Double cohomology of moment-angle complexes

TL;DR

This paper introduces a new double cohomology theory for moment-angle complexes, providing multiple definitions and explicit computations for certain classes of simplicial complexes, advancing understanding in algebraic topology.

Contribution

It defines the double cohomology for moment-angle complexes, offers three equivalent differential definitions, and computes it for specific simplicial complexes including clique complexes of chordal graphs.

Findings

Defined double cohomology for moment-angle complexes.

Provided three equivalent differential definitions.

Computed double cohomology for clique complexes of chordal graphs.

Abstract

We put a cochain complex structure ${CH}^*(\mathcal Z_K)$ on the cohomology of a moment-angle complex $\mathcal Z_K$ and call the resulting cohomology the double cohomology, ${HH}^*(\mathcal Z_K)$. We give three equivalent definitions for the differential, and compute ${HH}^*(\mathcal Z_K)$ for a family of simplicial complexes containing clique complexes of chordal graphs.