Problem 11.10 — (D.

Definitions and mathematical statement

For a space $C$, its integral homology has torsion in degree $i$ if some nonzero $x\in H_i(C;{\mathbb{Z}})$ satisfies $mx=0$ for a positive integer $m$. D. Thurston asks whether the particular configuration spaces used in source reference [237] have such torsion.

The second question asks whether that torsion can yield a torsion invariant of homology 3-spheres. Existence of torsion classes alone does not construct an invariant: one needs a topologically natural rule assigning appropriate torsion data, with independence of choices and the desired nontriviality. The source does not specify whether the target is a fixed torsion group, an obstruction class, or another torsion-sensitive object, and does not assert that every torsion class automatically gives an invariant.

Reference [237] and the configuration-space definitions are absent. Ordinary ordered configurations, unordered configurations, compactifications, relative homology groups and spaces assembled by gluing diagram faces can have different integral homology; none is substituted for the named construction. The configuration size, homological degree, dependence on the manifold, and intended homology-sphere category also need that reference. In the usual integral interpretation a homology 3-sphere has the integer homology of $S^3$, but this convention requires source confirmation. No finite-type degree or coefficient prime is specified.

Short English statement

Do the specific configuration spaces in reference [237] have integral homology torsion, and can it produce a torsion-sensitive invariant of homology 3-spheres? The spaces and the intended invariant construction are not supplied.

Sources

• Local UnsolvedMath registry, permanent ID 10400206, source code catalogue entry; statement attribution D. Thurston; proposer field and proposal year absent; update 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 11.10, PDF page 157: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access date 2026-07-29; PDF font-structure extraction. Provenance distinguishes 231 numbered bold headings. Present primary-literature status and redistribution terms require review.
• The crucial source reference [237] is named without its bibliography entry. The supplied summary gives no additional precise configuration-space definition or verified result.