# $\operatorname{SL}_2(\mathbb R)$ 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.

• On the Floer-theory proposal, defining a 3-manifold invariant via Lagrangian Floer theory in the character variety of a Heegaard surface (Atiyah-Floer's proposal) is tricky even in the SU(2) case, because no-one has found a direct method of coping with the singularities. A (too) simple example is the genus-1 splitting of the 3-sphere, where the variety is the pillowcase orbifold and the Lagrangians are arcs running from one singular point to another. Feb 25, 2010 at 17:31
• There are two known indirect methods - an infinite-dimensional one (Salamon-Wehrheim) and another (Manolescu-Woodward) which replaces the character variety by something bigger. Trying to work something out in the SL(2,C) or SL(2,R) case would certainly require bravery, especially since there's no known instanton analogue to guide the construction. Feb 25, 2010 at 17:33
• An SU(3) analog of Casson invariant can be found here front.math.ucdavis.edu/0006.5018 Apr 14, 2011 at 6:59
• @Gil the link in your comment is broken, here's a replacement: arxiv.org/abs/math/0006018 Mar 29 at 7:08

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.
• Hey Charlie, Thanks for the answer. I didn't know about Dennis Johnson's result, sounds intriguing. I'm not sure how the Boyer-Nicas' invariant is related to Cynthia Curtis' invariant (I guess hers doesn't need to be defined for homology spheres). I thought the fact that $SL(2,R)=Sp(2,R)$ might add a little subtlety to the theory, at least if one generalizes to rational-homology spheres or to general 3-manifolds. From what Perutz says, it's probably hard to get an analogous Floer theory. One could also consider homomorphisms to any cover of $PSL(2,R)$. -Ian Mar 1, 2010 at 16:44