Exercise 27, § 5

Let D be the unit disc in $\mathbf{C}$ and let X be a subspace of D which is a neighbourhood of $\mathbf{S}_1$. Let R be the least fine equivalence relation in X which, for $u, v\in \mathbf{S}_1$, identifies the points $f(u)$ and $f(v)$ if $f(u) =f(v)$. Let $p: X\rightarrow X/R$ be the canonical surjection. Let T be a compact topological space and let $f:\mathbf{S}_1\rightarrow T$ be a continuous surjective mapping. Put $a=f(1)$ and denote by $\alpha \in \pi_1(T, a)$ the class of the loop $t\mapsto f(e^{2\pi it})$ in T.

a) Prove that there exists a unique mapping $\sigma$ from T into $X/R$ such that $\sigma \circ f=p|_{\mathbf{S}_1}$. Prove that $\sigma$ defines a homeomorphism of T onto its image $p(\mathbf{S}_1)$.

b) Let $r$ be a real number such that $0< r <1$; suppose that X is the set of the $z\in D$ such that $r\leqslant |z|\leqslant 1$. Prove that $X/R$ is homeomorphic to the cylinder of the mapping $f$. Deduce that the homomorphism $\sigma_*$ is a group isomorphism of $\pi_1(T, a)$ onto $\pi_1(X/R, p(1))$.

c) Suppose that X = D. Prove that $X/R$ is homeomorphic to the cone of the mapping $f$. Deduce that the group homomorphism $\sigma_*:\pi_1(T, a)\rightarrow \pi_1(X/R, p(1))$ is surjective and that its kernel is the smallest normal subgroup of $\pi_1(T, a)$ which contains $\alpha$.

d) Let $r$ be a real number such that $0< r <1$; let $X_1$ be the set of the $z\in X$ such that $|z|\leqslant r$ and $X_2$ the set of the $z\in D$ such that $r\leqslant |z|\leqslant 1$; suppose that $X_2\subset X$. Let $\beta \in \pi_1(X_2, r)$ be the class of the loop given by $t\mapsto re^{2\pi it}$; let $\delta \in \varpi_{p(r),p(1))}$ be the class of the path given by $t\mapsto p((1-t)r+t)$.

Prove that there exists a unique group homomorphism $\mu$ from $\pi_1(T, a)*\pi_1(X_2, r)$ into $\pi_1(X/R, p(1))$ which coincides with the homomorphism $\sigma_*$ in $\pi_1(T, a)$ and with the homomorphism $v\mapsto \delta^{-1}p_*(v)\delta$ in $\pi_1(X_2, p(r))$. Prove that $\mu$ is surjective and that its kernel is the smallest normal subgroup which contains $\alpha \beta^{-1}$.