Let me first say that my background is theoretical physics so I find it hard to look at some of the mathematical literature.


The kind of problem I am interested in is the following one. Consider a 4-dimensional smooth, simply connected orientable manifold M and suppose A and B are immersed surfaces. Denote with $A\#B$ the connected sum of A and B (since M is simply connected I think there is only one way of doing this). Denote with $A\cdot B $ the intersection number of A and B in M. Is it true that the self-intersection number of $A\#B $ in M is given by $(A \# B) \cdot (A \# B) = A \cdot A + 2 A \cdot B + B \cdot B $?


If A, B are 2-spheres, looking at Freedman and Quinn, Topology of 4-Manifolds page 24 I would think so, but I am not sure I am understanding the book well.
What could be said, perhaps making additional assumptions, if A and B are surfaces of more complicated topology? 



Self-intersection of immersed surfaces and connected sum
The answer is: **yes**, the formula holds.


For any $4n$-dimensional manifold and $2n$-dimensional immersed submanifolds $A$ and $B$. (Everything must be oriented if you want consider an integer formula, intersections with signs. If you drop the condition on orientability, then you still can count the points of intersections mod 2.)
Now the formula holds, because intersection is dual to the cohomological multiplication, and $A\#B$ is equal to $A+B$.


The parity of the dimensions is necessary, because cohomological multiplication (as well as the intersection number) is not commutative, but anticommutative.


Actually one can generalize the statement further (the dimensions of the immersed submanifolds might be greater than the half of the dimension of the ambient manifold). In this case the intersection number one can be replaced by the homology class represented by the transversal intersection of the immersed submanifolds.


