Graded Rational Exterior Algebra

• Graded rational exterior algebra is a cohomology model defined by an exterior algebra on finitely many odd-degree generators and their ℤ-orders.
• It bridges algebra and topology by linking lattice theory with weighted polyhedral products to realize CW-complexes with prescribed integral cohomology.
• The explicit classification for the three-generator case enables a complete correspondence between algebraic orders and topological constructions.

A graded rational exterior algebra is a central object in the paper of cohomology rings of topological spaces, particularly those whose rational cohomology possesses the structure of an exterior algebra on finitely many generators. The classification and realization of integral orders in such algebras is both a classical algebraic question and a topologically significant one, closely linked to the realization problem in algebraic topology. The interplay of graded-commutative algebra, lattice theory, and geometric constructions such as weighted polyhedral products underpins recent advances, culminating in a complete solution for three generators with mild parity and degree constraints (So et al., 21 Nov 2025).

1. Construction of the Graded Rational Exterior Algebra

Let $S$ be a commutative ground ring. The free graded–commutative $S$-algebra on generators $x_1,\ldots,x_m$ of positive degrees $d_1,\ldots,d_m$ is denoted $F_S[x_1,\ldots,x_m]$, subject to the relations: $x_ix_j = -x_jx_i$ if both $|x_i|$ and $|x_j|$ are odd, and $x_i^2=0$ whenever $|x_i|$ is odd. Specifically, over $S = \mathbb{Q}$ with all $d_i$ odd, $F_\mathbb{Q}[x_1,\ldots,x_m]/(x_1^2,\ldots,x_m^2)$ yields the genuine exterior algebra $\Lambda_\mathbb{Q}(x_1,\ldots,x_m)$ with $|x_i| = d_i \in 2\mathbb{N}+1$.

The case of principal interest is $m=3$, where

$$R = \Lambda_\mathbb{Q}(x_1,x_2,x_3) = F_\mathbb{Q}[x_1,x_2,x_3] / (x_1^2,x_2^2,x_3^2),$$

with each $|x_i| = d_i > 0$ odd. This algebra provides a rational model for the cohomology ring of a product of odd-dimensional spheres.

2. Orders in $\Lambda_\mathbb{Q}(x_1,x_2,x_3)$

Given a finite-dimensional graded $\mathbb{Q}$-algebra $R$, an order $A\subset R$ (with $S = \mathbb{Z}$) is a graded subring that is a lattice in $R$: $A$ is a finitely generated free abelian group with $\mathbb{Q}$-span equal to $R$ (i.e., $\mathbb{Q}\otimes_{\mathbb{Z}} A \cong R$). In the case of $R = \Lambda_\mathbb{Q}(x_1,x_2,x_3)$, an order is a torsion-free graded ring that, upon rationalization, recovers the exterior algebra. Equivalently, $A$ admits a homogeneous $\mathbb{Q}$-basis for $R$ that also generates $A$ as a free $\mathbb{Z}$-module.

3. The Realization Problem for Orders

The realization problem asks: Given a graded algebra $A$, does there exist a CW-complex $X$ whose integral cohomology ring $H^*(X;\mathbb{Z})$ is isomorphic to $A$ as a graded $\mathbb{Z}$-algebra? When $$H^*(X;\mathbb{Q})\cong \Lambda_\mathbb{Q}(x_1,x_2,x_3)$$, $H^*(X;\mathbb{Z})$ is necessarily an order in the exterior algebra. The central question is then: For which orders $A\subset \Lambda_\mathbb{Q}(x_1,x_2,x_3)$ does there exist a CW-complex $X$ with $H^*(X;\mathbb{Z}) \cong A$?

