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.

• I fixed some of your formatting difficulties; I hope I didn't introduce any errors. – David E Speyer Apr 22 '10 at 22:51
• As regards the math, I don't really understand what you are asking, but does is this mathoverflow.net/questions/20336/… the same as your question? – David E Speyer Apr 22 '10 at 22:53
• Herb, to get you started, can you describe $H_2(\mathbb{CP}^2)$ as an abstract group? Can you write down a basis in terms of embedded oriented surfaces? – Tim Perutz Apr 23 '10 at 2:03
• Thanks, Tim. I know that somehow CP^1 is a basis, but I am trying to understand why. I think it is because CP^1 is the characteristic class for H_2, but I am not sure. Would you please suggest.? – Herb Apr 28 '10 at 5:41

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 $a$.