Homotopy types of moment-angle complexes associated to almost linear resolutions

TL;DR

This paper investigates the homotopy types of moment-angle complexes linked to almost linear resolutions, revealing their structure as wedges and connected sums of spheres, and establishing formality and loop space properties.

Contribution

It introduces a broader class of simplicial complexes related to Gorenstein ideals with almost linear resolutions, extending known homotopy and algebraic results.

Findings

Moment-angle complexes are wedges of spheres under certain linearity conditions.

Associated manifolds are formal and have rational homotopy types as connected sums of sphere products.

The loop space homotopy type involves products of spheres and their loop spaces.

Abstract

We show that the Hurewicz image in the homology of a moment-angle complex, when passed through an isomorphism with the Ext-module of the corresponding Stanley-Reisner ideal, contains the linear strand of this ideal. This recovers and refines results of various authors identifying the homotopy type of a moment-angle complex as a wedge of spheres when the underlying ideal satisfies certain linearity properties. Going further, we study the homotopy types of moment-angle manifolds associated to Gorenstein Stanley-Reisner ideals with (componentwise) almost linear resolutions. The simplicial complexes that give rise to these manifolds are part of an even larger class that we introduce, which generalises the homological behaviour of cyclic polytopes, stacked polytopes and odd-dimensional neighbourly sphere triangulations. For these simplicial complexes the associated moment-angle manifolds are shown to be formal, having the rational homotopy type of connected sums of sphere products, and the (integral) loop space homotopy type of products of spheres and loop spaces of spheres. Along the way we establish a number of purely algebraic results, in particular generalising a result of R\"omer characterising Koszul modules so that it can be applied to modules with almost linear resolutions.