On the homotopy decomposition for the quotient of a moment-angle complex and its applications

TL;DR

This paper establishes a homotopy decomposition for quotients of moment-angle complexes, generalizing known constructions, and applies it to compute equivariant cohomology and verify the weak Toral Rank Conjecture.

Contribution

It introduces a homotopy decomposition for quotients of moment-angle complexes and extends Davis-Januszkiewicz constructions to these quotients.

Findings

Homotopy equivalence of quotients to homotopy colimits of toric diagrams

Generalization of Davis-Januszkiewicz construction to quotients

Proof of the weak Toral Rank Conjecture for partial quotients

Abstract

In this paper we prove that the quotient of any real or complex moment-angle complex by any closed subgroup in the naturally acting compact torus on it is equivariantly homotopy equivalent to the homotopy colimit of a certain toric diagram. For any quotient we prove an equivariant homeomorphism generalizing the well-known Davis-Januszkiewicz construction for quasitoric manifolds and small covers. We deduce formality of the corresponding Borel construction space under the natural assumption on the group action in the complex case leading to the new description of the equivariant cohomology for the quotients by any coordinate subgroups. We prove the weak Toral Rank Conjecture for any partial quotient by the diagonal circle action. We give an explicit construction of partial quotients by circle actions having arbitrary torsion in integral cohomology.