I am using the following results:
Given
$M$ a smooth closed orientable manifold and $A$ and $B$ two smooth closed orientable submanifolds. $A$ and $B$ intersect transversely and $\dim A+\dim B=\dim M$.
Then $[A]^*\smile[B]^*=[A\cap B]^*$ holds, where $\smile$ denotes cup product and $[X]$ denotes the homology class of the manifold $[X]$ and $[X]^*$ denotes Poincar\'{e} dual to $[X]$
The accented e of Poincare is not appearing.
Sorry about that.