Research project: Symmetries of polyhedral products

In the work on symmetries of polyhedral products induced by symmetries of simplicial complexes, we observe that the unstable splitting of a polyhedral product is equivariant which set the ground for the study of representation stability in toric topology.

Over the past decades, there has been a growing interest in torus actions on moment-angle complexes, a special example of polyhedral products, and their related quotient spaces. Our recent work employs the Taylor resolution to the calculation of the cohomology of partial quotients of moment-angle complexes and builds connections with weighted simplicial homology. This method recovers important calculations of cohomology rings of quotient manifolds including quasi-toric manifolds and quotient of product of spheres under free circle actions and sets a way for cohomology calculations of more general toric manifolds.