Question 10.21 — (F.

Definitions and mathematical statement

For an abelian group $G$, the source calls a function $q$ quadratic when its polarization \[ b_q(x,y)=q(x+y)-q(x)-q(y) \] is bilinear. It is homogeneous if $q(nx)=n^2q(x)$ for every $n\in{\mathbb{Z}}$ and $x\in G$. The excerpt does not give the value group or identify $G$.

It reports a canonical assignment $\sigma\mapsto q_\sigma$ from $\operatorname{spin}^c$ structures on the intended 3-manifolds to quadratic functions, with $q_\sigma$ homogeneous if and only if $\sigma$ comes from a spin structure. Not every $\operatorname{spin}^c$ structure is induced by a spin structure; the question is not restricted to the homogeneous case.

F. Deloup asks whether $\arg\gamma(q_\sigma)$ admits a lift to a mod-16 invariant. The source states that such a lift would give a finite-type invariant of degree one in the $\operatorname{spin}^c$ Goussarov--Habiro theory. This is the requested lift and its reported consequence, not a claimed existing invariant.

The function $\gamma$, its Gauss-sum normalization and nonvanishing conditions, the scaling of $\arg$, and the codomain and reduction map for "mod 16" are missing. An ordinary argument of a nonzero complex number is valued in ${\mathbb{R}}/(2\pi{\mathbb{Z}})$; identifying it with a mod-8 class, and then choosing a reduction from a mod-16 target, requires extra conventions. In particular a ${\mathbb{Z}}/16{\mathbb{Z}}$ target is not silently assumed when rational-valued refinements may be intended. The underlying quadratic refinement, manifold category and degree-one surgery conventions also need the surrounding source.

Short English statement

Can the Gauss-phase quantity attached to a $\operatorname{spin}^c$ quadratic function be lifted to a mod-16 invariant, yielding the stated degree-one finite-type invariant? The phase normalization and lift map are absent, and nonhomogeneous $\operatorname{spin}^c$ cases must be retained.

Sources

• Local UnsolvedMath registry, permanent ID 10400196, source code catalogue entry; statement attribution F. Deloup; proposer field and proposal year absent; update 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Question 10.21, PDF page 154, including the supplied quadratic-function footnote 37: 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.
• Adjacent local ID 10400195 concerns closed $\operatorname{spin}^c$ 3-manifolds and $Y_d^c$-equivalence, but does not restore the missing phase or quadratic data. The supplied summary gives no exact resolving reference.