EDIT: Immediately after I wrote this question, I remembered the elegant paper "An intersection homology invariant for knots in a rational homology 3-sphere" by Frohman and Nicas, which I believe does exactly what I'm asking. As a new question, I would be interested in knowing if this perspective has ever been generalized to instanton Floer homology, in any way, say, in some special cases. I'll leave the rest of the original question here for now, with the hope of rewriting it soon.
This question is just idle speculation, since I know nearly nothing about (Goresky-Macpherson) intersection homology. As far as I understand, it is a set of homology groups associated to singular spaces of a certain form. My first question is
Suppose $M$ is a 3-manifold, and let $R_M$ denote the space of homomorphisms from $\pi_1(M)$ to $SU(2)$, modulo the action of $SU(2)$ by conjugation. Then
- Can intersection homology (IH) be defined for $R_M$?
- If the answer to 1. is "yes," has this IH been studied anywhere?
A preliminary google search revealed that IH is well defined if we take M to be a surface rather than a 3-manifold. The representation variety $R_M$ can be much more singular for a 3-manifold than a surface, but the extra singularities will at least be algebraic, as can be seen, for example, by considering a presentation for $\pi_1(M)$ with $n$ generators, from which $R_M$ can be written as a quotient of an algebraic subspace of $SU(2)^n$. This makes me think that $IH(R_M)$ might be well-defined.
On the other hand, this question is not very interesting if the answer is "no," so let me ask the motivating question. Generally speaking, I am wondering if any work has been done on finding a "choice-free" definition of the Casson invariant $\lambda(M)$.
In particular, for concreteness, let's consider either Taubes' gauge-theoretic defn., or Casson's original Heegaard-splitting defn. In these cases, one needs to choose perturbations, and/or a Heegaard splitting and perturbations, respectively, to define $\lambda(M)$. In Casson's construction, the choice of a Heegaard surface provides an ambient space in which to view $R_M$, and Taubes' construction does away with this "unnecessary" choice, at the price of allowing the ambient space to be infinite dimensional (the space of all gauge-equivalence-classes of $SU(2)$ connections on $SU(2)\times M$). However, both defn.'s still require perturbations.
Has there been any work on defining $\lambda(M)$ directly in terms of $R_M$, without any need for perturbations? If the answer to the previous questions is "yes," does IH have anything to do with $\lambda(M)$?
From this point of view, I guess what I'm really asking is a very general questions about generalizations of Morse theory: when can a smooth real-valued function on a manifold $M$ be used to gain information about the topology of $M$, without perturbing to make it a generic (Morse) function? Is the CS functional nice enough for this purpose? Relevant terms might be Kirwan's "minimal degeneracy" or Goresky-MacPherson "Stratified Morse Theory," but I've only heard these mentioned with respect to the Riemann surface case. Can they be used to define the Casson invariant?