Abstract
We construct two classes of wildly embedded space fillers of R3. First, every crumpled cube is shown to have an embedding in R3 that admits a monohedral tiling of R3. Second, a solid Alexander horned sphere with a topologically trivial interior is shown to admit a monohedral tiling of a cube and hence R3. By joining a solid horned sphere with compact polyhedral 3-submanifolds of R3 with one boundary component, we construct space fillers homeomorphic to the polyhedral submanifolds but of different embedding types. Using the suitably embedded crumpled cubes instead of a solid horned sphere, space fillers of even more different topological types can be produced.
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