On the Homotopy Decomposition for the Quotient of a Moment–Angle Complex and Its Applications

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 the formality of the corresponding Borel construction space under the natural assumption on the group action in the complex case, which leads to a new description of the equivariant cohomology for the quotients by any coordinate subgroups. We prove the weak toral rank conjecture for the partial quotient of a moment–angle complex by the diagonal circle action. We also give an explicit construction of partial quotients by circle actions to show that their integral cohomology may have arbitrary torsion.