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Given any Group $G$, is there a topological space whose fundamental group is exactly $G$?

If yes, what (which Theorem) is this result based on?

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up vote 7 down vote accepted

Given $G$ one can construct the so called classifying space $BG$ whose fundamental group is $G$ (and whose all other homotopy groups are trivial) If you just want the fundamental group, you can keep the $2$-skeleton of $BG$. Google for these keywords, and yiu will find details about this; it is also explained in textbooks on homotopy theory.

This result is not based on any theorem: it is just a construction.

There is a theorem, though, that tells us that if $X$ and $Y$ are two different spaces which satisfy the same conditions (all homotopy groups trivial except the fundamental group, and the latter isomorphic to $G$) then $X$ and $Y$ are in fact homotopy equivalent. This is Whitehead's theorem. (One needs sensible hypothesis on the spaces: that they be CW-complexes, for example) This has the consequence that, up to homotopy, there is exactly one space satisfying those conditions.

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If we write the group $G$ in terms of generators and relations, starting with one 0-cell, we add 1-cells corresponding to each generator and 2-cells corresponding to each relation. Then the fundamental group of this 2-dimensional cell complex is $G$.

See page 52 of Hatcher's Algebraic Topology for details.

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