MINIMISING THE BOUNDARIES OF PUNCTURED PROJECTIVE PLANES IN S³

This paper concerns 3-manifolds X obtained by Dehn surgery on a knot in S³, in particular those which contain embedded projective planes. Either, they are homeomorphic to the 3-real projeclive space ℝP³, or they are reducible.

Let p be the number of intersections of a projective plane in X with the core of the solid torus added during surgery. We prove here that either X is reducible or p is bigger than or equal to five.

Consequently, if X is homeomorphic to ℝP³ then all its projective planes are pierced at least in five points by the core of the surgery. This result is considered as a step towards showing that ℝP³ cannot be obtained by a Dehn surgery along a knot in S³.