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Topology of slices through the Sierpiński tetrahedron

Yuto Nakajima, Takayuki Watanabe

Abstract

We investigate slices of the Sierpiński tetrahedron from a topological viewpoint. For each $c\in[0,1]$, we study the Čech (co)homology group of the slice at height $c$. We show that the topology of the slice exhibits a sharp dichotomy. If $c$ is a dyadic rational, then the slice has finitely many connected components, infinite first Čech homology, and trivial higher homology. If $c$ is not a dyadic rational, then the slice is totally disconnected and all positive-degree Čech homology groups vanish.

Topology of slices through the Sierpiński tetrahedron

Abstract

We investigate slices of the Sierpiński tetrahedron from a topological viewpoint. For each , we study the Čech (co)homology group of the slice at height . We show that the topology of the slice exhibits a sharp dichotomy. If is a dyadic rational, then the slice has finitely many connected components, infinite first Čech homology, and trivial higher homology. If is not a dyadic rational, then the slice is totally disconnected and all positive-degree Čech homology groups vanish.
Paper Structure (8 sections, 16 theorems, 39 equations)

This paper contains 8 sections, 16 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Non-autonomous iterated function systems and Čech-Sumi homology group
  5. Proof
  6. Preliminary lemmas
  7. Proofs of main results
  8. The general dimensional setting

Key Result

Corollary 1.3

For Lebesgue almost every $c\in [0, 1]$ with $(a_j(c))_{i=1}^{\infty}$, we have

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • proof
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • ...and 23 more