I am thinking about the precise formulation of the Lefschetz duality for the relative cohomology. If I understand [this Wikipedia article](http://en.wikipedia.org/wiki/Lefschetz_duality) correctly, there is an isomorphism between $H^k(M, \partial M)$ and $H\_{n-k}(M)$ and hence (I suppose) a non-degenerate pairing $H^k(M, \partial M) \times H^{n-k}(M) \rightarrow \mathbb{R}$. However, I have trouble visualizing this pairing. Let $[(\alpha, \theta)] \in H^k(M, \partial M)$ and $[\beta] \in H^{n - k}(M)$, is it then true that
$$
 \left< [(\alpha, \theta)], [\beta] \right> =
\int\_M \alpha \wedge \beta + \int\_{\partial M}\theta \wedge \beta\_{|\partial M}
$$
or am I missing something? If unrelated to Lefschetz duality, does this pairing ever appear in topology?


I can understand how to define a pairing on the homology by counting intersections, but I really don't see how this works for cohomology. Also, a reference on Lefschetz cohomology or just analysis/topology on manifolds with boundary would be greatly appreciated!



Pairing used in Lefschetz duality
Yes, your formula is right. For the intuitive understanding just compute it for 1- and 2- dimensional half-spaces.


See Bott & Tu, Differential forms in Algebraic topology, $\S 5$, Poincaré duality.


I give only sketch of proof for your question.


1. First of all you need pairing between $H\_c^k(M, \partial M)$ and $H^{n-k}(M)$.
2. Just consider $M= [0,+\infty)$, find $H\_c^k(M), H\_c^k(M,\partial M), H^k(M), H^k(M,\partial M) $ and check that you have non-generating pairing.
3. By induction, expand previous statement to $\mathbb R\_{+}^n = \{(x\_1,x\_2\dots,x\_n)|x\_1\geqslant 0\}$ (read Bott & Tu $\S 4$ and do the same things).
4. Prove that there is Mayer-Vietoris sequence for $H\_c^k(M,\partial M)$ similar to Mayer-Vietoris sequence for $H\_c^k(M)$.
5. Prove duality the same way as in $\S 5$ (check the commutativity of diagram and apply 5-lemma).


That's all, I performed these actions without any troubles. 


