This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the trivial SU(2) bundle over the 3-manifold M, modulo gauge transformation. In the stationary phase approximation as the level k-->infinity, we write this integral as the sum of functions f_i(k), where the sum is over the set {c_i} of critical points of the chern-simons functional (which is basically the log of the integrand). The critical points are the flat connections. In his analysis of the function f_tr(k) corresponding to the trivial flat connection (product connection), Rozansky (as well as previous authors, I believe) found the Casson invariant (as the second coefficient if you write f_i as a power series in 1/k, or something like that).

Now, at first glance it strikes me as odd that the casson invariant shows up here, because the casson invariant can, by definition, be thought of as a (signed) sum over ALL flat connections, EXCEPT the trivial connection. So my question is: what gives? Why does the contribution from the trivial connection give the casson invariant? Why isn't it instead, say, the sum of the 2nd coefficients of the contributions from all other flat connections? is there some explanation for this? Have contributions from other connections, say reducible connections for a homology 3-sphere, been analyzed? Does the casson invariant show up there as well?

If you've gotten this far, here's another question: right now, i can write |H_1(M)|*(Casson-Walker invariant of M), say for lens spaces M=L(p,q) as the sum over certain INTEGERS, one per conjugacy class of reducible flat connection on SU(2)XM, PLUS the signature of the 2-bridge knot b(p,q). I'm trying to identify this decomposition of the casson invariant with others in the literature, of which there are many, but I'm having the problem that none of these are (always) integral decompositions, even though they ADD UP to an integer. Well this is a long shot, but any ideas would be appreciated!