Modify the construction of the Alexander horned sphere to produce an embedding \(S^2 \hookrightarrow \mathbb{R}^3\) for which neither component of \(\mathbb{R}^3 - S^2\) is simply-connected.
Solution
The standard Alexander construction can be localized in a small \(3\)-ball meeting an otherwise tame sphere in a disk. We use two disjoint such balls, one on each side of the sphere.
1. Start with a round sphere \(S_0\subset\mathbb R^3\) and choose two disjoint closed disks\[ D_+,D_-\subset S_0. \]Choose disjoint small \(3\)-balls \(N_+,N_-\) with\[ N_\pm\cap S_0=D_\pm, \]where \(N_+\) lies mainly on the exterior side of \(S_0\) and \(N_-\) mainly on the interior side.
Take two small disjoint coordinate balls centered at distinct points of the sphere and shrink them so that each meets the sphere in a single tame disk.
2. Inside \(N_+\) replace the disk \(D_+\) by a localized Alexander horned disk whose horns accumulate from the exterior side. Inside \(N_-\) make the mirror-image replacement, with horns accumulating from the interior side. Leave the sphere unchanged outside \(D_+\cup D_-\).
Delete from the standard Alexander horned sphere a small tame disk disjoint from the horn accumulation set. The remainder is a disk with the same wild horn structure, embedded in a \(3\)-ball and agreeing with a standard disk near its boundary. This is the required localized horned disk. Reflecting the construction interchanges its two sides.
3. The resulting surface \(S\) is homeomorphic to \(S^2\).
Each replacement is a topological disk attached along the same boundary circle as the disk it replaces, and it agrees with the old disk in a collar of that boundary. Thus replacing \(D_+\) and \(D_-\) does not change the abstract surface: we have removed two disks from \(S^2\) and glued back two disks along their boundary circles. Hence the result is again \(S^2\).
4. The exterior component of \(\mathbb R^3-S\) is not simply-connected.
Inside \(N_+\) the exterior-side complement contains exactly the localized complement used in Hatcher’s Alexander-horned-sphere argument. Let \(\alpha_+\) be a meridian loop linking the first-stage handle. Hatcher’s van Kampen calculation identifies the successive stage groups with free groups and sends each old meridian injectively to a commutator of two new meridians. Hence \(\alpha_+\) is nontrivial in the local exterior complement.
Choose \(N_+\) so that its intersection with the rest of the exterior component is a collar of a disk, hence simply-connected. Van Kampen then expresses the fundamental group of the full exterior as a free product with the local exterior group as a factor. In particular the inclusion of the local exterior complement is injective on \(\pi_1\), so \([\alpha_+]\) remains nontrivial globally.
5. The bounded component of \(\mathbb R^3-S\) is also not simply-connected.
Apply the same argument to the reflected horned disk in \(N_-\). A meridian \(\alpha_-\) linking its first-stage inward handle is nontrivial in the local interior complement by the identical free-group/commutator calculation. The local piece meets the rest of the bounded component through a simply-connected collar, so van Kampen again makes the local fundamental group inject into the global one. Thus \([\alpha_-]\ne1\).
Therefore \[ \boxed{\text{neither component of }\mathbb R^3-S\text{ is simply-connected}.} \]