I am trying to work with the intersection form in 4-manifolds. Specifically,
I am working with $CP^2$ (complex projective 2-space.), whose form is given by $(1)$.
Now, I know how to compute an actual numerical value when we work with the form in cohomology: we cup-product two cochains a,b , and then evaluate $a \cup b$ on the fundamental class.
But when we work in homology (using Poincare Duality) , I am not too clear on how we actually get a number by starting with a matrix (we always have representative surfaces for 2-homology in a 4-manifold.). What do we evaluate this matrix in.?