Vanishing for Non-$p$-Nilpotent Compact Lie Groups

Definitions and mathematical statement

Let $p$ be a prime and let $G$ be a compact Lie group of dimension $d>0$. Write $G^\circ$ for its identity component and $\pi_0(G)=G/G^\circ$ for its finite group of components. A finite group is $p$-nilpotent if it has a normal subgroup of order prime to $p$ and index a power of $p$. Assume that $\pi_0(G)$ is not $p$-nilpotent. Write $BG$ for the classifying space of $G$, $BG^\wedge_p$ for its $p$-completion, and $\Omega(BG^\wedge_p)$ for the based loop space of that completion.

Retain the source notation \[ A=C_*(\Omega(BG^\wedge_p)),\qquad \varepsilon\colon\Gamma_k A\longrightarrow A \] for the chain object and the map underlying the displayed question. Here $\varepsilon$ is only a label for that map. The supplied record does not specify the coefficient object $k$, the coefficient convention for $C_*$, the definition of $\Gamma_k$, or the construction of this map. These are unresolved parts of the formulation; in particular, no particular field or cellularization convention is assumed here.

Question (conditional on those conventions being supplied). For every such $G$, is the induced graded map \[ \pi_*(\varepsilon)\colon \pi_*(\Gamma_k C_*(\Omega(BG^\wedge_p))) \longrightarrow \pi_*(C_*(\Omega(BG^\wedge_p))) \] zero? Equivalently, in the intended spectral interpretation, is \[ \pi_n(\varepsilon)=0\qquad\text{for every }n\in{\mathbb{Z}}? \] The intended range of coefficient objects $k$ and the precise homotopical setting remain unspecified, so this is a conditional transcription rather than a fully determined mathematical formulation.

The accompanying excerpt says that a suspension makes a map denoted $\nu_*$ potentially nonzero in degrees from $0$ through $d$, and reports that Benson's squeezed resolutions show that, for finite groups, $\nu_*$ is an isomorphism in the $p$-nilpotent case and zero otherwise. These are supplied contextual claims. The excerpt does not define $\nu_*$ or explicitly identify it with the displayed map, so neither the identification nor a restriction of the question to degrees $0,\ldots,d$ is imposed here.

Short English statement

The question asks whether the displayed map induces zero on all homotopy groups whenever $G$ is a positive-dimensional compact Lie group whose component group is not $p$-nilpotent. The supplied record labels this question open. Its coefficient object $k$, functor $\Gamma_k$, and underlying map require clarification before the statement is fully precise.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30006658$, source code catalogue entry; supplied complete source record, titled "Vanishing for Non-$p$-Nilpotent Compact Lie Groups." The original excerpt labels the question "Question 2.2."
• Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, pp. 541--610, DOI 10.4171/owr/2026/9. Supplied source link: https://doi.org/10.4171/owr/2026/9.
• Supplied literature reference: Benson, "The Omega-Tate spectrum," Oberwolfach Report 9/2026, https://doi.org/10.4171/owr/2026/9. The record does not separately identify a proposer.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the compact-Lie-group question open, reports the finite-group analogue as known, and says that no proof or counterexample for the stated general case was found. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and states that it was reconstructed and checked against the official report. That is supplied provenance, not a verification performed for this draft. The excerpt also cites Benson's squeezed resolutions as reference [2], without supplying the corresponding bibliographic entry.