On the polyhedral product functor: a method of decomposition for moment-angle complexes

Spaces which are now called (generalized) moment-angle complexes or values of the “polyhedral product functor” have been studied by topologists since the 1960's thesis of G. Porter. In the 1970's E. B. Vinberg developed some of their features. In the late 1980's S. Lopez de Medrano developed beautiful properties of intersections of quadrics with recent further developments in joint work with S. Gitler.
In seminal work during the early 1990's, M. Davis and T. Januszkiewicz introduced manifolds now often called quasi-toric manifolds. They showed that every quasi-toric manifold is the quotient of a moment-angle complex by the free action of a real torus. The moment-angle complex is denoted $Z(K;(D^2,S^1))$ where $K$ is a finite simplicial complex.
The integral cohomology of the spaces $Z(K;(D^2,S^1))$ has been studied by Goresky-MacPherson, Buchstaber-Panov, Panov, Baskakov, Hochster, and Franz. Among others who have worked extensively on generalized moment-angle complexes are Notbohm-Ray, Grbic-Theriault, Strickland and Kamiyama-Tsukuda. Buchstaber-Panov synthesized several different developments in this subject. The direction of this lecture is guided by work of Denham-Suciu.
This lecture is a survey of recent work on generalized moment-angle complexes as well as related spaces. One of the results given here is a natural decomposition for the suspension of the generalized moment-angle complex, the value of the suspension of the "polyhedral product functor".
Since the decomposition is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. This last decomposition specializes to the homological decompositions in the work of several authors cited above. Furthermore, this decomposition gives an additive decomposition for the Stanley-Reisner ring of a finite simplicial complex extended to other natural settings. Applications to the real K-theory of moment-angle complexes as well as associated cup-product structures are given. Applications to robotic motion are illustrated via video clips.
This lecture is based on joint work with A. Bahri, M. Bendersky, and S. Gitler. The application to robotics is based on joint work with D. Koditschek and G. C. Lynch.