Realization of Oriented Dimension Functions by Equivariant Spheres

Definitions and mathematical statement

Let $G$ be a finite group, let $p$ be a prime, and let $k$ be a field of characteristic $p$. The $p$-orbit category $O_p(G)$ has objects $G/H$, where $H\leq G$ is a $p$-subgroup, and morphisms the $G$-equivariant maps. Its $k$-linearization allows the study of contravariant orbit-category modules, equivalently functors from $O_p(G)^{\mathrm{op}}$ to $k$-vector spaces.

The source asks whether the following equivalence holds for every oriented dimension function $f$ in the sense of the report: \[ \exists\text{ a }kO_p(G)^{\mathrm{op}}\text{-sphere }C \text{ realizing }f \quad\Longleftrightarrow\quad f\text{ satisfies the Borel--Smith conditions}. \] A dimension function records the sphere dimension at each subgroup object and is invariant under conjugation. In the usual homological interpretation, evaluation at $G/H$ has reduced homology concentrated in the degree prescribed by $f(H)$. An orientation requirement typically controls the action of the Weyl group $N_G(H)/H$ on the top homology, but the exact definition of "oriented dimension function" in this problem is not supplied and is not replaced here by a guessed parity or action condition.

The extraction also omits the precise meaning of an orbit-category sphere: whether it is a finite free or projective chain complex, what additional realization axioms it satisfies, the permitted dimension range and degree shift, and the full list of Borel--Smith relations. These definitions are necessary to turn the displayed equivalence into a self-contained theorem statement. In particular, realization by an algebraic orbit-category sphere must not be silently replaced by realization by a genuine sphere with a $G$-action, a representation sphere, or a spherical fibration. The source asks an exact realization equivalence; it does not state merely realization after taking joins or multiplying the dimension function.

Short English statement

Are the Borel--Smith conditions exactly the criterion for realizing every oriented dimension function by a sphere over the $p$-orbit category? The precise algebraic sphere and orientation definitions must be recovered from the report.

Sources

• Local UnsolvedMath registry, permanent ID $30000846$, source code catalogue entry, "Realization of Oriented Dimension Functions by Equivariant Spheres." No individual proposer is supplied.
• Homotopy Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/45, pp. 2671--2726: https://doi.org/10.4171/owr/2007/45.
• Supplied reference, "Okay--Yalcin, Dimension functions for spherical fibrations": https://arxiv.org/abs/1710.07315.
• Supplied literature triage, checked 2026-08-21: status "partially solved." It reports necessity of Borel--Smith conditions, sufficiency for classes including many small groups and groups with normal Sylow subgroups, and further necessary conditions after joins in later work. The exact special-case hypotheses are absent. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.