Noncrossing partitions of a marked surface

TL;DR

This paper introduces noncrossing partitions for marked surfaces, establishing their lattice structure, topological rank, and isomorphisms, including cases with symmetry and double points.

Contribution

It defines noncrossing partitions on marked surfaces, proves their lattice properties, and explores their topological and combinatorial structures, including symmetric cases with double points.

Findings

The lattice of noncrossing partitions is graded and forms a topologically described lattice.

Lower intervals are isomorphic to products of noncrossing partition lattices of other surfaces.

Symmetry and double points influence the structure similarly to punctures.

Abstract

We define noncrossing partitions of a marked surface without punctures (interior marked points). We show that the natural partial order on noncrossing partitions is a graded lattice and describe its rank function topologically. Lower intervals in the lattice are isomorphic to products of noncrossing partition lattices of other surfaces. We similarly define noncrossing partitions of a symmetric marked surface with double points and prove some of the analogous results. The combination of symmetry and double points plays a role that one might have expected to be played by punctures.