Exercise 32, § 5

Let A be a closed subset of $\mathbf{R}^3$ homeomorphic to $[0,1]$. The extremities of A are the two points $p$ of A such that A$-\{p\}$ is connected (cf. III, p. 280, th. 2); the other points of A will be called interior.

Let $p$ be a point of A; one says that A is moderate at $p$ if there exist a neighbourhood V of $p$ in $\mathbf{R}^3$ and a homeomorphism of V onto the unit ball of $\mathbf{R}^3$ which maps $A\cap V$ onto a segment.

a) Let $p$ be a point of A such that A is moderate at $p$. Let $(V_n)$ be a decreasing sequence of neighbourhoods of $p$ in $\mathbf{R}^3$ such that $\bigcap_nV_n=\{p\}$; for every $n$, let $a_n$ be a point of $V_n-$ A and denote by $j_n:\pi_1(V_n-A, a)\rightarrow \pi_1(V_1-A, a)$ the group homomorphism deduced from the injection of $V_n$ into $V_1$. If $p$ is an extremity of A (resp. an interior point), prove that there exists an integer $m$ such that for every integer $n\geqslant m$, the image of the homomorphism $j_n$ is reduced to the identity element (resp. is abelian).

b) Suppose that A is moderate at every point. Prove that there exists a homeomorphism of $\mathbf{R}^3$ onto itself which maps A onto a segment. Deduce that $\mathbf{R}^3-$ A is connected and simply arcwise connected.