Generic Intersection of non-planar Surfaces in R^4

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Bacle
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Hi, everyone:

How do we show that 2 planar surfaces in R^4 intersect at points (possibly empty

sets of points, but not in lines, etc.).

I am curious to see how we justify the Poincare dual of the intersection form in

cohomology being modular, i.e., integer-valued.?

I am confused because the same does not seem to apply to, e.g., lines, which,

when embedded in R^2 or R^3 and R^4 , intersect (if at all) at points.

Thanks.
 

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It's transversality. If the intersection is transverse, they intersect in a submanifold with the appropriate dimension (dim A +dim B - dim M). Here the dimension is 0, so they generically intersect in isolated points, if at all.
 
Just to say thanks, Zhentil; you have been very helpful.