2021/07/26 by Thuswaldner, Jörg M., Zhang, Shu-Qin · 1 citation
#52C22 #54F65 #57M50. Secondary: 51M20 #FOS: Mathematics #Geometric Topology (math.GT) #Primary: 28A80
paper · doi:10.48550/arxiv.2107.12076
Let M be a 3× 3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D ⊂ ℤ3 be a digit set containing |det M| elements. Then the unique nonempty compact set T=T(M,D) defined by the set equation MT=T+D is called an integral self-affine tile if its interior is nonempty. If D is of the form D=\0,v,…, (|det M|-1)v\ we say that T has a collinear digit set. The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets. In particular, we prove that a large class of these tiles is homeomorphic to a closed 3-dimensional ball. Moreover, we show that in this case T carries a natural CW complex structure that is defined in terms of the intersections of T with its neighbors in the lattice tiling \T+z : z∈ ℤ3\ induced by T. This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.