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The University of Southampton
Mathematical Sciences

Research project: Symmetries of polyhedral products

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A polyhedral product is a homotopy theoretical generalisation of moment-angle complexes, one of the main objects in toric topology. As functorial construction polyhedral products are determined a finite simplicial complex and a sequence of topological pairs. Thus symmetries of polyhedral products arise in two different ways: symmetries of polyhedral products induced by symmetries of simplicial complexes; group actions on polyhedral products induced by group actions on topological spaces.

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.

Related research groups

Pure Mathematics
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