Functorial Maps Between p-Local Finite-Group Classifying Spaces

Definitions and mathematical statement

Fix a prime $p$. A $p$-local finite group is a triple $(S,\mathcal F,\mathcal L)$, with $S$ a finite $p$-group, $\mathcal F$ a saturated fusion system on $S$, and $\mathcal L$ an associated centric linking system. Its classifying space is the Bousfield--Kan $p$-completion $B\mathcal G=|\mathcal L|^\wedge_p$ of the nerve, with canonical map $\theta:BS\to B\mathcal G$. Use primed notation for a second such triple and $\theta':BS'\to B\mathcal G'$.

Let $\phi:S\to S'$ be fusion-preserving. In the usual morphism-preserving interpretation this means that for every $P,Q\leq S$ and $\psi\in\operatorname{Hom}_{\mathcal F}(P,Q)$, there is $\psi'\in\operatorname{Hom}_{\mathcal F'}(\phi(P),\phi(Q))$ such that \[ \phi|_Q\circ\psi=\psi'\circ\phi|_P. \] The source does not state injectivity of $\phi$, so none is added.

Does every such $\phi$ admit a continuous map \[ f:|\mathcal L|^\wedge_p\longrightarrow|\mathcal L'|^\wedge_p \quad\text{with}\quad f\circ\theta\simeq\theta'\circ B\phi? \] This homotopy-commuting equation spells out extension of the map $B\phi:BS\to BS'$ after passage to the two canonical classifying-space maps. The question is existence of an actual map of spaces, not merely a map of suspension spectra or a map after replacing the target by a wreath-product construction. It does not request uniqueness or a coherent functorial choice of extensions for all homomorphisms.

The report-specific definitions of fusion preservation and completion conventions should be confirmed. The original record says a stable result is known; its dated assessment describes a stable extension after passing to a $p$-local wreath product. These reported stable statements are not silently identified with the direct unstable extension asked for here.

Short English statement

Does a fusion-preserving homomorphism of the underlying finite $p$-groups extend to an actual map between the associated $p$-completed linking-system classifying spaces, making the canonical maps commute up to homotopy?

Sources

• Local UnsolvedMath registry, permanent ID $30001052$, source code catalogue entry, "Functorial Maps Between p-Local Finite-Group Classifying Spaces." No individual proposer is supplied.
• Topologie (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/43, pp. 2419--2476: https://doi.org/10.4171/owr/2008/43.
• Supplied reference, "Castellana--Libman": https://doi.org/10.1016/j.aim.2009.02.011.
• Supplied literature triage, checked 2026-08-21: status "partially solved." It reports a stable extension after passage to a $p$-local wreath product, while saying the direct extension question is not settled in general. The raw record also says the report knows a stable result. This is supplied dated curation, not independently verified; time-sensitive status, exact stable hypotheses, and cited claims require independent verification.