

International conference "GEOMETRY, TOPOLOGY, ALGEBRA and NUMBER THEORY, APPLICATIONS" dedicated to the 120th anniversary of Boris Delone (1890–1980)
August 16, 2010 10:00, Moscow






On the polyhedral product functor: a method of decomposition for momentangle complexes
Frederick Cohen^{} 
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Abstract:
Spaces which are now called (generalized) momentangle 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 quasitoric manifolds. They showed that every quasitoric manifold is the quotient of a momentangle complex by the free action of a real torus. The momentangle 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 GoreskyMacPherson, BuchstaberPanov, Panov, Baskakov, Hochster, and Franz. Among others who have worked extensively on generalized momentangle complexes are NotbohmRay, GrbicTheriault, Strickland and KamiyamaTsukuda. BuchstaberPanov synthesized several different developments in this subject. The direction of this lecture is guided by work of DenhamSuciu.
This lecture is a survey of recent work on generalized momentangle complexes as well as related spaces. One of the results given here is a natural decomposition for the suspension of the generalized momentangle complex, the value of the suspension of the "polyhedral product functor".
Since the decomposition is geometric, an analogous homological decomposition for a generalized momentangle 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 StanleyReisner ring of a finite simplicial complex extended to other natural settings. Applications to the real Ktheory of momentangle complexes as well as associated cupproduct 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.
Language: English

