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I believe the following is well known after talking to some experts, but I am unable to find a reference for the case with boundary.

Fix a field $F$ and an oriented $n$-manifold $(M,\partial M)$. We have a pairing $H_*(M) \otimes H_{n-*}(M,\partial M) \rightarrow F$ given by taking transverse representatives of the cycles and counting the oriented intersections.

Claim: This pairing is the same as the Poincaré–Lefschetz duality pairing.

Is there a standard reference for this? Or at least an argument that is only a few lines, possibly relying on the closed version of the statement?

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    $\begingroup$ The proof of the statement is fairly direct from the old-fashioned PL triangulations approach to the proof (i.e. Poincare's approach). When using singular homology the proof is more subtle. Bredon's "Geometry and Topology" is a good source. $\endgroup$
    – Ryan Budney
    Commented Jan 10, 2022 at 18:05
  • $\begingroup$ @RyanBudney Thanks Ryan; it seems like 11.10 is the best Bredon has for non-submanifolds in which he concludes there is some geometric intersection of subspaces if there is a nontrivial cup product. $\endgroup$
    – Connor Malin
    Commented Jan 10, 2022 at 19:12
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    $\begingroup$ Right, Bredon is concerned about Steenrod realizability. For the pairing on all homology classes perhaps there isn't a write-up beyond the classical Poincare perspective. That said, I must be wrong -- perhaps the work of Greg Friedman is where to look? faculty.tcu.edu/gfriedman $\endgroup$
    – Ryan Budney
    Commented Jan 10, 2022 at 19:42
  • $\begingroup$ The theorem you are looking for is Theorem 95.15 in my lecture notes: uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/… $\endgroup$
    – Stefan Friedl
    Commented Jan 10, 2022 at 19:57
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    $\begingroup$ sorry, I should have referred to Theorem 95.14. $\endgroup$
    – Stefan Friedl
    Commented Jan 10, 2022 at 20:26

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This is proven for smooth manifolds by Goresky in "Whitney Stratified Chains and Cochains".

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