Topology of slices through the Sierpi\'nski tetrahedron

TL;DR

This paper explores the topological properties of slices through the Sierpiński tetrahedron, revealing a dichotomy based on whether the slice height is a dyadic rational or not, affecting connectivity and homology.

Contribution

It provides a detailed topological analysis of the slices, establishing a clear dichotomy in their homological properties depending on the height's rationality.

Findings

Dyadic rational slices have finitely many connected components and infinite first homology.

Non-dyadic rational slices are totally disconnected with trivial homology.

The topology of slices varies sharply with the height parameter.

Abstract

We investigate slices of the Sierpi\'nski tetrahedron from a topological viewpoint. For each $c\in[0,1]$, we study the \v{C}ech (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 \v{C}ech homology, and trivial higher homology. If $c$ is not a dyadic rational, then the slice is totally disconnected and all positive-degree \v{C}ech homology groups vanish.