Generalized Smith Questions for Compact Lie Groups

Definitions and mathematical statement

Let $h_*$ and $h'_*$ be generalized reduced homology theories on based spaces, and let $G$ be a compact Lie group. A finite based $G$-CW complex means a $G$-CW complex with a distinguished $G$-fixed basepoint and finitely many equivariant cells outside that basepoint; cells have the form $G/H\times D^k$, with $H\leq G$ closed. Its fixed-point space is the based space \[ X^G=\{x\in X:gx=x\text{ for every }g\in G\}. \] Here the homology theories are interpreted as nonequivariant theories evaluated on the underlying based spaces; the source does not explicitly specify an equivariant interpretation.

Determine precisely for which triples $(h_*,h'_*,G)$ there exists a finite based $G$-CW complex $X$ such that \[ \left(\forall n\in{\mathbb{Z}},\ h_n(X)=0\right) \quad\text{and}\quad \left(\exists m\in{\mathbb{Z}},\ h'_m(X^G)\neq 0\right). \] Thus the original existence question is asked for each given pair of theories and each given compact Lie group; no universal existence assertion is supplied. No restrictions such as a choice of prime, localization, connectivity, or a particular family of homology theories are specified.

The supplied curation labels the problem partially solved and reports results for abelian compact Lie groups and several other cases. It does not state the precise hypotheses or scope of those results, so that assessment does not supply a classification theorem for the unrestricted formulation above.

Short English statement

For given generalized reduced homology theories $h_*$ and $h'_*$ and a compact Lie group $G$, determine whether a finite based $G$-CW complex can have zero $h_*$-homology in every degree while its $G$-fixed points have nonzero $h'_*$-homology in some degree. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004199$, title: Generalized Smith Questions for Compact Lie Groups; exact source problem code: catalogue entry. The record supplies proposed year 2019 and no named proposer. Its corrected_verified statement label records supplied editorial verification, not independent mathematical verification here.
• Supplied report citation: Homotopy Theory (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/36, pp. 2183--2256. https://doi.org/10.4171/owr/2019/36. The report is described as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "original Oberwolfach report": https://doi.org/10.4171/OWR/2019/36.
• Supplied literature source: On the Balmer Spectrum for Compact Lie Groups. https://arxiv.org/abs/1810.04698.
• Supplied literature source: Chromatic Fixed Point Theory for Extraspecial Two-Groups. https://arxiv.org/abs/2008.00330.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that the generalized Smith question is solved for abelian compact Lie groups and several other cases, while no general classification for arbitrary compact Lie groups is known. This is supplied dated curation derived from the OWR material, not independently verified here; time-sensitive status, cited claims, and their hypotheses require independent verification.