your group, form a bouquet of circles consisting of one circle for each generator in your presentation, and then attach one disk for each relation in the obvious way. – Andreas Blass Jul 22 at 1:59
The etale fundamental group of the punctured complex plane is Z-hat. – John Wiltshire-Gordon Jul 22 at 3:21
If you care about the topology on Z-hat, you should learn about classifying space BG. – John Wiltshire-Gordon Jul 22 at 3:22
Another way of putting the preceding comment might be: is Z-hat a group, or is it a topological group, or ...? It depends in what context you are working. The question of profinite completions of spaces and in particular of the profinite completion of the circle is not that far away here. Personally I feel John's previous comment is slightly off target as the étale fundamental group is not the same as the fundamental group... but it does indicate the link with algebraic geometry. For the general case, look at books on combinatorial group theory for starters. – Tim Porter Jul 22 at 7:14
show 1 more comment