Topological Modular Forms for Cyclic Groups

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}_{\geq 1}$, and let $C_n={\mathbb{Z}}/n{\mathbb{Z}}$ be the cyclic group of order $n$. The source uses $\operatorname{TMF}$ for topological modular forms and asks about the spectrum denoted by $\operatorname{TMF}^{C_n}$.

Requested calculation. For every positive integer $n$, explicitly determine the equivalence type of the spectrum $\operatorname{TMF}^{C_n}$, or at least determine all its integer-graded homotopy groups \[ \pi_k\left(\operatorname{TMF}^{C_n}\right),\qquad k\in{\mathbb{Z}}. \] For an ordinary spectrum $X$, the notation $\pi_k(X)=[\Sigma^k\mathbb S,X]$ denotes stable homotopy classes of maps from the $k$-fold suspension of the sphere spectrum.

Unresolved qualification. The supplied record does not specify the equivariant construction underlying $\operatorname{TMF}^{C_n}$. In particular, it does not identify a $C_n$-action or a genuine $C_n$-equivariant refinement, or state whether the superscript means categorical fixed points, homotopy fixed points, or another construction. It also does not specify a localization or completion. Consequently, the displayed request preserves the source notation but defines a unique calculation only after these conventions have been supplied. The integer grading above applies if the intended result is an ordinary fixed-point spectrum; if the intended target is an equivariant spectrum itself, the grading and homotopy invariant must also be specified.

Short English statement

For every positive integer $n$, calculate the spectrum written $\operatorname{TMF}^{C_n}$, or at least its homotopy groups in every degree. The supplied record does not define the equivariant construction or fixed-point convention, so the intended spectrum requires clarification. Its partially solved status is supplied dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30004205$, supplied record entitled Topological Modular Forms for Cyclic Groups; source problem code catalogue entry. The project registry files are data/unsolvedmath/problems.json and the supplementary catalog/problems.json. No individual proposer is identified in the supplied record; its proposed year is 2019.
• Supplied source citation: Homotopy Theory (2020), Oberwolfach Reports, pp. 2183--2256, DOI 10.4171/owr/2019/36, https://doi.org/10.4171/owr/2019/36. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2020 label and its 2019 DOI component are retained without resolving their relationship.
• Supplied literature source, "original Oberwolfach report": https://doi.org/10.4171/OWR/2019/36.
• Supplied literature source, C-two-equivariant Topological Modular Forms: https://arxiv.org/abs/2101.06841.
• Supplied literature source, Genuine C-n-equivariant Topological Modular Forms: https://arxiv.org/abs/2510.23602.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports computations or decompositions for small cyclic groups such as $C_2$ and $C_3$, while reporting no calculation for general $n$. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This records the supplied editorial assessment and does not establish mathematical review of this draft.