Spectral sequences of a Morse shelling

TL;DR

This paper explores the relationship between tilings of simplicial complexes, shellability, and spectral sequences, revealing how shellable tilings support acyclic quivers and induce spectral sequences converging to the complex's (co)homology.

Contribution

It introduces a novel connection between tilings, shellability, and spectral sequences, extending discrete Morse theory to new combinatorial structures.

Findings

Acyclic quivers are supported by shellable tilings.

Shellings induce two spectral sequences converging to (co)homology.

Spectral sequences' first pages are free modules over critical tiles.

Abstract

We recently introduced a notion of tilings of geometric realizations of finite relative simplicial complexes and related those tilings to the discrete Morse theory of R. Forman, especially when they have the property of being shellable, a property shared by the classical shellable complexes. We now observe that every such tiling supports a quiver which is acyclic precisely when the tiling is shellable and then, that every shelling induces two spectral sequences which converge to the relative (co)homology of the complex. Their first pages are free modules over the critical tiles of the tiling.