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Casson's invariant is an invariant of a homology 3-sphere, obtained by ``counting" representations of the fundamental group into $SU(2)$. I was wondering if there is an analogous invariant counting representations into $SL(2,R)$? Curtis has an invariant counting representations into $SL(2,C)$. 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 $SL(2,R)$ representations? One would throw away higher dimensional components of intersection of the $SL(2,R)$ 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.

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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. –  Tim Perutz Feb 25 '10 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. –  Tim Perutz Feb 25 '10 at 17:33
    
An SU(3) analog of Casson invariant can be found here front.math.ucdavis.edu/0006.5018 –  Gil Kalai Apr 14 '11 at 6:59
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1 Answer

  1. 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)

  2. 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.

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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 –  Ian Agol Mar 1 '10 at 16:44
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