Intersection of two homology classes 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?
 A: Recall that $H_k(X)$ is a subquotient of $C_k(X)$, the dimension-$k$ singular simplices in $X$, and that $H^k(X)$ is a subquotient of $C^k(X)=\mbox{Hom}(C_k(X),\mathbb{Z})$.  So a $k$-dimensional cohomology class is represented by a cocycle (up to coboundaries), and its evaluation on a homology class is easily seen to be well-defined: if $[\alpha]\in H^k(X)$ and $[c]\in H_k(X)$, then $ \langle \alpha+\delta \beta, c + \partial d \rangle = \langle \alpha, c \rangle$ since $\delta \alpha=0$ and $\partial c=0$.  (Recall the adjunction $\langle \delta \gamma , e \rangle = \langle \gamma, \partial e \rangle $ -- this is in fact nothing more than the definition of $\delta$.)
As for your second question, the wedge product of differential forms is the de Rham analogue of the cup product, and under Poincare duality, cup product in cohomology corresponds to intersection product in homology.  In the de Rham setting, this is proved relatively early on in Bott & Tu's "Differential Forms in Algebraic Topology".
