Problem 10.16 — (T.

Definitions and mathematical statement

An integral homology $3$-sphere is a closed $3$-manifold $M$ with integral homology isomorphic to that of $S^3$. Let $\mathcal H$ denote their classes in the category intended by the source. T. Ohtsuki asks for a product \[ \circ:\mathcal H\times\mathcal H\longrightarrow\mathcal H, \qquad(M_1,M_2)\longmapsto M_1\circ M_2, \] that corresponds, through the source's equation (50), to the product of Jacobi diagrams given by connected sum.

A Jacobi diagram here is a vertex-oriented trivalent diagram modulo the source's diagram relations. To express the requested compatibility schematically, if $\Phi$ is the correspondence in (50) and $\mathbin{\#}_{\mathrm{diag}}$ its diagram connected-sum operation, the intended requirement has the form \[ \Phi(M_1\circ M_2) =\Phi(M_1)\mathbin{\#}_{\mathrm{diag}}\Phi(M_2), \] with any grading, coefficients, completion or correction terms specified by (50). Neither that equation nor these operations are supplied in the record, so this display only identifies the requested correspondence and is not an asserted reconstruction of (50). The precise equivalence classes of manifolds, whether the product really takes values in manifolds or a linearized/completed space, and any required product laws must be recovered from the source context. No associativity, unit or commutativity axiom is added.

Short English statement

Construct a product of integral homology $3$-spheres that corresponds, under the missing equation (50), to connected sum of Jacobi diagrams.

Sources

• T. Ohtsuki, attributed Problem 10.16 in Problems on invariants of knots and 3-manifolds (2002), supplied PDF page 152. https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. The statement refers explicitly to equation (50), which is absent from the supplied record. Recorded access 2026-07-29; extraction pdf-font-structure.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10400191$, exact source code catalogue entry; research classification OPEN-TRIAGE.