$\DeclareMathOperator\SL{SL}$Casson's invariant is an invariant of a homology 3-sphere, obtained by “counting” representations of the fundamental group into $\operatorname{SU}(2)$. I was wondering if there is an analogous invariant counting representations into $\SL(2,\mathbb R)$? Curtis has an invariant counting representations into $\SL(2,\mathbb C)$. These invariants are obtained by taking a Heegaard splitting of the manifold, and considering the intersection of the representation varieties of the two handlebodies in the representation variety of the Heegaard surface. Casson has to perturb the resulting varieties to make them transverse, then counts the intersections. Curtis counts only the finite points of intersection using algebraic geometry to resolve any singularities, and ignoring any higher dimensional components of the intersection. Then they both have to show that this count is invariant under stabilization of Heegaard splittings, and therefore an invariant of the manifold. I was wondering whether one could combine the two approaches to get an analogous invariant in the case of $\SL(2,\mathbb R)$ representations? One would throw away higher dimensional components of intersection of the $\SL(2,\mathbb R)$ varieties of the two handlebodies, and perturb near the isolated intersections to get a count of intersection points.

If this works, what about making an analogous Floer theory, by counting holomorphic disks between finite intersection points?

I have't done a literature search, but I suspect this is an open question.