Abstract
As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing non-trivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in moment-angle complexes. As a result, we obtain explicit constructions of infinitely many non-formal manifolds that appear in topology, complex geometry and algebraic geometry.
Highlights
A moment-angle complex ZK over a simplicial complex K on m vertices is built from ordered products of discs and circles in Cm that are glued together along the face category of K
In the context of commutative algebra, supposing k is a field or Z, the cohomology algebra of ZK is isomorphic to the Tor-algebra Tork[m](k[K], k) of the face ring k[K], due to [10] and [4, Theorem 1]
From the perspective of complex geometry, by identifying ZK with the complement U (K) of a coordinate subspace arrangement corresponding to K, moment-angle complexes are LVM manifolds [6, 23] when K is the boundary of the dual of a simple polytope
Summary
A moment-angle complex ZK over a simplicial complex K on m vertices is built from ordered products of discs and circles in Cm that are glued together along the face category of K. It is important to emphasise that we do not impose any restrictions on n-arity of these Massey products, on the choice of simplicial complexes Ki for any i, nor on the dimension of classes in the Massey product This construction generalises Baskakov’s [5] family of non-trivial triple Massey products in the cohomology of moment-angle complexes, taking triangulations of spheres for K1, K2 and K3. Our framework constructs infinitely many families of such examples, confirming that non-trivial higher Massey products are much more common in moment-angle complexes and moment-angle manifolds than previously thought. These techniques do not just apply to moment-angle complexes. Using this decomposition and our constructions, it is possible to produce non-trivial Massey products in (CA, A)K by incorporating cohomological classes of A to the classes we construct in the cohomology of full subcomplexes of K in order to create Massey products in ZK
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