Rep-Tiles

TL;DR

This paper demonstrates that any smooth compact n-dimensional manifold with connected boundary can be topologically isotoped to a rep-tile, providing a classification and explicit constructions of rep-tiles in all dimensions.

Contribution

It proves that all such manifolds are topologically isotopic to rep-tiles and provides explicit constructions for them in any finite CW complex homotopy type.

Findings

Every smooth compact n-manifold with connected boundary is isotopic to a rep-tile.

All rep-tiles are classified up to isotopy in all dimensions.

Explicit constructions of rep-tiles in the homotopy type of finite bouquets of spheres.

Abstract

An $n$-dimensional rep-tile is a compact, connected submanifold of $\mathbb{R}^n$ 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 $\mathbb{R}^n$ 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.