Paper 2022 Q1

Let $X$ be a path-connected topological space and $b, b^{\prime} \in X$. We write $I=[0,1]$. (a) Given a path $w: I \rightarrow X$ with $w(0)=b$ and $w(1)=b^{\prime}$, describe the map $$ w_{\sharp}: \pi_1(X, b) \rightarrow \pi_1\left(X, b^{\prime}\right) $$ induced by $w$, and prove that it is well-defined and an isomorphism. How does this isomorphism depend on $w$ ? [You may assume that, if $u$ and $v$ are paths in $X$ such that $u(1)=v(0)$ and $u^{\prime}$ is homotopic to $u$ and $v^{\prime}$ is homotopic to $v$ relative to $\partial I$, then uv and $u^{\prime} v^{\prime}$ are homotopic relative to $\partial I$.] (b) Given loops $\ell$ and $\ell^{\prime}$ in $X$ based at $b$, we say that $\ell$ and $\ell^{\prime}$ are freely homotopic if they are homotopic when viewed as maps from $S^1$ to $X$, not fixing $b$. Give an algebraic necessary and sufficient condition for $\ell$ and $\ell^{\prime}$ to be freely homotopic in terms of their homotopy classes in $\pi_1(X, b)$. (c) Give an example of a space $X$ and distinct elements $l, m \in \pi_1(X, b)$ such that $l$ and $m$ are freely homotopic.