Moment angle complexes and duality for tight manifolds

TL;DR

This paper investigates the homology of Moment-Angle complexes associated with triangulated manifolds, establishing inequalities and duality theorems that characterize tight manifold triangulations.

Contribution

It introduces a new duality theorem in Double Homology and characterizes tight triangulations via homology inequalities.

Findings

Homology rank satisfies a specific inequality for tight manifolds.

Equality in the inequality characterizes F-tightly triangulated manifolds.

A new duality theorem in Double Homology is established for tight manifolds.

Abstract

For a field $\mathbb{F}$ and a triangulated compact $\mathbb{F}$-orientable manifold, consider the homology of the associated Moment-Angle ccomplex $H_*(\mathcal{Z}_{\mathcal{K}})$. We show the total homology rank $\beta(\mathcal{Z}_{\mathcal{K}})$ satisfies the inequality $\beta(\mathcal{Z}_{\mathcal{K}};\mathbb{F})\geq 2^{m-1}(\beta(\mathcal{K};\mathbb{F})-2)+2$, with equality occurring exactly when the triangulation is $\mathbb{F}$-tight. Using Lefschetz duality, we introduce a short exact sequence of functors that, in turn, introduces a new duality theorem in Double Homology for tight manifold triangulations.