SB-labelings and posets with each interval homotopy equivalent to a sphere or a ball

TL;DR

This paper introduces SB-labelings for finite lattices, showing that such lattices have intervals homotopy equivalent to spheres or balls, with applications to well-known lattice structures.

Contribution

The paper defines SB-labelings and proves that finite lattices with these labelings have intervals homotopy equivalent to spheres or balls, expanding understanding of lattice topology.

Findings

Finite lattices with SB-labelings have intervals homotopy equivalent to spheres or balls.

Examples include weak order, Tamari lattice, and finite distributive lattices.

SB-labelings provide a new framework for analyzing lattice topologies.

Abstract

We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.