# A connected sum and wild cells

Can we find such a connected sum of two spheres (in any dimension) that is not homeomorphic to the sphere? $\def\R{\mathbb R}$

It seems that there should be examples like that, because there are lots of examples of “wild spheres” or “wild cells”, the best known being the Alexander horned sphere (such a tricky embedding of a closed 3-ball [and its boundary sphere] in $\R^3$ that its complement is not homeomorphic to the complement of the standard unit ball – the fundamental group of this complement is not only nontrivial but also not finitely generated!). If we embed $\R^3$ in the sphere $S^3=\R^3\cup \{\infty\}$ then glueing complements of such objects (which are completely different from disks) should result in something not homeomorphic with the sphere.

However – surprise, surprise – this article from 1951 claims that two solid horned spheres glued via the identity map on the boundary are homeomorphic with the sphere…

Update: note that we need two “wild cells” to produce a counterexample – if only one embedding of disk to the sphere is nonstandard, then the operation of connected sum just fills this nonstandard hole to reconstruct the sphere.
Offtopic: even more is true in dimension $3$: the hole could be “cut” by a $2$-dimensional sphere (not necessarily being a $3$-disc!) and still by filling it with a disc we get the $3$-sphere as is cited in this article (1st sentence of the third paragraph).

I suppose that an example may be nontrivial, but maybe someone here knows the reference.