Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices

TL;DR

This paper explores the relationship between free actions of subgroups on real moment-angle complexes and subtori actions on complex moment-angle complexes for certain piecewise linear spheres, using toric wedge induction and binary matroid structures.

Contribution

It introduces a novel application of toric wedge induction and binary matroid analysis to characterize free subgroup actions on moment-angle complexes for low-vertex spheres.

Findings

Subgroups of $bZ_2^m$ acting freely are induced by subtori of $T^m$.

The method combines toric wedge induction with combinatorial binary matroid analysis.

Results apply to $(n-1)$-dimensional spheres with at most $n+4$ vertices.

Abstract

Let $K$ be an $(n-1)$-dimensional piecewise linear sphere on $[m]$, where $m\leq n+4$. There are a canonical action of $m$-dimensional torus $T^m$ on the moment-angle complex $\mathcal{Z}_K$, and a canonical action of $\mathbb{Z}_2^m$ on the real moment-angle complex $\mathbb{R}\mathcal{Z}_K$, where $\mathbb{Z}_2$ is the additive group with two elements. We prove that any subgroup of $\mathbb{Z}_2^m$ acting freely on $\mathbb{R}\mathcal{Z}_K$ is induced by a subtorus of $T^m$ acting freely on $\mathcal{Z}_K$. The proof primarily utilizes a suitably modified method of toric wedge induction and the combinatorial structure of a specific binary matroid of rank $4$.