Higher order Massey products and applications

In this survey, we discuss two research areas related to Massey’s higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley–Reisner rings of simplicial complexes.

Toric objects associated with the dodecahedron

In this paper we illustrate a tight interplay between homotopy theory and combinatorics within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related small cover. We finish by studying Massey products in the cohomology ring of moment-angle manifolds over the dodecahedron and how the existence of nontrivial Massey products influences the behaviour of the Poincar? series of the corresponding Pontryagin algebra.

Wedge Operations and Torus Symmetries II

A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex K(J) obtainable by a sequence of wedgings from K.The main idea was that characteristic maps on K theoretically determine all possible characteristic maps on a wedge of K.We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere K of dimension n-1 with m vertices, the Picard number Pic(K) of K is m-n. We call K a seed if K cannot be obtained by wedgings. First, we show that for a fixed positive integer 𝓁, there are at most finitely many seeds of Picard 𝓁 number supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.Secondly, we investigate a systematicmethod to find all characteristic maps on K(J) using combinatorial objects called (realizable) puzzles that only depend on a seed K. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.