Problem 10.1 — Can the Casson invariant of an integral homology 3-sphere M be characterized by the signature of a certain 4-manifold…

Definitions and mathematical statement

Let $M$ be an integral homology $3$-sphere, meaning a closed connected $3$-manifold with $H_*(M;{\mathbb{Z}})\cong H_*(S^3;{\mathbb{Z}})$, equipped with the orientation needed for the Casson invariant $\lambda(M)$. For a compact oriented $4$-manifold $W$ with oriented boundary $\partial W\cong M$, write $\sigma(W)$ for the signature of its real intersection pairing, after quotienting by its radical if necessary.

Question. Can $\lambda(M)$ be characterized by the signature of a certain $4$-manifold bounded by $M$? More precisely, the requested characterization must specify the admissible class or construction of such a $W$, together with a relation recovering $\lambda(M)$ from $\sigma(W)$ and any explicitly required correction or auxiliary data. The source gives no such construction or formula. It does not ask for an identity for the signature of every bounding $4$-manifold: signatures change under interior connected sum with closed oriented $4$-manifolds while the boundary stays fixed. The intended extra restriction on the word "certain" is therefore essential.

The supplied research summary asserts that the Casson invariant is one half of a "signature defect." No definition of that defect, normalization, construction, or supporting citation is given. This assertion is retained as an unverified editorial description and does not specify an answer to the source question. The smooth versus topological category and the precise normalization of $\lambda$ remain unspecified.

Short English statement

Find a geometrically specified bounding $4$-manifold whose signature characterizes the Casson invariant of an integral homology $3$-sphere. The required construction and any signature-defect correction are missing; an arbitrary bounding manifold cannot supply a boundary invariant by its signature alone.

Sources

• Local UnsolvedMath registry, permanent ID 10400176, source code catalogue entry, Problem 10.1; no individual proposer is supplied.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 10.1, PDF page 142. 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 registry labels the item partially_solved and PARTIAL-PROGRESS. Its summary asserts a relationship with a bounding $4$-manifold and the formula "Casson invariant = 1/2 the signature defect," without a definition or separate reference. The background retains NEEDS_REVIEW for a current primary-literature status audit and for redistribution terms of the publicly accessible author/publisher copy.