Loop space decompositions of moment-angle complexes associated to flag complexes

TL;DR

This paper studies the loop space structure of moment-angle complexes derived from flag complexes, showing they decompose into products of spheres and loops, with a general result on the class of such spaces.

Contribution

It establishes that the loop space of these complexes belongs to a class of spaces homotopy equivalent to products of spheres and loops, and proves this class is closed under retracts.

Findings

Loop spaces of moment-angle complexes decompose into products of spheres and loops.

The class of spaces homotopy equivalent to such products is closed under retracts.

Provides new tools for analyzing the topology of moment-angle complexes.

Abstract

We prove that the loop space of the moment-angle complex associated to the $k$-skeleton of a flag complex belongs to the class $\mathcal{P}$ of spaces homotopy equivalent to a finite type product of spheres and loops on simply connected spheres. To do this, a general result showing $\mathcal{P}$ is closed under retracts is proved.