The PSL(2,C) Casson invariant decomposition conjecture

Let $\Sigma$ be a closed, orientable $3$-manifold. Write $\lambda_{\operatorname{PSL}_2(\mathbb C)}(\Sigma)$ and $\lambda_{\operatorname{SL}_2(\mathbb C)}(\Sigma)$ for the respective Casson invariants, and let $X^*(W_1)$ and $X^*(W_2)$ be the character varieties associated with the two handlebodies in a Heegaard splitting. For an isolated character $\chi\in X^*(W_1)\cap X^*(W_2)$, let $w_2(\chi)$ denote its second Stiefel–Whitney obstruction, and let $n_\chi$ denote its local intersection multiplicity. The PSL(2,C) Casson invariant decomposition conjecture. For any closed, orientable $3$-manifold $\Sigma$,

$$\lambda_{\operatorname{PSL}_2(\mathbb C)}(\Sigma)=\frac{1}{|H^1(\Sigma;\mathbb Z_2)|}\lambda_{\operatorname{SL}_2(\mathbb C)}(\Sigma)+\sum_\chi n_\chi,$$

where the sum is taken over all isolated characters $\chi\in X^*(W_1)\cap X^*(W_2)$ with $w_2(\chi)\neq 0$. The formula expresses the contribution from characters with vanishing obstruction in terms of the $\operatorname{SL}_2(\mathbb C)$ invariant, with the remaining isolated non-liftable characters contributing the correction terms $n_\chi$; the source presents it among observations and conjectures whose detailed analysis is deferred to future work.