Topology of slices through the Sierpi\'nski tetrahedron

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.