Александровские чтения-2016 | 22-26 мая 2016
Участник: Ivan Limonchenko
Имя: Ivan Limonchenko
Название: On cohomology operations for polyhedral products
Город, страна: Moscow, Russia
Организация: Moscow State University
Starting with a simplicial complex K and a pair of spaces (X,A) V.M.Buchstaber and T.E.Panov constructed a topological space with a compact toruc action called a polyhedral product. It was motivated by a topologycal study of projective toric varieties due to M.Davis and T.Januszkiewicz and generalizes the notion of a moment-angle-manifold Z_P of a simple polytope P, introduced by them in 1991. In 1998 V.M.Buchstaber and T.E.Panov introduced a construction of a CW complex Z_K called a moment-angle-complex and proved it to be equivariantly homeomorphic to Z_P when K is a nerve complex of a convex simple polytope P. They also computed cohomology ring of Z_K in terms of the Tor-algebra of K, a well known notion in combinatorial commutative algebra, therefore linking toric topology with algebra. The topology of Z_K can be very complicated in general; e.g., recently the case when the face ring of K is a Golod ring was studied intensively, in that case Z_K is a rational wedge of spheres and thus a rationally formal space. In 2008 T.E.Panov and N.Ray proved that quasitoric manifolds of Davis and Januszkiewicz (and, therefore, all projective toric varieties) are formal. However, moment-angle-manifolds can be nonformal, having nontrivial higher Massey operations in cohomology. The first example was constructed in 2003 by I.V.Baskakov. In this talk we are going to introduce combinatorial conditions for graph-associahedra and generalized associahedra P under which there are nontrivial higher Massey operations in cohomology of Z_P. We also determine a family of n-dimensional flag nestohedra P starting with a simple 3-polytope dual to the Baskakov 2-sphere, which are not graph-associahedra, s.t. there is a nontrivial n-fold Massey product in cohomology of Z_P for any n starting with 3. The latter polytopes are 2-truncated cubes, a class of polytopes introduced and studied by V.M.Buchstaber and V.D.Volodin (any such a polytope can be realized as a sequence of codimension 2 face truncations starting with an n-cube).

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