Exercise 11, § 5

Let $X$ be a path-connected topological space and let $f: X\rightarrow X$ be a homeomorphism. Let T be the quotient space of $X\times \mathbf{I}$ by the finest equivalence relation for which the points $(x,0)$ and $(f(x),1)$ are equivalent, for all $x\in X$. We denote by $p: X\times \mathbf{I}\rightarrow T$ the canonical mapping; we denote by $j: X\rightarrow T$ the mapping given by $x\mapsto p(x,1/2)$. Let $a$ be a point of X, let $c\in \Lambda_{f(a),a}(X)$ be a path of origin $f(a)$ and of term $a$, and let $\gamma = [c]$.

a) Prove that the mapping from $\mathbf{I}$ into T given by $t\mapsto p(a,\frac{1}{2}-t)$ for $t\in [0,\frac{1}{2}]$ and $t\mapsto p(c(2t-1),\frac{3}{2}-t)$ for $t\in ]\frac{1}{2},1]$ is a loop at $j(a)$ in T. We denote its class by $\delta$.

b) Let $\mathsf{S}$ be the unique group homomorphism from $\pi_1(X, a)*\mathbf{Z}$ into $\pi_1(T, j(a))$ which maps a class $v\in \pi_1(X, a)$ onto $j_*(v)$ and the element $t= 1$ of $\mathbf{Z}$ onto the class $\delta$. Prove that the homomorphism $\mathsf{S}$ is surjective and that its kernel is the smallest normal subgroup which contains the elements $vt(\gamma^{-1}f_*(v)\gamma )^{-1}t^{-1}$, for $v\in \pi_1(X, a)$.

c) For $v\in \pi_1(X, a)$, put $\varphi (v) =\gamma^{-1}f_*(v)\gamma$; prove that $\varphi$ is an automorphism of the group $\pi_1(X, a)$. Let $\varphi :\mathbf{Z}\rightarrow$ Aut($\pi_1(X, a)$) be the unique group homomorphism which maps 1 onto $\varphi$. Construct an isomorphism of the group $\pi_1(T, j(a))$ onto the external semidirect product of $\mathbf{Z}$ by $\pi_1(X, a)$ with respect to $\varphi$.