Product decompositions of moment-angle manifolds and $B$-rigidity

TL;DR

This paper investigates the relationship between the algebraic structure of cohomology rings of moment-angle manifolds and the geometric decompositions of these manifolds, revealing new properties of B-rigid polytopes and their stability under products.

Contribution

It demonstrates that tensor product decompositions of cohomology rings correspond to geometric product decompositions of moment-angle manifolds, and shows B-rigid polytopes are closed under products.

Findings

Tensor product decompositions correspond to manifold decompositions.

B-rigid polytopes are closed under products.

Koszul homology decomposes via joins of simplicial complexes.

Abstract

A simple polytope $P$ is called $B$-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal{Z}_P$ over $P$. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that $B$-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley-Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.