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?



Reference request for Poincaré–Lefschetz duality as an intersection pairing
This is proven for smooth manifolds by Goresky in "Whitney Stratified Chains and Cochains".


