Crumpled cube and solid horned sphere space fillers (Q701794)

A closed subspace \(M\) of \(\mathbb{R}^3\) admits a \textit{monohedral tiling} of \(\mathbb{R}^3\), if \(\mathbb{R}^3\) is covered by congruent copies of \(M\) any two of which have disjoint interiors. A \textit{crumpled cube} is a space homeomorphic to the closure of the bounded complementary domain of a wildly embedded \(2\)-sphere in \(\mathbb{R}^3\). Using techniques of \textit{C. C. Adams} [Math. Intell. 17, No. 2, 41--51 (1995; Zbl 0847.52020)] and \textit{W. Kuperberg} [Discrete Comput Geom. 13, 561--567 (1995; Zbl 0824.57016)], the author shows that for any crumpled cube \(K\) there is an embedding \(h\) of \(K\) into \(\mathbb{R}^3\) such that \(h(K)\) admits a monohedral tiling of \(\mathbb{R}^3\). Then he shows that a solid Alexander horned sphere with topologically trivial interior admits a monohedral tiling of a cube and hence \(\mathbb{R}^3\). Finally he shows that for a given compact polyhedral \(3\)-submanifold \(M\) of \(\mathbb{R}^3\) with one boundary component, there is a homeomorphic copy \(M'\) of \(M\) in \(\mathbb{R}^3\) of different embedding type, and such that \(M'\) admits a monohedral tiling of \(\mathbb{R}^3\).