Question

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.

Answer 1

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 $a$.

Answer 2

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.