Working with Intersection Forms in Homology. Computation. Hi, everyone:
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.?
Thanks.  
 A: If you have 2 surfaces in a 4-manifold, they represent two elements (say, $a$ and $b$) in the degree 2 homology.  If you have a matrix representation $M$ of the intersection form, this means you have already chosen a basis of degree 2 homology, and you can express $a$ and $b$ as column vectors with respect to this basis.  You get an integer by taking $a^TMb$, where $a^T$ denotes the transpose of $a$.
A: Thanks , David, both for the formatting and the ref. Unfortunately, I think my
question may be much simpler than your refs: I have a matrix representation
of a form in cohomology , which can be dualized (and I give this dualized form)
to homology. This form/matrix is supposed to output an integer value; this value
is the number of points of intersection of two submanifolds of a 4-manifold, with
the sign having to see with the orientation of the two submanifolds.
I am just not clear on how I can get this integer value from the matrix, i.e.,
how I can get the intersection number using
this form; I know I need to evaluate this matrix on some 2x1 vector, I just have
no idea of what this vector would be.
Herb.
