Massey Products in the Cohomology of the Moment-Angle Manifolds Corresponding to Pogorelov Polytopes

Abstract

In this work we construct nontrivial Massey products in the cohomology of moment-angle manifolds corresponding to polytopes from the Pogorelov class. This class includes the dodecahedron and all fullerenes, i. e. simple 3-polytopes with only 5-gonal and 6-gonal facets. The existence of a nontrivial Massey product implies the nonformality of the space in the sense of rational homotopy theory.

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“…The Pogorelov class is large and includes all fullerenes, whose facets are pentagons and hexagons. Zhuravleva [26,Theorem 3.2] showed that for any Pogorelov polytope P , K P = ∂(P * ) has a full subcomplex K as shown in Figure 15a. This full subcomplex was used to explicitly construct a nontrivial Massey product α 1 , α 2 , α 3 ⊂ H * (Z P ) where α 1 ∈ H 0 (K 567 ), α 2 ∈ H 0 (K 2b0...bn ) and α 3 ∈ H 0 (K 34 ).…”

“…For example, from each of the obstruction graphs in the classification of lowest-degree triple Massey products in moment-angle complexes [12,15], we obtain infinite families of non-trivial triple Massey products of higher dimensional classes. We give an alternative proof of known examples of non-trivial triple Massey products in moment-angle manifolds, such as those associated with Pogorelov polytopes [26] and permutahedra or stellohedra [19,20] using "stretched" obstruction graphs. Also, the two constructions, Constructions 3.5 and 4.6, can be combined to create new higher Massey products.…”

Toric objects associated with the dodecahedron

“…There is a short exact sequence 1 → Ker r * → π 1 (R (1) For more on the combinatorics of D and P 4 , and the hyperbolic structures on small covers and real moment-angle manifolds over dodecahedron and 120-cell, see [6,14,37]. (2) Zhuravleva [59] constructed a nontrivial triple Massey product in H * (Z K ) when K = K P is a nerve complex of an arbitrary Pogorelov polytope P. The latter class includes all fullerenes and, in particular, the dodecahedron. Now we turn to a discussion of Poincaré series of face rings and related topics.…”