The Golod characterization of co-H moment-angle complexes

Let $K$ be a simplicial complex. The associated moment-angle complex is $(D^2,S^1)^K$ , where $(D^2,S^1)$ is the standard polyhedral product pair. A simplicial complex $K$ is Golod if all products and higher Massey products vanish in the appropriate Tor-algebra of its Stanley–Reisner ring.

Golod characterization. The moment-angle complex $(D^2,S^1)^K$ is a co- $H$ -space if and only if $K$ is Golod.

This conjecture proposes a topological characterization of Golod complexes through the homotopy type of their moment-angle complexes. Considerable work on the homotopy theory of moment-angle complexes over Golod complexes motivates the statement, but its resolution is not specified here.

The conjecture asks whether $(D^2,S^1)^K$ is a co- $H$ -space exactly when the simplicial complex $K$ is Golod. The retrieved literature records substantial partial results, but no resolution for arbitrary $K$ .

Known results

For flag complexes, Beben and Grbić proved the equivalence between Golodness, the co- $H$ -space condition, and trivial multiplication in positive Tor.
For several restricted classes, including triangulated $d$ -spheres in specified dimensions, results relate $m$ -Golodness to $\operatorname{cat}(Z_K)\le m$ .
$Z_K$ homotopy equivalent to a suspension implies that $K$ is Golod; the converse fails, but this does not settle the weaker co- $H$ question.
Wedge-of-spheres and connected-sum decompositions are known in further special cases, without proving the general characterization.

Current status (as of September 2026): The characterization is established for flag complexes and some restricted families, while the equivalence for arbitrary simplicial complexes remains open; no claimed general proof or counterexample was retrieved.

The Golod characterization of co- $H$ moment-angle complexes

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