4. Algebraic Classification and Topological Realization

The case $m=3$, with each $d_i>1$ and repeated degrees assumed odd, is entirely resolved by two principal theorems:

A. Algebraic Classification (Theorem 4.6):

Every order $A\subset\Lambda_\mathbb{Q}(x_1,x_2,x_3)$ is isomorphic to a weighted exterior order $A(\mathbf{c},\mathbf{d})$, determined by a coefficient sequence $\mathbf{c} = (c_\sigma)$ indexed by nonempty subsets $\sigma\subset\{1,2,3\}$, with each $c_\sigma\in\mathbb{Z}_{>0}$ subject to divisibility conditions (see Definition 2.4 in (So et al., 21 Nov 2025)). Concretely,

$$A = \mathbb{Z} \oplus L(1) \oplus L(2) \oplus L(3),$$

where each $L(k)$ is a degree-graded subgroup, and the multiplication is diagonalized via appropriately chosen bases. The structural constants $c_{ij}, c_{123}$ govern products $b_i b_j = c_{ij} b_{ij}$ and generation of the top degree component.

B. Topological Realization (Theorem 3.1):

Given any valid coefficient sequence and degree list $d_1,d_2,d_3>1$, there exists an explicit CW-complex $X(\mathbf{c},\mathbf{d})$ whose integral cohomology ring is $A(\mathbf{c},\mathbf{d})$. The construction utilizes weighted polyhedral products and a single attaching map in the top dimension, modifying the attaching map of a standard Whitehead product to control torsion and realize the prescribed order.

Combined, these results yield: For $m=3$, $d_i>1$ (with repeated $d_i$ odd), every $\mathbb{Z}$-order in $\Lambda_\mathbb{Q}(x_1,x_2,x_3)$ arises as $H^*(X;\mathbb{Z})$ for some CW-complex $X$ (So et al., 21 Nov 2025).

5. Proof Strategy and Methodology

The algebraic-to-geometric correspondence leverages:

• Weighted Sphere Product Algebras: Ordinary exterior algebras are realized by products of spheres; introducing integer weights $c_\sigma$ yields weighted polyhedral products, which encode the multiplication structure of the order.
• Cellular and Homological Analysis: The construction is carried out by inductively building CW-complexes, attaching cells corresponding to the faces of $\partial\Delta^2$ with self-maps that multiply degrees by $c_\sigma$. For arbitrary coefficient sequences, a modification in the top attaching map cancels surplus torsion and ensures the final cohomology is precisely $A(\mathbf{c},\mathbf{d})$.
• Smith Normal Form and Diagonalization: On the algebraic side, orders are brought into weighted form by diagonalizing the multiplication structure within the successive powers of the augmentation ideal, using techniques reminiscent of the Smith normal form.

The hypothesis on the parity of repeated degrees secures vanishing squares and the surjectivity of alternating-square maps, both algebraically and topologically essential.

6. Examples, Non-Examples, and Implications

Every simply-connected order in $\Lambda_\mathbb{Q}(x_1,x_2,x_3)$ with all $d_i$ odd is a weighted exterior order and thus realizable. For instance, any lattice of the prescribed form determined by a valid coefficient sequence can be constructed as the cohomology ring of a corresponding CW-complex.

A non-allowable order arises if two generators share an even degree (e.g., both $x_2$ and $y_2$ in degree 2): one can build an order with a rank-2 $\mathbb{Z}$-summand in degree 2 and $\mathbb{Z}\oplus\mathbb{Z}/2$ in degree 5, whose multiplication structure cannot be diagonalized as a weighted exterior order. The realizability of such objects remains open (So et al., 21 Nov 2025).

The comprehensive algebraic classification and explicit topological construction settle the realization problem for torsion-free rings rationally modeled on a three-generator exterior algebra, subject to elementary degree and parity requirements.

7. Significance and Further Directions

The complete resolution of the realization problem for graded rational exterior algebras with three generators, as embodied in (So et al., 21 Nov 2025), establishes a precise correspondence between algebraic data (orders, coefficient sequences) and geometric constructions (CW-complexes, weighted polyhedral products) in rational homotopy theory. The methodology underscores the efficacy of explicit cell attachments and weighted constructions in controlling the ring structure of cohomology.

A plausible implication is that analogous treatment may be viable for higher numbers of generators, conditional on further structural insights. Certain cases, such as orders involving even-degree generators or more intricate torsion phenomena, remain avenues for ongoing inquiry.