Substituting compact disks in stable planes

Given two stable planes of dimension 2l containing compact topological 2l-disks whose boundaries are isomorphic unitals, we show that it is possible to substitute one of these disks with the other, thus changing the line system in one of the planes, and that this produces another stable plane. The proof uses intersection theory in singular homology in order to show that existence of interior intersection points of two secants can be detected by looking at their traces in the unital. We conclude with a class of concrete examples based on P. Sperner's 4-dimensional affine planes with group .