Problem 10.13 — (J.

Definitions and mathematical statement

J. Roberts asks: "What is the space of 3-manifolds?" The supplied item is a foundational question about how to organize 3-manifolds, not a specified classification theorem or an assertion about a defined topological space.

To make such a question mathematical, one must specify the objects, their equivalence, and the additional structure intended by "space". For example, a set of homeomorphism or diffeomorphism classes, a category with cobordisms as morphisms, a topological or metric organization, and a formal algebraic space generated by manifolds are different targets. The excerpt selects none of them. It also does not restrict to closed, compact, oriented, connected, smooth or geometric manifolds.

Consequently no particular topology, quotient, metric, category or invariant can be designated as the requested space from this record. A satisfactory formulation would first state what structure is meant and what mathematical purpose or universal property it should serve. Those criteria are absent. The original qualitative question is therefore retained explicitly rather than replaced by classifying all 3-manifolds or by studying metrics on one fixed manifold.

Short English statement

Roberts asks for a meaningful notion of a space of 3-manifolds. The source does not define the objects, equivalence or structure, so it remains a qualitative foundational programme.

Sources

• Local UnsolvedMath registry, permanent ID 10400188, source code catalogue entry; statement attribution J. Roberts; proposer field and proposal year absent; update 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 10.13, PDF page 150: 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 supplied summary explicitly identifies this as a philosophical/meta question and gives no precise additional criterion or resolving reference.