Problem 10.2 — (V.

Definitions and mathematical statement

Turaev asks to relate the surgery formula for the Casson--Walker--Lescop invariant described immediately before this problem to Lescop's formula [251]. If $D$ is surgery data in a common admissible class presenting a three-manifold $M_D$, and $F(D)$ and $F_{\mathrm{Lescop}}(D)$ denote the two expressions after translating conventions, the requested comparison should explain their relationship and recover the same normalized invariant. The preceding formula, Lescop's formula and their admissible surgery-data classes are absent; $F$ and $F_{\mathrm{Lescop}}$ are placeholders and no particular link formula is fabricated.

The supplied normalization footnotes are essential: \[\lambda_{\mathrm{CW}}(M)=2\lambda_{\mathrm C}(M) \quad\text{for an integral homology three-sphere }M,\] \[\lambda_{\mathrm{CWL}}(M)=\frac{|H_1(M;{\mathbb{Z}})|}{2}\lambda_{\mathrm{CW}}(M) \quad\text{for a rational homology three-sphere }M.\] An integral homology three-sphere has integral homology that of $S^3$; a rational homology three-sphere has rational homology that of $S^3$, and then $|H_1(M;{\mathbb{Z}})|$ is the order of its finite first homology group. These formulas imply that the stated conventions give $\lambda_{\mathrm{CWL}}=\lambda_{\mathrm C}$ on integral homology spheres. Orientation conventions and any wider class of three-manifolds covered by the omitted surgery formulas must still be specified.

The following paragraph, attributed to C. Lescop, supplies context rather than another explicit question: it describes Casson's 1984 integer-valued invariant as an appropriate count of $SU(2)$-representations of the fundamental group, citing [3,159], and reports that Cappell, Lee and Miller [76] extended the counting approach to compact Lie groups to obtain further invariants of integral homology spheres. The precise counting and Lie-group hypotheses are not supplied, and this report is not used to invent a second target.

Short English statement

Explain the connection between the preceding surgery formula and Lescop's formula, preserving the stated factors of two and homology-group order. Both formulas themselves are missing from the extracted record.

Sources

• Local UnsolvedMath registry, permanent ID $10400177$, exact source code catalogue entry.
• V. Turaev, attribution in Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 10.2, PDF page 142, normalization footnotes 32--33: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• Lescop [251] is the comparison reference. The following C. Lescop paragraph cites [3,159] for Casson's construction and Cappell--Lee--Miller [76]. Full entries are not supplied.