The Refined Unstable-Module Realization Conjecture

Definitions and mathematical statement

Let $\mathcal U$ be the category of unstable modules over the mod-$2$ Steenrod algebra: for a homogeneous element $x$, the operations satisfy $\operatorname{Sq}^i x=0$ when $i>\deg x$, as well as the Steenrod relations. Let $\mathcal U_0$ be its locally finite subcategory, meaning that each element generates a finite-dimensional Steenrod submodule. Let $X$ be a space and set $M=H^*(X;{\mathbb{F}}_2)$, with $M\notin\mathcal U_0$.

The supplied refined realization conjecture asserts both the unbounded strong realization conjecture for this cohomology module and the inequality \[ S(M)\leq N(M). \] The source describes $S(M)$ as the first nilpotence-filtration level whose quotient belongs to none of the subcategories $\mathcal U_n$, and $N(M)$ as the report's invariant determined by Steenrod operations. It supplies neither the filtration and precise quotient convention, the subcategories $\mathcal U_n$ for $n>0$, the formula for $N$, nor the content of the named unbounded strong realization conjecture. The existence and value convention for a first such level also need to be specified. Thus the displayed inequality and non-locally-finite hypothesis are preserved exactly, but cannot yet constitute a fully defined independent conjecture; no substitute invariant or realization statement is invented.

Short English statement

For mod-$2$ cohomology that is not locally finite as an unstable Steenrod module, the source asks for its named strong realization assertion and the bound $S\leq N$. The report-specific invariants and the named assertion are not defined in the extraction.

Sources

• Local UnsolvedMath registry, permanent ID $30001050$, source code catalogue entry, "The Refined Unstable-Module Realization Conjecture." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Topologie (2009), Oberwolfach Reports, pp. 2419--2476, DOI https://doi.org/10.4171/owr/2008/43; supplied literature label "Gaudens report."
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled partially solved; says the source report proves the refined realization conjecture only for $N=0,1,2$, and the general assertion remains unresolved. This is supplied dated curation, not independently verified, and the exact theorem hypotheses require review.