2024/12/28 by Ryan Blair, Blair, Ryan, Patricia Cahn +5 · 1 voice · 1 citation
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.2412.19986
arxiv published 2024/12/28 · arxiv updated 2025/09/30
An n-dimensional rep-tile is a compact, connected submanifold of ℝn with non-empty interior which can be decomposed into pairwise isometric rescaled copies of itself whose interiors are disjoint. We show that every smooth compact n-dimensional submanifold of ℝn with connected boundary is topologically isotopic to a polycube that tiles the n-cube, and hence is topologically isotopic to a rep-tile. It follows that there is a rep-tile in the homotopy type of any finite CW complex. In addition to classifying rep-tiles in all dimensions up to isotopy, we also give new explicit constructions of rep-tiles, namely examples in the homotopy type of any finite bouquet of spheres.