$\DeclareMathOperator\SL{SL}$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.
Best Answer
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 As Charlie's answer suggests, the approach is inspired by Johnson's unpublished work.