Toral rank conjecture for moment-angle complexes

TL;DR

This paper proves the toral rank conjecture for moment-angle complexes by introducing a doubling operation on simplicial complexes, linking combinatorial and topological properties to estimate cohomology ranks.

Contribution

It introduces the doubling operation on simplicial complexes and applies it to prove the toral rank conjecture for general moment-angle complexes.

Findings

Proved the toral rank conjecture for Z_K using the doubling operation.

Established a lower bound for the cohomology rank of RZ_K.

Extended previous results from polytopes to general complexes.

Abstract

We consider an operation K \to L(K) on the set of simplicial complexes, which we call the "doubling operation". This combinatorial operation has been recently brought into toric topology by the work of Bahri, Bendersky, Cohen and Gitler on generalised moment-angle complexes (also known as K-powers). The crucial property of the doubling operation is that the moment-angle complex Z_K can be identified with the real moment-angle complex RZ_L(K) for the double L(K). As an application we prove the toral rank conjecture for Z_K by estimating the lower bound of the cohomology rank (with rational coefficients) of real moment-angle complexes RZ_K$. This paper extends the results of our previous work, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.