Let $X$ be a topological space and let $x_{0}$ and $x_{1}$ be points of $X$.

(a) Explain how a path $u:[0,1] \rightarrow X$ from $x_{0}$ to $x_{1}$ defines a map $u_{\sharp}: \pi_{1}\left(X, x_{0}\right) \rightarrow$ $\pi_{1}\left(X, x_{1}\right)$.

(b) Prove that $u_{\sharp}$ is an isomorphism of groups.

(c) Let $\alpha, \beta:\left(S^{1}, 1\right) \rightarrow\left(X, x_{0}\right)$ be based loops in $X$. Suppose that $\alpha, \beta$ are homotopic as unbased maps, i.e. the homotopy is not assumed to respect basepoints. Show that the corresponding elements of $\pi_{1}\left(X, x_{0}\right)$ are conjugate.

(d) Take $X$ to be the 2-torus $S^{1} \times S^{1}$. If $\alpha, \beta$ are homotopic as unbased loops as in part (c), then exhibit a based homotopy between them. Interpret this fact algebraically.

(e) Exhibit a pair of elements in the fundamental group of $S^{1} \vee S^{1}$ which are homotopic as unbased loops but not as based loops. Justify your answer.