Topology of complements of skeletons

TL;DR

This paper investigates the topology of complements of skeletons in polytopal complexes, establishing exact sequences and characterizations for various complex types, and explores applications to CAT(0) cubical complexes and crossing complexes.

Contribution

It introduces a long exact sequence relating homologies of skeleton complements and links, providing new characterizations of complex classes and linking CAT(0) cubical complexes with crossing complexes.

Findings

Established a long exact sequence for homologies of skeleton complements.

Characterized Cohen-Macaulay and Leray complexes via skeleton complements.

Identified new similarities between CAT(0) cubical complexes and crossing complexes.

Abstract

Given a polytopal complex $X$, we examine the topological complement of its $k$-skeleton. We construct a long exact sequence relating the homologies of the skeleton complements and links of faces in $X$, and using this long exact sequence, we obtain characterisations of Cohen-Macaulay and Leray complexes, stacked balls, and neighbourly spheres in terms of their skeleton complements. We also apply these results to CAT(0) cubical complexes, and find new similarities between such a complex and an associated simplicial complex, the crossing complex.