Realizing orders in rational sphere product algebras with three generators
Tseleung So, Donald Stanley, Stephen Theriault, Ben Williams
TL;DR
The paper advances the cohomology realization problem for orders inside 3-generator sphere-product $\mathbb{Q}$-algebras by introducing weighted sphere-product rings $A(\mathfrak{c},\underline{d})$ and proving both topological realizations and algebraic classifications. It develops weighted polyhedral products and CW-power-coupled constructions to realize arbitrary $A(\mathfrak{c},\underline{d})$ when degrees exceed 1 (with no two even-degree generators sharing the same degree), and shows that every order in a 3-generator sphere-product $\mathbb{Q}$-algebra is isomorphic to some $A(\mathfrak{c},\underline{d})$. The key results are a Topological existence theorem (constructing CW-complexes realizing these algebras) and an Algebraic classification theorem (reducing lattice-ordered structures to coefficient sequences $\mathfrak{c}$). The findings generalize realizability in the simply-connected/Euler-odd setting and provide concrete, scalable methods for constructing spaces whose integral cohomology rings match prescribed lattice-orders. The work connects polyhedral-product topology with rational homotopy-theoretic realization problems, offering new tools for understanding how lattice-structured cohomology rings arise from topological spaces.
Abstract
The realization problem asks which algebras can be realized as the cohomology of spaces. We study this problem in the context of the orders in a graded rational exterior algebra on three generators. An order is a subring whose underlying additive group is a lattice. We give conditions for when such an order is realizable, and in particular show that in the simply-connected case any order is realizable if the generators of the exterior algebra are odd.