Can one set and use some axioms for calculating intersection forms of manifolds?

Question

I want to calculate the intersection form of some (four?) manifolds, and I wonder is there any axioms that one can compute the intersection form of (four?) manifolds just by them? like axioms of homology group.

for example some possible candidates for that axioms (if exist) are: $Q_{M\#N}=Q_M\oplus Q_N$, $Q_{S^4}=0$, etc.

I am asking this because I am not much easy with computing intersection form directly using cup product and I am looking for a more intuitive and easy way for that.

Answer 1

Summary of comments: The intersection form is defined in terms of cup and cap products. There is axiomatized in tom Dieck's book Algebraic Topology see §17.2 for cup products and §18.1 for cap products.

Intersection form Using cohomology: One can define intersection form using de Rham cohomology. This approach has been discussed in

Moore, John Douglas, Lectures on Seiberg-Witten invariants, Lecture Notes in Mathematics 1629. Berlin: Springer (ISBN 3-540-41221-2/pbk). viii, 121 p. (2001). ZBL1036.57014.