Realizing orders in rational sphere product algebras with three generators

TL;DR

This paper investigates which algebraic structures called orders in rational exterior algebras with three generators can be realized as cohomology rings of topological spaces, providing specific conditions for realizability.

Contribution

It establishes conditions for the realizability of orders in graded rational exterior algebras with three generators, especially in the simply-connected case with odd generators.

Findings

Any order in the simply-connected case with odd generators is realizable.

Provides necessary and sufficient conditions for realizability of orders.

Advances understanding of the realization problem in algebraic topology.

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.