Suppose we have a (closed, oriented) 3-manifold M with a Heegard surface F of genus g. Let F* denote F with a puncture. Then the space H of representations of pi_1(F*) on SU(2) is just SU(2)^2g, and the representation spaces of the two handlbodies sit inside H. Call these spaces Q_1 and Q_2 — we will always think of them as subspaces of H. Finally, the intersection R = Q_1 \cap Q_2 is the representation space for M (Note: we haven't quotiented out by conjugation or anything).
Question 1: In the paper http://www.jstor.org/pss/2001712, Boyer and Nicas claim that if M is a \Q-homology sphere, the homological intersection [Q_1 . Q_2 ] is equal to |H_1(M)|, and they say it's easy to prove. I can't seem to figure out how to do it though, and I've tried for a bit… it seems like there's some bit of theory I must be missing. Can anyone see how to prove it?
Question 2: Is the Euler characteristic of R (that is, Q_1 \cap Q_2) also |H_1(M)|? If so, how could we prove this? In particular, is there a general relationship between the intersection pairing between two complementary submanifolds, and the Euler characteristic of their intersection (even when the intersection is not a finite number of points)?
The above is reminiscent of Morse-Bott theory, where the differential forms on the critical set of your morse function give a basis for the chain groups of your homology, and therefore the Euler characteristic of the critical set is the Euler characteristic of the manifold (or something like that… do I have this right?) This requires Morse-Bott non degeneracy of the critical set.
Final Question: What's an explicit relationship between the morse theory and the inersection theory? And when we just care about Euler characteristic, can we relax the Morse-Bott non-degeneracy? It seems that Q_1 and Q_2 don't always intersect "non-degenerately" (not only non-transversely, but the intersection might not even be smooth, for example) but Boyer and Nicas still claim that the intersection number is something nice (and computable on general grounds). Under what conditions could the same thing happen with a non-Morse-Bott morse function?
Thanks! I hope there aren't too many questions here…
Best Answer
It might be helpful to look at the book of Akbulut and McCarthy on Casson's Invariant. I think the answer to Question 1 is fairly clearly explained in Proposition 1.1b of of Chapter III.