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- Jan 26, 2012

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**Problem**: Recall that the fundamental group $\pi_1(X,x_0)$ is the set of homotopy classes of base point preserving maps $(S^1,s_0)\rightarrow (X,x_0)$. Let $[S^1,X]$ be the set of free homotopy classes of maps without conditions on base points. There there exists a map $\Phi:\pi_1(X,x_0)\rightarrow [S^1,X]$ obtained by ignoring base points.

Show that:

(a) $\Phi$ is onto if $X$ is path connected.

(b) $\Phi([f])=\Phi([g])$ if and only if $[f]$ and $[g]$ are conjugate in $\pi_1(X,x_0)$.

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