On self-affine tiles whose boundary is a sphere

Let M be a 3×3 integer matrix each of whose eigenvalues is greater than 1 in modulus and let D ⊂ Z ³ be a set with |D| = | det M|, called a digit set. The set equation MT = T + D uniquely defines a nonempty compact set T ⊂ R ³. If T has positive Lebesgue measure it is called a 3-dimensional self-affine tile. In the present paper we study topological properties of 3-dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form D = {0, v, 2v, . . ., (| det M| − 1)v} for some v ∈ Z ³ \ {0}. We prove that the boundary of such a tile T is homeomorphic to a 2-sphere whenever its set of neighbors in a lattice tiling which is induced by T in a natural way contains 14 elements. The combinatorics of this lattice tiling is then the same as the one of the bitruncated cubic honeycomb, a body-centered cubic lattice tiling by truncated octahedra. We give a characterization of 3-dimensional self-affine tiles with collinear digit set having 14 neighbors in terms of the coefficients of the characteristic polynomial of M. In our proofs we use results of R. H. Bing on the topological characterization of spheres.