Fat-wedge filtration and decomposition of polyhedral products

A Golod complex with non-suspension moment-angle complex

“…Theorem 1.3 provides a counter example to this expectation. In fact, if Z K is homotopy equivalent to a suspension then the fat wedge filtration of Z K is trivial, by Theorem 1.3 of [15]. By Theorem 6.9 of the same paper, we see that K is stably homotopy Golod, which contradicts our result.…”

“…By strengthening this condition, K is said to be (stably) homotopy Golod [15] if the maps |ι I,J | : |K I∪J | → |K I * K J | for I ∩ J = ∅ are (stably) null homotopic and H * (Z K ; k) has trivial higher Massey products for any fields k. By definition, the following implication holds: stably homotopy Golod =⇒ Golod.…”

“…In general, Z K can have torsion in its integral homology groups even if K is Golod. The 6 vertex triangulation of RP 2 is such an example, see for example [9] or Example 10.10 of [15].…”

“…They demonstrated that the momentangle complex associated with a shifted simplicial complex is homotopy equivalent to a wedge of spheres. In [15], Kishimoto and the first author extended this result to dual sequentially Cohen-Macaulay complexes, and provided some new Golod complexes. In these studies, the following theorem concerning the decomposition of polyhedral products (see Definition 2.1), as introduced by Bahri, Bendersky, Cohen, and Gitler [2], plays an essential role.…”

A non-Golod ring with a trivial product on its Koszul homology

“…[BP02,Theorem 7.7]. A lot of research has been devoted to study the relation between the structure of H * (K R ) and the topology of Z ∆ , see for example [DS07;IK14;IK15;GPTW16]. In terms of moment-angle complexes, our example gives rise to a moment-angle complex which is not formal but has a trivial cup-product in its cohomology.…”