Open Problems and Questions about Geodesics — Question 10 (Hingston)

Definitions and mathematical statement

Let $M$ be a compact manifold with a Riemannian metric $g$, or with a Finsler metric in the Finsler version, and let $G$ be a finitely generated abelian group. Write $\Lambda M$ for its free loop space and $L_g(\gamma)$ for the length of a loop $\gamma$. For $a\geq 0$, set \[ \Lambda_a M=\{\gamma\in\Lambda M:L_g(\gamma)\leq a\}, \] and let $i_a:\Lambda_a M\hookrightarrow\Lambda M$ be the inclusion. For $Y\in H_*(\Lambda M;G)$, define its critical level by \[ \operatorname{cr}_g(Y)=\inf\left\{a\geq 0:Y\in\operatorname{im}\left((i_a)_*:H_*(\Lambda_a M;G)\longrightarrow H_*(\Lambda M;G)\right)\right\}. \] In particular, this definition gives $\operatorname{cr}_g(0)=0$. For a nonzero class $Y$, the supplied source also gives the minimax description \[ \operatorname{cr}_g(Y)=\inf_{\substack{c\text{ a singular cycle}\\ [c]=Y}}\ \sup_{\gamma\in |c|}L_g(\gamma), \] where $|c|$ is the union of the images of the singular simplices occurring with nonzero coefficients in $c$. The source does not specify the regularity or topology of $\Lambda M$; a precise analytic model supporting these length sublevels and the stated minimax description remains to be fixed in review.

For $M=S^n$, with $n$ retained as an unspecified dimension parameter, the question attributed to Hingston is whether there exist a metric $g$, a class $0\neq X\in H_*(\Lambda S^n;{\mathbb{Z}})$, and a positive integer $m$ such that \[ 0<\operatorname{cr}_g(mX)<\operatorname{cr}_g(X). \] Here $mX$ is the integer multiple of the homology class $X$, rather than an operation that iterates loops. The positive lower bound necessarily excludes $mX=0$. The supplied record does not specify the intended range or quantifier of $n$, or whether the final existence question is intended separately for Riemannian and Finsler metrics. These qualifications remain unresolved.

Short English statement

For a sphere of the intended dimension, can multiplying an integral homology class of its free loop space by a positive integer strictly lower its minimax length while keeping that length positive? The dimension quantifier, the precise loop-space model, and the intended Riemannian versus Finsler scope require clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800930$, source code catalogue entry, collection AMR Open Problem Lists. The record labels the problem open; this is a supplied status label, not an independently verified assessment.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as identified in the supplied record), Question 10. Supplied source URL: https://arxiv.org/abs/1308.5417. The record attributes the question/context to Hingston.
• The supplied provenance reports access on 2026-10-16 and extraction by arxiv_latex. These provenance details are reproduced from the record; no independent source visit or literature verification is asserted.