Topological Complexity of Polygon Spaces and Three-Manifolds

Definitions and mathematical statement

For a path-connected space $X$, use the unreduced convention: $\operatorname{TC}(X)$ is the least number of open sets covering $X\times X$ on each of which the endpoint map \[ \operatorname{ev}:X^{[0,1]}\to X\times X,\qquad \gamma\mapsto(\gamma(0),\gamma(1)), \] admits a continuous section. The reduced convention subtracts one. The record does not specify which normalization it uses.

Let $n\geq3$ and $\ell=(\ell_1,\ldots,\ell_n)\in{\mathbb{R}}_{>0}^n$. Under the usual genericity convention, $\sum_i\varepsilon_i\ell_i\ne0$ for every $\varepsilon_i\in\{1,-1\}$. The labelled planar polygon space modulo translations and rotations can be written \[ M_\ell=\left\{(u_1,\ldots,u_n)\in(S^1)^n: \sum_{i=1}^n\ell_i u_i=0\right\}/\operatorname{SO}(2), \] where rotations act diagonally. Reflections are not included in this quotient, nor are arbitrary permutations of edges; self-intersection is not excluded by the supplied statement. The first request is to determine $\operatorname{TC}(M_\ell)$ for arbitrary generic length data in the intended nonempty, connected domain. Nonemptiness requires the polygon inequalities, and genericity alone does not ensure that the space is nonempty or path-connected. With the endpoint-map definition on all of $X\times X$, a space with multiple path components has no finite topological complexity because some pairs cannot be joined by a path; restricting to components or also quotienting by reflections would be different problems.

The second question asks whether, for closed connected three-manifolds $M,N$, \[ \pi_1(M)\cong\pi_1(N)\quad\Longrightarrow\quad \operatorname{TC}(M)=\operatorname{TC}(N). \] No orientability condition is explicitly supplied. In particular, for $p\geq2$ and $\gcd(p,q)=1$, write \[ L(p,q)=S^3/\left((z_1,z_2)\sim(\zeta z_1,\zeta^qz_2)\right), \qquad \zeta=e^{2\pi i/p}. \] Since these lens spaces have fundamental group ${\mathbb{Z}}/p{\mathbb{Z}}$, the explicit special question is \[ \operatorname{TC}(L(p,q))\stackrel{?}{=}\operatorname{TC}(L(p,1)) \quad\text{for all such }p,q. \] The lens-space comparison is a necessary instance of the broader group-dependence assertion, not an asserted equivalent reformulation.

Short English statement

The record asks for the motion-planning complexity of generic planar polygon spaces and whether that complexity for a closed three-manifold is determined by its fundamental group. Lens spaces with the same cyclic fundamental group provide the stated test case. Normalization and connectedness conventions matter for the polygon question.

Sources

• Local UnsolvedMath registry, permanent ID $30000827$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied.
• Mini-Workshop: Topology of closed one-forms and Cohomology Jumping Loci (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/40, pp. 2321--2360. https://doi.org/10.4171/owr/2007/40.
• Supplied reference labelled "planar polygon spaces": https://arxiv.org/abs/1504.00848.
• The supplied corrected_verified note reports checking mathematical content against the official report. The assessment checked 2026-08-21 reports computations for important polygon-space families and sharp alternatives in broad classes, while arbitrary length data and the general three-manifold group-dependence question remain unsettled. This and the partially solved label are dated supplied triage requiring independent verification.