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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...

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Could you please explain what do you mean by |H_1(M)|? If M is a Q-holomlogy sphere, is not this zero? Or you conisder some coefficients different from $R, Q$? It will be also helpfull if you specify the paper of Boyer and Nicas. –  Dmitri Jan 4 '10 at 22:48
|H_1(M)| means the number of elements of the first homology of M with Z coefficients, which is finite iff M is a rational homology sphere. –  Kevin Walker Jan 6 '10 at 22:13
Sam I added the reference, hopefully the correct one :) –  Dmitri Jan 6 '10 at 22:40
yup that's correct, thanks –  Sam Lewallen Jan 7 '10 at 6:33
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2 Answers 2

up vote 2 down vote accepted

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.

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Thanks, this is indeed what Boyer + Nicas were referring to. –  Sam Lewallen Feb 6 '10 at 20:06
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I used to understand this stuff pretty well, but it's been a long time. I think the following answer is correct, but I'm not certain.

Since M is a rational homology sphere, the irreducible points of R are separated from the reducible points, so we can treat them separately.

[EDIT: This isn't true in general (consider the case where $M$ is a connect sum). $M$ being a QHS only guarantees that the "very reducible" points of $R$, with image in the center of $SU(2)$, are isolated from the irreducibles. So the argment below only works in a special case.]

I claim that (1) the irreducible part of R contributes zero to the homological intersection number of the Q's, and (2) the contribution of the reducible part of R is $H_1(M)$. I think claim (1) follows from

(a) the answer to your question 2, which is that the contribution of a submanifold of intersection is equal to the Euler characteristic of its normal bundle (normal to both the Q's);

(b) using symplectic structure to show that the normal bundle is isomorphic to the tangent bundle in this case; and

(c) the observation that SU(2) acts freely, so the Euler characteristic of an irred. component of R is zero.

For claim (2), the idea is to show that the intersection number is equal to the number of homomorphisms $f$ from the finite abelian group $H_1(M)$ to $S^1$. If the the image of $f$ is $\pm 1$, then it corresponds to a unique transverse point of $R$. Otherwise, both $f$ and $-f$ lie on a 2-sphere component of $R$. By an argument similar to the one above, this 2-sphere contributes its Euler characteristic, namely 2, to the homological intersection number.

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Thanks, this looks really good! But I don't think being a QHS is enough to guarantee that the reducibles and irreducibles are separated? Sometimes the irreducibles limit to the reducibles (this is discussed in the paper of Boyer and Nicas). That was the case where I really couldn't see what to do... do you have any ideas there? Thanks again! –  Sam Lewallen Jan 7 '10 at 6:32
Oops -- you're right. That's a silly thing for me to get wrong. I don't immediately see a way to fix my argument above to work in the general case. Maybe by understanding the tangent spaces of Q_1 and Q_2 in more detail one can cook up a perturbation which gets rid of the irreducibles while not changing the reducibles very much? Or maybe there's a simpler, better argument which identifies the homology classes of Q_1 and Q_2 inside H_*(H)? –  Kevin Walker Jan 7 '10 at 16:18
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