Lipschitz homotopy groups and a quantitative size filtration

Definitions and mathematical statement

The source asks for an approach to homotopy groups adapted to Alexandrov spaces, without specifying desired axioms or quantitative features. The supplied partial answer uses Lipschitz representatives and a quantitative refinement of ordinary homotopy groups.

Let $(X,x_0)$ be a pointed finite-dimensional Alexandrov space, in the usual length-space, lower-curvature-bound convention, and let $k\geq1$. Equip $S^k$ with the unit round geodesic metric and a basepoint $*$. Define $\pi_k^{\mathrm{Lip}}(X,x_0)$ using based Lipschitz maps $S^k\to X$, modulo based Lipschitz homotopies with the product metric. The aid reports published comparison isomorphisms \[ \pi_k^{\mathrm{Lip}}(X,x_0)\cong\pi_k(X,x_0). \] The precise comparison theorem, including completeness and boundary conventions, is not cited in full.

Via this comparison, for $\alpha\in\pi_k(X,x_0)$ define \[ \sigma_{k,X}(\alpha)=\inf\{\operatorname{Lip}(f):f:(S^k,*)\to(X,x_0) \text{ Lipschitz and }[f]=\alpha\}. \] The aid states that this is finite for every class and zero for the trivial class. For every based Lipschitz map $h:(X,x_0)\to(Y,y_0)$ between spaces in this setting, \[ \sigma_{k,Y}(h_*\alpha)\leq\operatorname{Lip}(h)\,\sigma_{k,X}(\alpha). \] The proposed size is not asserted a group norm with an exact triangle inequality. The reported product control is, with a suitable dimension-dependent Lipschitz pinch-map constant $C_k$, \[ \sigma_{k,X}(\alpha\beta)\leq C_k\left(\sigma_{k,X}(\alpha)+\sigma_{k,X}(\beta)\right). \] Here multiplication is the usual possibly nonabelian product for $k=1$, and may be written additively for $k\geq2$. The exact chosen pinch map and constant are not supplied.

For the positive lower bound, assume $U=B(x_0,\rho)$, $\rho>0$, admits a strong local Lipschitz contraction to $x_0$, fixing $x_0$ and preserving every smaller concentric ball. In particular, the intended homotopy $H$ has $H(z,0)=z$, $H(z,1)=x_0$, $H(x_0,t)=x_0$, and $H(B(x_0,r)\times[0,1])\subseteq B(x_0,r)$ for $r<\rho$, with the Lipschitz regularity needed on compact restrictions. Since the based unit sphere has radius at most $\pi$ from $*$, the aid's bound is \[ \alpha\ne0\quad\Longrightarrow\quad\sigma_{k,X}(\alpha)\geq\rho/\pi. \] The normalization of the domain metric is essential to that constant.

The associated fixed-degree homotopy systole is \[ \operatorname{sys}_k(X,x_0)=\inf_{\alpha\ne0}\sigma_{k,X}(\alpha), \] with value $+\infty$ when $\pi_k=0$. Under the stated contraction it is at least $\rho/\pi$, and is finite if there is a nonzero class. For every metric rescaling $d_X\mapsto c\,d_X$, $c>0$, both $\sigma_{k,X}$ and this systole scale by $c$. The same uniform lower bound applies if a global systole is defined by taking the infimum across all nontrivial degrees. These are the supplied quantitative conclusions, conditional on the contraction and comparison hypotheses, rather than a new characterization of Alexandrov homotopy types.

Short English statement

The supplied approach transfers Lipschitz representative sizes to ordinary homotopy classes. It gives functorial bounds, controlled products, and a positive homotopy systole under a specified strong local contraction. The constant $\rho/\pi$ uses the unit round sphere and is not independent of domain normalization.

Sources

• Local UnsolvedMath registry, permanent ID $20003142$, exact source code catalogue entry; canonical aim-topology-notes.json, notes[229], source tag problem.
• AIM workshop list "Open Problems in Non-negative Sectional Curvature," workshop "Manifolds with nonnegative sectional curvature," Problem 17: https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. The supplied verification calls the canonical question exact. The PDF was not fetched.
• The research summary refers to published work comparing Lipschitz and ordinary homotopy groups of finite-dimensional Alexandrov spaces, but supplies no authors, titles, identifiers, or full authoritative citations. The original literature field is empty.
• Research attempt 1: result partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low; classification PARTIAL-PROGRESS, registry status partially_solved. Created and updated 14 August 2026. These are supplied assessments requiring review.
• Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. The machine-generated research aid is not peer reviewed and requires independent verification before citation.