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COMMON PART

Project Number22-71-00106

Project titlePolyhedral products in topology, geometry and combinatorics

Project LeadLimonchenko Ivan

AffiliationNational Research University Higher School of Economics,

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-105 - Topology

Keywordspolyhedral product, moment-angle complex, graph product, toric variety, quasitoric manifold, simple polytope, higher Massey product, Kan-Thurston functor, Dror functor, Milgram construction, coformal space

PROJECT CONTENT

Annotation
The project is focused on studying topological spaces with rich symmetry arising from the compact torus actions, geometric and combinatorial structures on the sets of orbits. The research is carried out in the framework of toric topology, a rapidly developing field of mathematics, which lies at the crossroads of topology, geometry, combinatorics, polytope theory, combinatorial commutative algebra, algebraic and symplectic geometry and is characterized by deep interconnections of the ideas and methods arising in these areas. In toric topology we study compact torus actions on smooth manifolds and cellular spaces, whose orbit spaces carry rich combinatorial structures. The subject was established in the late 1990s as a study of topological counterparts of algebraic toric varieties by means of algebraic topology. The central objects of study are topological analogues of nonsingular projective toric varieties - quasitoric manifolds - and their generalizations (torus manifolds), as well as moment-angle complexes and manifolds, and their generalizations (polyhedral products). Moment-angle complexes are cellular spaces formed by gluing together products of 2-dimensional discs and circles according to a recipe determined by an underlying simplicial complex. In the case when the simplicial complex is a polytopal sphere, its moment-angle-complex is a smooth manifold. The notions of a moment-angle complex and a moment-angle manifold unify several constructions from various areas of mathematics, including: the intersection of special real and Hermitian quadrics studied in topology and holomorphic dynamics, level sets for moment maps in the construction of Hamiltonian toric manifolds via symplectic reduction, and complements of coordinate subspace arrangements in a real or complex Euclidean space. A polyhedral product is a functorial generalization of a moment-angle complex in which the 2-dimensional discs and circles are replaced by arbitrary cellular topological pairs. The construction of a polyhedral product leads us to the new applications of toric topology, including: the Whitehead filtration in homotopy theory, the study of asphericity in group cohomology, the study of right-angled Artin groups, right-angled Coxeter groups and graph products in geometric group theory, multigraded versions of persistent homology and bar-codes in topological data analysis, and the study of robotics and arachnid mechanisms in topological robotics. Toric topology has been developing rapidly by the moment and it attracts a number of researchers from all over the world. In January-June 2020 at the Fields Institute for Research in Mathematical Sciences (Toronto, Canada) the thematic program "Toric Topology and Polyhedral Products": http://fields.utoronto.ca/activities/19-20/toric took place. The head of this project was an active participant of this event as a Fields-Ontario postdoctoral fellow at the Fields Institute and the University of Toronto. He was also one of the organizers of a postdoc seminar for young researchers at the Fields Institute: http://www.fields.utoronto.ca/activities/19-20/postdoc-seminar. In 2022 a special volume of the Fields Institute Communications journal is going to be published; its title will be "Toric Topology and Polyhedral Products". The focus program "Toric Topology, Geometry and Polyhedral Products" at the Fields Institute is already planned for the continuation of the thematic semester in 2024: http://www.fields.utoronto.ca/activities/24-25/toric Furthermore, since 2020 an international seminar "International Polyhedral Products Seminar" has been working at Princeton University (Princeton, USA) and in 2022 the head of this project and the members of the research group are planning to take part in the international research project Heilbronn Focused Research Group based at the University of Southampton (Southampton, UK). Thus, we plan to closely coordinate the research on the project with the above mentioned research activities at the Fields Institute, the University of Southampton, and Princeton University. This will contribute to the implementation of the project results, obtaining new applications and deepening scientific cooperation with our colleagues in Russia and abroad.

Expected results
We plan to study homotopy invariants of polyhedral products (X,A)^K, including those of moment-angle complexes Z_K=(D^2,S^1)^K, by means of toric topology and combinatorial commutative algebra. We are going to develop a theory of a new combinatorial invariant of a simplicial complex - brunnian subsets in the cohomology rings of Z_K (or equivalently, in Koszul homology of the Stanley-Reisner ring of the simplicial complex K). That is, such a set of cohomology classes that it carries a non-trivial higher Massey product, whereas any its proper subset is not brunnian itself. It is planned to study the Milnor-Moore spectral sequence for moment-angle complexes: to describe some of its elements and study the problem of non-trivial differentials. A sufficient, though not a necessary, condition for this spectral sequence to collapse is the coformality of the moment-angle complex. We are going to construct examples of non-coformal moment-angle complexes and describe the families of simplicial complexes for which their moment-angle complexes are coformal. We will construct a generalization of a moment-angle complex to the case of an infinite simplicial complex K as a colimit of moment-angle complexes corresponding to a certain filtration of the simplicial complex K. We plan to describe cohomology rings of those infinite moment-angle complexes and, based on this description, to introduce and study a new invariant of a discrete group G: the cohomology of an infinite moment-angle complex over its classifying space BG, viewed as an infinite simplicial complex with the help of the functorial Milgram construction. It is planned to study the presentations of Cartesian subgroups in graph products of discrete groups. Minimal sets of generators for those groups were described in the work of Panov and Verjovkin (2016). A particularly important case is given by commutator subgroups of the right-angled Coxeter groups RC_K, which are known to be fundamental groups of real moment-angle complexes (D^1,S^0)^K. In that case, Li Cai has recently proposed an alternative set of generators and a new approach to describing the relations between them. We plan to formalize and generalize this method to Cartesian subgroups in graph products. We will obtain upper and lower bounds for the number of relations in the minimal presentations. We are also going to describe and realize an algorithm for computing those relations as well as to compare the generators obtained by Panov-Veryovkin and Cai. We plan to generalize the construction of a polyhedral product in toric topology by applying the Kan-Thurston and Dror functors to it and, in particular, to moment-angle complexes. For the generalization obtained, in which some spaces are replaced by the homologically equivalent ones, we are going to study the homotopy invariants by means of toric topology and combinatorial commutative algebra.

REPORTS