Give an example of a topological space $X$ such that $\pi_1(X)=S_3$, where $\pi_1(X)$ is the fundamental group.
Give an example of a topological space $X$ such that $\pi_1(X)=S_3$.
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algebraic-topology
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5Note that this is true for [any group](http://math.stackexchange.com/questions/939856/every-group-is-a-fundamental-group), not just $S_3$. – 2017-01-09
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0See [this question](http://math.stackexchange.com/questions/1053234/topological-space-with-given-fundamental-group) as well – 2017-01-09
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0For instance, see http://math.stackexchange.com/questions/228240/given-any-group-g-is-there-a-topological-space-whose-fundamental-group-is-exact – 2017-01-09
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0@Arthur Do you know an easy example of a topological space $X$ such that $\pi(X)=S_3$? Could be a Cayley graph? – 2017-01-11
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2You can make it the following way: Since $S^3 = \langle t_1, t_2 \mid t_1^2 = t_2^2 = (t_1t_2)^3 = e\rangle$, we can start with a figure-$8$ (the two loops representing $t_1$ and $t_2$ respectively), and three closed discs (representing the three relations in the presentation above). For each disc, glue the edge to the figure-eight in a way corresponding to the given relation. This means that one disc has its edge glued to go twice around one of the loops in the $8$, one disc goes twice around the other, and one disc has its edge go three whole laps around the entire $8$. – 2017-01-11