Studying the first pages of Gompf-Stipsicz's 4-Manifolds & Kirby Calculus forced me to worry about the geometric meaning of homology and cohomology classes; in particular page 7 contains the following sentence, which I feel quite obscure:
Let $X$ be a compact, oriented, topological 4-manifold. When $X$ is oriented, it admits a fundamental class $[X]\in H_4(X,\partial X; \mathbb Z)$ [which is, as explained before, a top-degree homology class generating $H^4(X,\partial X; \mathbb Z)$].
Definition. The symmetric bilinear form $Q_X\colon H^2(X,\partial X;\mathbb Z)\times H^2(X,\partial X;\mathbb Z) \to \mathbb Z$ defined by $Q_X(a,b)=\langle a\cup b,[X]\rangle$ is called the intersection form of $X$.
My first question is: how's that pairing $\langle-,-\rangle$ defined?
I think I have a slight confidence with the idea of homology-cohomology classes as geometric objects (immersed submanifold, cycles as sub-simplices...).
I'm also able to forsee that all the different pairing definable in various (co)homology theories can someway can someway be reconduced to a single idea.
So my second question is in fact a modified version of the first one: how can I link the former "intersection" pairing to the similar one $ \langle -,-\rangle_{dR}\colon H^k_{dR}(M)\times H^{n-k}_{dR}(M)\to \mathbb R\colon (a,b)\mapsto \int_M a\land b $ in de Rham cohomology?