Problem 10.18 — (F.

Definitions and mathematical statement

Retain $q_\sigma$ as the source's notation for a family of quadratic forms. The supplied excerpt does not define $\sigma$, the domains or value groups of these forms, their quadratic identities, or any nondegeneracy hypotheses.

Problem (F. Deloup). Classify the monoid of isomorphism classes of the quadratic forms $q_\sigma$, with operation induced by orthogonal sum: \[ \mathcal Q=\{[q_\sigma]\},\qquad [q_\sigma]+[q_\tau]=[q_\sigma\perp q_\tau]. \] Here an isomorphism is an isomorphism of the underlying algebraic objects that preserves the quadratic form and any required structure. When the forms have additive domains $A_\sigma,A_\tau$ and a common additive target, the usual orthogonal sum is \[ (q_\sigma\perp q_\tau)(x,y)=q_\sigma(x)+q_\tau(y), \qquad(x,y)\in A_\sigma\oplus A_\tau. \] This formula describes the operation conditionally; it does not identify the missing family. A classification should determine the isomorphism classes and their sum, for example through complete invariants or a presentation of the monoid, including its neutral element in the intended setup. The problem concerns actual isomorphism classes under orthogonal sum; it does not specify passing to a Witt group or a stable-equivalence quotient. Until $q_\sigma$ is defined, a particular monoid cannot be reconstructed from this record.

Short English statement

Deloup asks to classify the quadratic forms denoted $q_\sigma$, together with their orthogonal-sum operation. The family itself is not defined in the excerpt, so its underlying domains and quadratic-form conventions must be recovered before a concrete classification problem is fixed.

Sources

• Local UnsolvedMath registry, permanent ID 10400193, source code catalogue entry; attribution F. Deloup. The stored title is truncated after "Problem 10.18 --- (F.".
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 10.18, PDF page 153. Supplied URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The collection distinguishes 231 numbered bold problem, conjecture, and question headings.
• The supplied status field is open and research_classification is OPEN-TRIAGE, whereas the research summary begins "Classification: PARTIAL-PROGRESS." No specific progress result is supplied. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.