Problem 10.4 — (M.

Definitions and mathematical statement

For a closed oriented smooth $3$-manifold $M$, let $\operatorname{Spin}(M)$ be the set of isomorphism classes of spin structures, namely lifts of its oriented frame bundle through $\operatorname{Spin}(3)\to\operatorname{SO}(3)$. This set is a torsor for $H^1(M;{\mathbb{Z}}/2{\mathbb{Z}})$: it has a free transitive action of that group but no specified distinguished element.

M. Polyak asks for an invariant $\lambda(M,\sigma)$ of a closed $3$-manifold together with a spin structure such that, for every manifold in the intended class, \[ \lambda_{\mathrm{CWL}}(M) =\sum_{\sigma\in\operatorname{Spin}(M)}\lambda(M,\sigma), \] where $\lambda_{\mathrm{CWL}}$ is the Casson--Walker--Lescop invariant in the source's normalization. The desired invariant should be invariant under the appropriate isomorphisms of spin manifolds. The target additive group, orientation conventions and any further requirements distinguishing a substantive spin refinement from an arbitrary decomposition of the sum are not supplied.

The source additionally says that $\operatorname{Spin}^{c}(M)$ is a torsor over $H^1(M;{\mathbb{Z}})$. This conflicts with the standard torsor group $H^2(M;{\mathbb{Z}})$ for spin-$c$ structures. The printed assertion is recorded as a source issue, rather than silently used as a definition or substituted into the target equation. Orientation and smoothness above make the standard spin terminology explicit; the original excerpt says only closed $3$-manifold and does not spell out those conventions.

Short English statement

Construct a spin-structure refinement of the Casson--Walker--Lescop invariant whose sum over all spin structures recovers the original invariant for each closed $3$-manifold.

Sources

• M. Polyak, attributed Problem 10.4 in T. Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), supplied PDF page 143. https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. No separate construction or normalization reference is supplied.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10400179$, exact source code catalogue entry; research classification OPEN-TRIAGE.