Cut-and-pastes of incompressible surfaces in 3-manifolds (Q1802992)

Let \(F_ 1\) and \(F_ 2\) be two orientable incompressible surfaces of genus \(g_ 1\) and \(g_ 2\), respectively, properly embedded in a compact orientable 3-manifold \(M\) intersecting transversely. The author studies when cut-and-paste (CP) operations on components of the intersection of \(F_ 1\) and \(F_ 2\) yield incompressible surfaces \(F\). First he shows that if \(g_ 1\) or \(g_ 2=1\), then one can always do CP operations to obtain an incompressible \(F\). On the other hand he shows that for any \(g_ 1\) and \(g_ 2\) greater than one there is a closed 3-manifold \(M\) (containing \(F_ 1\) and \(F_ 2\) intersecting transversely) such that every CP operation yields a compressible \(F\); in fact each component of \(F\) bounds a handlebody.