A sufficient condition for the genus of an amalgamated 3-manifold not to go down

摘要

Let M i be a connected, compact, orientable 3-manifold, F i a boundary component of M i with g(F i ) 2, i = 1, 2, and F 1 ≌ F 2 . Let  : F 1 → F 2 be a homeomorphism, and M = M 1 ∪  M 2 , F = F 2 = (F 1 ). Then it is known that g(M ) g(M 1 ) + g(M 2 ) - g(F ). In the present paper, we give a sufficient condition for the genus of an amalgamated 3-manifold not to go down as follows: Suppose that there is no essential surface with boundary (Q i1 , Q i ) in (M i1 , F i ) satisfying χ(Q i ) 3 - 2g(M i ), i = 1, 2. Then g(M ) = g(M 1 ) + g(M 2 ) - g(F ).