I'm wondering if anyone can point me to a reference on how the various Lefschetz-Poincare dual torsion pairings of a manifold with boundary fit together.

To explain in more detail, consider a connected compact oriented n-manifold $M$ with boundary. Then we have the various dualities with rational coefficients $H_i(M;Q)=H_{n-i}(M, \partial M; Q)$ and $H_i(\partial M;Q)=H_{n-i-1}(\partial M; Q)$, but we also obtain, from some easy tinkering with the long exact sequence of the pair and other basic observations that, e.g. $im(H_i(M)\to H_i(M, \partial M))$ is dual to $im(H_{n-i}(M)\to H_{n-i}(M, \partial M))$ (in fact this is how we define signatures on manifolds with boundary) and similar results hold for the other "shared" terms in the long exact sequence.

I'd like to know more about the generalization of this over Z and, in particular, about what torsion pairings to Q/Z exist on the various images, cokernels, etc. of the long exact sequence and which are nonsingular. Does anyone know of any place in the literature where this is worked out?

Thanks!

Updated: Thanks, Tom, for your response. I've been thinking about it, and while I agree that the situation is much murkier over Z, I think there are still some things that can be said.

For example, it's true that the map $H_n(M)\to H_n(M, bd M)$ (let's assume $n$ is the middle dimension to simplify the discussion), only yields a nondegenerate pairing on the image (which you call $A$) mod torsion and not a perfect (nonsingular) pairing. But now since $H_n(M)\to A/\text{torsion}$ has a free group as its image, we have a (non-unique) splitting that lets us consider $A/\text{torsion}$ as a direct summand of $H_n(M)$. Let's fix this summand for now (the choice turns out not to matter). Since $H_n(M)/\text{torsion}$ and $H_n(M, bd M)/\text{torsion}$ are (perfectly) Z-dual, $A/\text{torsion}$ must be dual to something in $H_n(M, bd M)/\text{torsion}$, and I claim that the thing it's dual to is isomorphic to the kernel of $H_n(M,bd M)\to H_{n-1}(bd M)/\text{torsion}$ (mod torsion). Notice that any non-torsion element of this kernel has a multiple represented by a cycle in $M$, and so it must have non-zero intersection number with any cycle in the boundary, and this shows that as far as intersection pairings are concerned, the choice of $A/\text{torsion}$ as a summand of $H_n(M)$ doesn't matter. I don't really want to get into the details of the rest of my claim here, but the basic idea should be that these are the things that should be dual once we tensor with Q but over Z we need to make sure we have all the appropriate Z-primitives, which I'm pretty sure this does (since if a multiple of $x\in H_n(M,bd M)$ is in $A$, then $x$ becomes torsion in $H_{n-1}(M)$).

So what's the point of all this? Well, even over Z we can say that there is a nonsingular pairing between a certain cokernel and a certain kernel, as long as we mod out the torsion in the right places. I suspect that there's a more subtle version of this for the torsion linking pairings where instead of quotienting out all torsion we just kill certain torsion subgroups. I think I've seen things somewhat of this nature in papers relating Witt groups of Z pairings to Witt groups of Q/Z pairings.