Exercise 5, § 5

Let $P$ be the subspace of $\mathbf{R}^2$ defined as in Exercise 5, I, p. 146, b) ; let us also denote by $P^+$ (resp. $P^-$) the set of points $(x, y)\in P$ such that $y\geqslant 0$ (resp. $y\leqslant 0$). Let $a$ be the point $(0,0,1)$ of $\mathbf{R}^3$ and let A be the union of the segment $[a,0]$ and of the segments $[a,(2r_n,0)]$, for $n\in \mathbf{N}$, Put $B^+= (P^+\times \{0\})\cup A$, $B^-= (P^-\times \{0\})\cup A$ and $B = (P\times \{0\})\cup A$.

a) Prove that the space $B^+$ is contractible and that the Poincaré group $\pi_1(B^+, a)$ is a free group on $\mathbf{N}$.

b) Prove that A is contractible at $a$.

c) Prove that the canonical homomorphism from $\pi_1(B^+, a)*\pi_1(B^-, a)$ into $\pi_1(B, a)$, deduced from the canonical inclusions of $B^+$ and $B^-$ in B, is not surjective.