Connected sums of sphere products and minimally non-Golod complexes (Q6613453)

Toric topology is an emerging field in algebraic topology, which has been founded by Buchstaber, Panov, etc. As in most sub-areas of topology, one of the most important problems is to determine the homotopy types or combinatorial types under some certain conditions.\N\NGolod and minimally non-Golod are concepts from homological algebra. For a simplicial complex, these two concepts are determined by the Poincaré series of its Stanley-Reisner ring. More explicitly, a simplicial complex $K$ is Golod if and only if the cup product and all higher Massey products are trivial, a simplicial complex $K$ is minimally non-Golod if and only if $K$ is not Golod but deleting any vertex in $K$ results in a complex which is Golod.\N\NThe problem solved in this paper was raised in [\textit{J. Grbić} et al., Trans. Am. Math. Soc. 368, No. 9, 6663--6682 (2016; Zbl 1335.55007)]. The key technique in this paper is to analyse the inclusion map $j:Z_{K-{i}}\rightarrow Z_{K}$ and a retraction induced by the theory of stable splitting of polyhedral products. Note that in the paper \textit{I. Yu. Limonchenko} [Dal'nevost. Mat. Zh. 15, No. 2, 222--237 (2015; Zbl 1334.52015)], Proposition 2.8 proposes that such a result can be obtained under the sense of homotopy equivalence when $K$ is a triangulated sphere. \N\NThis paper gives a nice counterexample to the claim that the condition of homeomorphic is necessary when $K$ is not a triangulated sphere and generalizes it to real moment-angle complexes. The author also verifies that when a (real) moment angle complex $Z_{K}(RZ_{K})$ is homotopy equivalent to a connected sum of sphere products with two spheres in each product, then $K$ is a join of the standard simplex and a complex $L$, where $L$ is Gorenstein* and minimally non-Golod (in the real case Gorenstein* is not necessary). The solution of the main theorem can be also derived from this result.\N\NFor the entire collection see [Zbl 1540.57001].