$\operatorname{SL}_2(\mathbb R)$ Casson invariant? $\DeclareMathOperator\SL{SL}$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. 
 A: *

*Boyer and Nicas defined an SL(2,C) Casson invariant. The idea is to just ignore the noncompact components of intersection and you get a well defined invariant. My guess is their
proof carries over verbatum to SL(2,R)

*Dennis Johnson  defined a geometric casson invariant.  He never published it, but
it is the sum over the irreducible representations of the Reidemeister torsion of the
complex corresponding to cohomology of the manifold M with coefficients in ad of the representation when it is defined, and zero otherwise. He called it a geometric casson's invariant because he arrived at the torsion by computing the "angle" between the chaaracter varieties of two handlebodies from a Heegaard splitting of the manifold, inside the character variety of the splitting surface.
A: A 2020 arxiv posting of Nosaka (An $SL_2(\mathbb{R})$-Casson invariant and Reidemeister torsions) defines an $SL(2,\mathbb{R})$ Casson invariant. As Charlie's answer suggests, the approach is inspired by Johnson's unpublished work. 
